Introduction To Business Statistics
Uploaded by
Mady PunsalanIntroduction To Business Statistics
Uploaded by
Mady PunsalanCaritas Business College BUSINESS STATISTICS – OAD 221
Module 1: AN INTRODUCTION TO BUSINESS STATISTICS
Topic : Introduction to Business Statistics
OBJECTIVE: The aim of the present lesson is to enable the students to understand
the meaning, definition, nature, importance and limitations of statistics.
STRUCTURE:
1.1 Introduction
1.2 Meaning and Definitions of Statistics
1.3 Types of Data and Data Sources
1.4 Types of Statistics
1.5 Scope of Statistics
1.6 Importance of Statistics in Business
1.7 Limitations of statistics
1.8 Summary
1.9 Self-Test Questions
1.10 Suggested Readings
1.1 INTRODUCTION
For a layman, ‘Statistics’ means numerical information expressed in quantitative
terms. This information may relate to objects, subjects, activities, phenomena, or
regions of space. As a matter of fact, data have no limits as to their
reference, coverage, and scope. At the macro level, these are data on gross national
product and shares of agriculture, manufacturing, and services in GDP (Gross
Domestic Product).
At the micro level, individual firms, howsoever small or large, produce extensive
statistics on their operations. The annual reports of companies contain variety of data
on sales, production, expenditure, inventories, capital employed, and other activities.
These data are often field data, collected by employing scientific survey techniques.
Unless regularly updated, such data are the product of a one-time effort and have
limited use beyond the situation that may have called for their collection. A student
knows statistics more intimately as a subject of study like economics, mathematics,
chemistry, physics, and others. It is a discipline, which scientifically deals with data,
and is often described as the science of data. In dealing with statistics as data,
statistics has developed appropriate methods of collecting, presenting, summarizing,
and analysing data, and thus consists of a body of these methods.
-1 -
Caritas Business College BUSINESS STATISTICS – OAD 221
1.2 MEANING AND DEFINITIONS OF STATISTICS
In the beginning, it may be noted that the word ‘statistics’ is used rather curiously in
two senses plural and singular. In the plural sense, it refers to a set of figures or data.
In the singular sense, statistics refers to the whole body of tools that are used
to collect data, organise and interpret them and, finally, to draw conclusions from
them. It should be noted that both the aspects of statistics are important if the
quantitative data are to serve their purpose. If statistics, as a subject, is inadequate
and consists of poor methodology, we could not know the right procedure to extract
from the data the information they contain. Similarly, if our data are defective or that
they are inadequate or inaccurate, we could not reach the right conclusions even
though our subject is well developed.
A.L. Bowley has defined statistics as: (i) statistics is the science of counting, (ii)
Statistics may rightly be called the science of averages, and (iii) statistics is
the science of measurement of social organism regarded as a whole in all its
manifestations. Boddington defined as: Statistics is the science of estimates and
probabilities. Further, W.I. King has defined Statistics in a wider context, the science
of Statistics is the method of judging collective, natural or social phenomena from the
results obtained by the analysis or enumeration or collection of estimates.
Seligman explored that statistics is a science that deals with the methods of
collecting, classifying, presenting, comparing and interpreting numerical data
collected to throw some light on any sphere of enquiry. Spiegal defines statistics
highlighting its role in decision-making particularly under uncertainty, as follows:
statistics is concerned with scientific method for collecting, organising, summa rising,
presenting and analyzing data as well as drawing valid conclusions and making
reasonable decisions on the basis of such analysis. According to Prof. Horace
Secrist, Statistics is the aggregate of facts, affected to a marked extent by multiplicity
of causes, numerically expressed, enumerated or estimated according to reasonable
standards of accuracy, collected in a systematic manner for a pre-determined
purpose, and placed in relation to each other.
-2 -
Caritas Business College BUSINESS STATISTICS – OAD 221
From the above definitions, we can highlight the major characteristics of statistics as
follows:
(i) Statistics are the aggregates of facts. It means a single figure is not statistics.
For example, national income of a country for a single year is not statistics but
the same for two or more years is statistics.
(ii) Statistics are affected by a number of factors. For example, sale of a product
depends on a number of factors such as its price, quality, competition, the
income of the consumers, and so on.
(iii) Statistics must be reasonably accurate. Wrong figures, if analysed, will lead to
erroneous conclusions. Hence, it is necessary that conclusions must be
based on accurate figures.
(iv) Statistics must be collected in a systematic manner. If data are collected in a
haphazard manner, they will not be reliable and will lead to misleading
conclusions.
(v) Collected in a systematic manner for a pre-determined purpose
(vi) Lastly, Statistics should be placed in relation to each other. If one collects
data unrelated to each other, then such data will be confusing and will not
lead to any logical conclusions. Data should be comparable over time and
over space.
1.3 TYPES OF DATA AND DATA SOURCES
Statistical data are the basic raw material of statistics. Data may relate to an activity
of our interest, a phenomenon, or a problem situation under study. They derive
as a result of the process of measuring, counting and/or observing. Statistical data,
therefore, refer to those aspects of a problem situation that can be measured,
quantified, counted, or classified. Any object subject phenomenon, or activity that
generates data through this process is termed as a variable. In other words, a
variable is one that shows a degree of variability when successive measurements are
recorded. In statistics, data are classified into two broad categories: quantitative data
-3 -
Caritas Business College BUSINESS STATISTICS – OAD 221
and qualitative data. This classification is based on the kind of characteristics that are
measured.
Quantitative data are those that can be quantified in definite units of measurement.
These refer to characteristics whose successive measurements yield quantifiable
observations. Depending on the nature of the variable observed for measurement,
quantitative data can be further categorized as continuous and discrete data.
Obviously, a variable may be a continuous variable or a discrete variable.
(i) Continuous data represent the numerical values of a continuous variable. A
continuous variable is the one that can assume any value between any two
points on a line segment, thus representing an interval of values. The
values are quite precise and close to each other, yet distinguishably different.
All characteristics such as weight, length, height, thickness, velocity,
temperature, tensile strength, etc., represent continuous variables. Thus,
the data recorded on these and similar other characteristics are called
continuous data. It may be noted that a continuous variable assumes the
finest unit of measurement. Finest in the sense that it enables measurements
to the maximum degree of precision.
(ii) Discrete data are the values assumed by a discrete variable. A discrete
variable is the one whose outcomes are measured in fixed numbers. Such
data are essentially count data. These are derived from a process of
counting, such as the number of items possessing or not possessing a certain
characteristic. The number of customers visiting a departmental store
everyday, the incoming flights at an airport, and the defective items in a
consignment received for sale, are all examples of discrete data.
Qualitative data refer to qualitative characteristics of a subject or an object. A
characteristic is qualitative in nature when its observations are defined and noted in
terms of the presence or absence of a certain attribute in discrete numbers. These
data are further classified as nominal and rank data.
-4 -
Caritas Business College BUSINESS STATISTICS – OAD 221
(i) Nominal data are the outcome of classification into two or more categories of
items or units comprising a sample or a population according to some quality
characteristic. Classification of students according to sex (as males and
females), of workers according to skill (as skilled, semi-skilled, and unskilled),
and of employees according to the level of education (as matriculates,
undergraduates, and post-graduates), all result into nominal data. Given any
such basis of classification, it is always possible to assign each item to a
particular class and make a summation of items belonging to each class. The
count data so obtained are called nominal data.
(ii) Rank data, on the other hand, are the result of assigning ranks to specify
order in terms of the integers 1,2,3, ..., n. Ranks may be assigned according
to the level of performance in a test. a contest, a competition, an
interview, or a show. The candidates appearing in an interview, for example,
may be assigned ranks in integers ranging from I to n, depending on their
performance in the interview. Ranks so assigned can be viewed as the
continuous values of a variable involving performance as the quality
characteristic.
Data sources could be seen as of two types, viz., secondary and primary. The two
can be defined as under:
(i) Secondary data: They already exist in some form: published or unpublished -
in an identifiable secondary source. They are, generally, available from
published source(s), though not necessarily in the form actually required.
(ii) Primary data: Those data which do not already exist in any form, and thus
have to be collected for the first time from the primary source(s). By their very
nature, these data require fresh and first-time collection covering the whole
population or a sample drawn from it.
1.4 TYPES OF STATISTICS
There are two major divisions of statistics such as descriptive statistics and inferential
-5 -
Caritas Business College BUSINESS STATISTICS – OAD 221
statistics. The term descriptive statistics deals with collecting, summarizing,
and simplifying data, which are otherwise quite unwieldy and voluminous. It seeks to
achieve this in a manner that meaningful conclusions can be readily drawn from the
data. Descriptive statistics may thus be seen as comprising methods of bringing out
and highlighting the latent characteristics present in a set of numerical data. It
not only facilitates an understanding of the data and systematic reporting thereof in a
manner; and also makes them amenable to further discussion, analysis, and
interpretations.
The first step in any scientific inquiry is to collect data relevant to the problem in
hand. When the inquiry relates to physical and/or biological sciences, data
collection is normally an integral part of the experiment itself. In fact, the very manner
in which an experiment is designed, determines the kind of data it would require
and/or generate. The problem of identifying the nature and the kind of the relevant
data is thus automatically resolved as soon as the design of experiment is finalized. It
is possible in the case of physical sciences. In the case of social sciences, where the
required data are often collected through a questionnaire from a number of carefully
selected respondents, the problem is not that simply resolved. For one thing,
designing the questionnaire itself is a critical initial problem. For another, the number
of respondents to be accessed for data collection and the criteria for selecting
them has their own implications and importance for the quality of results obtained.
Further, the data have been collected, these are assembled, organized, and
presented in the form of appropriate tables to make them readable. Wherever
needed, figures, diagrams, charts, and graphs are also used for better presentation
of the data. A useful tabular and graphic presentation of data will require that the raw
data be properly classified in accordance with the objectives of investigation and the
relational analysis to be carried out. .
-6 -
Caritas Business College BUSINESS STATISTICS – OAD 221
A well thought-out and sharp data classification facilitates easy description of the
hidden data characteristics by means of a variety of summary measures. These
include measures of central tendency, dispersion, skewness, and kurtosis, which
constitute the essential scope of descriptive statistics. These form a large part of
the subject matter of any basic textbook on the subject, and thus they are being
discussed in that order here as well.
Inferential statistics, also known as inductive statistics, goes beyond describing a
given problem situation by means of collecting, summarizing, and meaningfully
presenting the related data. Instead, it consists of methods that are used for drawing
inferences, or making broad generalizations, about a totality of observations on the
basis of knowledge about a part of that totality. The totality of observations about
which an inference may be drawn, or a generalization made, is called a population
or a universe. The part of totality, which is observed for data collection and analysis
to gain knowledge about the population, is called a sample.
The desired information about a given population of our interest; may also be
collected even by observing all the units comprising the population. This total
coverage is called census. Getting the desired value for the population through
census is not always feasible and practical for various reasons. Apart from time and
money considerations making the census operations prohibitive, observing each
individual unit of the population with reference to any data characteristic may at times
involve even destructive testing. In such cases, obviously, the only recourse available
is to employ the partial or incomplete information gathered through a sample for the
purpose. This is precisely what inferential statistics does. Thus, obtaining a particular
value from the sample information and using it for drawing an inference about the
entire population underlies the subject matter of inferential statistics. Consider a
-7 -
Caritas Business College BUSINESS STATISTICS – OAD 221
situation in which one is required to know the average body weight of all the college
students in a given cosmopolitan city during a certain year. A quick and easy way to
do this is to record the weight of only 500 students, from out of a total strength of,
say, 10000, or an unknown total strength, take the average, and use this average
based on incomplete weight data to represent the average body weight of all the
college students. In a different situation, one may have to repeat this exercise for
some future year and use the quick estimate of average body weight for a
comparison. This may be needed, for example, to decide whether the weight of the
college students has undergone a significant change over the years compared.
Inferential statistics helps to evaluate the risks involved in reaching inferences or
generalizations about an unknown population on the basis of sample information. for
example, an inspection of a sample of five battery cells drawn from a given lot may
reveal that all the five cells are in perfectly good condition. This information may be
used to conclude that the entire lot is good enough to buy or not.
Since this inference is based on the examination of a sample of limited number of
cells, it is equally likely that all the cells in the lot are not in order. It is also possible
that all the items that may be included in the sample are unsatisfactory. This may be
used to conclude that the entire lot is of unsatisfactory quality, whereas the fact may
indeed be otherwise. It may, thus, be noticed that there is always a risk of an
inference about a population being incorrect when based on the knowledge of a
limited sample. The rescue in such situations lies in evaluating such risks. For this,
statistics provides the necessary methods. These centres on quantifying in
probabilistic term the chances of decisions taken on the basis of sample information
being incorrect. This requires an understanding of the what, why, and how of
probability and probability distributions to equip ourselves with methods of drawing
statistical inferences and estimating the degree of reliability of these inferences.
-8 -
Caritas Business College BUSINESS STATISTICS – OAD 221
1.5 SCOPE OF STATISTICS
Apart from the methods comprising the scope of descriptive and inferential branches
of statistics, statistics also consists of methods of dealing with a few other issues of
specific nature. Since these methods are essentially descriptive in nature, they have
been discussed here as part of the descriptive statistics. These are mainly concerned
with the following:
(i) It often becomes necessary to examine how two paired data sets are related.
For example, we may have data on the sales of a product and the
expenditure incurred on its advertisement for a specified number of years.
Given that sales and advertisement expenditure are related to each other, it is
useful to examine the nature of relationship between the two and quantify the
degree of that relationship. As this requires use of appropriate statistical
methods, these falls under the purview of what we call regression and
correlation analysis.
(ii) Situations occur quite often when we require averaging (or totalling) of data
on prices and/or quantities expressed in different units of measurement. For
example, price of cloth may be quoted per meter of length and that of wheat
per kilogram of weight. Since ordinary methods of totalling and averaging do
not apply to such price/quantity data, special techniques needed for the
purpose are developed under index numbers.
(iii) Many a time, it becomes necessary to examine the past performance of an
activity with a view to determining its future behaviour. For example, when
engaged in the production of a commodity, monthly product sales are
an important measure of evaluating performance. This requires compilation
and analysis of relevant sales data over time. The more complex the
activity, the
-9 -
Caritas Business College BUSINESS STATISTICS – OAD 221
more varied the data requirements. For profit maximising and future sales
planning, forecast of likely sales growth rate is crucial. This needs careful
collection and analysis of past sales data. All such concerns are taken care of
under time series analysis.
(iv) Obtaining the most likely future estimates on any aspect(s) relating to a
business or economic activity has indeed been engaging the minds of all
concerned. This is particularly important when it relates to product sales and
demand, which serve the necessary basis of production scheduling and
planning. The regression, correlation, and time series analyses together help
develop the basic methodology to do the needful. Thus, the study of methods
and techniques of obtaining the likely estimates on business/economic
variables comprises the scope of what we do under business forecasting.
Keeping in view the importance of inferential statistics, the scope of statistics may
finally be restated as consisting of statistical methods which facilitate decision--
making under conditions of uncertainty. While the term statistical methods is often
used to cover the subject of statistics as a whole, in particular it refers to methods by
which statistical data are analysed, interpreted, and the inferences drawn for
decision- making.
Though generic in nature and versatile in their applications, statistical methods have
come to be widely used, especially in all matters concerning business and
economics. These are also being increasingly used in biology, medicine, agriculture,
psychology, and education. The scope of application of these methods has started
opening and expanding in a number of social science disciplines as well. Even a
political scientist finds them of increasing relevance for examining the political
behaviour and it is, of course, no surprise to find even historians statistical data, for
history is essentially past data presented in certain actual format.
-10
-
Caritas Business College BUSINESS STATISTICS – OAD 221
1.6 IMPORTANCE OF STATISTICS IN BUSINESS
There are three major functions in any business enterprise in which the statistical
methods are useful. These are as follows:
(i) The planning of operations: This may relate to either special projects or to
the recurring activities of a firm over a specified period.
(ii) The setting up of standards: This may relate to the size of employment,
volume of sales, fixation of quality norms for the manufactured product,
norms for the daily output, and so forth.
(iii) The function of control: This involves comparison of actual production
achieved against the norm or target set earlier. In case the production
has fallen short of the target, it gives remedial measures so that such a
deficiency does not occur again.
A worth noting point is that although these three functions-planning of operations,
setting standards, and control-are separate, but in practice they are very much
interrelated.
Different authors have highlighted the importance of Statistics in business. For
instance, Croxton and Cowden give numerous uses of Statistics in business such as
project planning, budgetary planning and control, inventory planning and control,
quality control, marketing, production and personnel administration. Within these also
they have specified certain areas where Statistics is very relevant. Another author,
Irwing W. Burr, dealing with the place of statistics in an industrial organisation,
specifies a number of areas where statistics is extremely useful. These are: customer
wants and market research, development design and specification, purchasing,
-11
-
Caritas Business College BUSINESS STATISTICS – OAD 221
production, inspection, packaging and shipping, sales and complaints, inventory and
maintenance, costs, management control, industrial engineering and research.
Statistical problems arising in the course of business operations are multitudinous.
As such, one may do no more than highlight some of the more important ones to
emphasis the relevance of statistics to the business world. In the sphere of
production, for example, statistics can be useful in various ways.
Statistical quality control methods are used to ensure the production of quality goods.
Identifying and rejecting defective or substandard goods achieve this. The sale
targets can be fixed on the basis of sale forecasts, which are done by using varying
methods of forecasting. Analysis of sales affected against the targets set earlier
would indicate the deficiency in achievement, which may be on account of several
causes: (i) targets were too high and unrealistic (ii) salesmen's performance has
been poor (iii) emergence of increase in competition (iv) poor quality of company's
product, and so on. These factors can be further investigated.
Another sphere in business where statistical methods can be used is personnel
management. Here, one is concerned with the fixation of wage rates, incentive norms
and performance appraisal of individual employee. The concept of productivity is
very relevant here. On the basis of measurement of productivity, the productivity
bonus is awarded to the workers. Comparisons of wages and productivity are
undertaken in order to ensure increases in industrial productivity.
Statistical methods could also be used to ascertain the efficacy of a certain product,
say, medicine. For example, a pharmaceutical company has developed a new
medicine in the treatment of bronchial asthma. Before launching it on commercial
basis, it wants to ascertain the effectiveness of this medicine. It undertakes an
experimentation involving the formation of two comparable groups of asthma
-12
-
Caritas Business College BUSINESS STATISTICS – OAD 221
patients. One group is given this new medicine for a specified period and the
other one is treated with the usual medicines. Records are maintained for the two
groups for the specified period. This record is then analysed to ascertain if there is
any significant difference in the recovery of the two groups. If the difference is really
significant statistically, the new medicine is commercially launched.
1.7 LIMITATIONS OF STATISTICS
Statistics has a number of limitations, pertinent among them are as follows:
(i) There are certain phenomena or concepts where statistics cannot be used.
This is because these phenomena or concepts are not amenable to
measurement. For example, beauty, intelligence, courage cannot be
quantified. Statistics has no place in all such cases where quantification is not
possible.
(ii) Statistics reveal the average behaviour, the normal or the general trend. An
application of the 'average' concept if applied to an individual or a particular
situation may lead to a wrong conclusion and sometimes may be disastrous.
For example, one may be misguided when told that the average depth of
a river from one bank to the other is four feet, when there may be some points
in between where its depth is far more than four feet. On this understanding,
one may enter those points having greater depth, which may be hazardous.
(iii) Since statistics are collected for a particular purpose, such data may not be
relevant or useful in other situations or cases. For example, secondary
data (i.e., data originally collected by someone else) may not be useful for the
other person.
(iv) Statistics are not 100 per cent precise as is Mathematics or Accountancy.
Those who use statistics should be aware of this limitation.
-13
-
Caritas Business College BUSINESS STATISTICS – OAD 221
(v) In statistical surveys, sampling is generally used as it is not physically
possible to cover all the units or elements comprising the universe. The
results may not be appropriate as far as the universe is concerned. Moreover,
different surveys based on the same size of sample but different sample units
may yield different results.
(vi) At times, association or relationship between two or more variables is studied
in statistics, but such a relationship does not indicate cause and effect'
relationship. It simply shows the similarity or dissimilarity in the movement of
the two variables. In such cases, it is the user who has to interpret the results
carefully, pointing out the type of relationship obtained.
(vii) A major limitation of statistics is that it does not reveal all pertaining to a
certain phenomenon. There is some background information that statistics
does not cover. Similarly, there are some other aspects related to the problem
on hand, which are also not covered. The user of Statistics has to be well
informed and should interpret Statistics keeping in mind all other aspects
having relevance on the given problem.
Apart from the limitations of statistics mentioned above, there are misuses of it. Many
people, knowingly or unknowingly, use statistical data in wrong manner. Let us see
what the main misuses of statistics are so that the same could be avoided when one
has to use statistical data. The misuse of Statistics may take several forms some of
which are explained below.
(i) Sources of data not given: At times, the source of data is not given. In the
absence of the source, the reader does not know how far the data are
reliable. Further, if he wants to refer to the original source, he is unable to do
so.
-14
-
Caritas Business College BUSINESS STATISTICS – OAD 221
(ii) Defective data: Another misuse is that sometimes one gives defective data.
This may be done knowingly in order to defend one's position or to prove a
particular point. This apart, the definition used to denote a certain
phenomenon may be defective. For example, in case of data relating to
unem- ployed persons, the definition may include even those who are
employed, though partially. The question here is how far it is justified to
include partially employed persons amongst unemployed ones.
(iii) Unrepresentative sample: In statistics, several times one has to conduct a
survey, which necessitates to choose a sample from the given population or
universe. The sample may turn out to be unrepresentative of the universe.
One may choose a sample just on the basis of convenience. He may collect
the desired information from either his friends or nearby respondents in his
neighbourhood even though such respondents do not constitute a
representative sample.
(iv) Inadequate sample: Earlier, we have seen that a sample that is
unrepresentative of the universe is a major misuse of statistics. This apart, at
times one may conduct a survey based on an extremely inadequate
sample. For example, in a city we may find that there are 1, 00,000
households. When we have to conduct a household survey, we may take a
sample of merely 100 households comprising only 0.1 per cent of the
universe. A survey based on such a small sample may not yield right
information.
(v) Unfair Comparisons: An important misuse of statistics is making unfair
comparisons from the data collected. For instance, one may construct an
index of production choosing the base year where the production was much
less. Then he may compare the subsequent year's production from this
low base.
-15
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Such a comparison will undoubtedly give a rosy picture of the production
though in reality it is not so. Another source of unfair comparisons could be
when one makes absolute comparisons instead of relative ones. An absolute
comparison of two figures, say, of production or export, may show a good
increase, but in relative terms it may turnout to be very negligible. Another
example of unfair comparison is when the population in two cities is different,
but a comparison of overall death rates and deaths by a particular disease is
attempted. Such a comparison is wrong. Likewise, when data are not properly
classified or when changes in the composition of population in the two years
are not taken into consideration, comparisons of such data would be unfair as
they would lead to misleading conclusions.
(vi) Unwanted conclusions: Another misuse of statistics may be on account of
unwarranted conclusions. This may be as a result of making false
assumptions. For example, while making projections of population in the
next five years, one may assume a lower rate of growth though the past two
years indicate otherwise. Sometimes one may not be sure about the changes
in business environment in the near future. In such a case, one may use an
assumption that may turn out to be wrong. Another source of unwarranted
conclusion may be the use of wrong average. Suppose in a series there are
extreme values, one is too high while the other is too low, such as 800
and 50. The use of an arithmetic average in such a case may give a wrong
idea. Instead, harmonic mean would be proper in such a case.
(vii) Confusion of correlation and causation: In statistics, several times one
has to examine the relationship between two variables. A close relationship between
the two variables may not establish a cause-and-effect-relationship in the sense
that one
-16
-
Caritas Business College BUSINESS STATISTICS – OAD 221
variable is the cause and the other is the effect. It should be taken as something that
measures degree of association rather than try to find out causal relationship..
1.8 SUMMARY
In a summarized manner, ‘Statistics’ means numerical information expressed in
quantitative terms. As a matter of fact, data have no limits as to their reference,
coverage, and scope. At the macro level, these are data on gross national product
and shares of agriculture, manufacturing, and services in GDP (Gross Domestic
Product). At the micro level, individual firms, howsoever small or large, produce
extensive statistics on their operations. The annual reports of companies contain
variety of data on sales, production, expenditure, inventories, capital employed, and
other activities. These data are often field data, collected by employing scientific
survey techniques. Unless regularly updated, such data are the product of a one-time
effort and have limited use beyond the situation that may have called for their
collection. A student knows statistics more intimately as a subject of study like
economics, mathematics, chemistry, physics, and others. It is a discipline, which
scientifically deals with data, and is often described as the science of data. In dealing
with statistics as data, statistics has developed appropriate methods of collecting,
presenting, summarizing, and analysing data, and thus consists of a body of these
methods.
1.9 SELF-TEST QUESTIONS
1. Define Statistics. Explain its types, and importance to trade, commerce
and business.
2. “Statistics is all-pervading”. Elucidate this statement.
3. Write a note on the scope and limitations of Statistics.
4. What are the major limitations of Statistics? Explain with suitable examples.
5. Distinguish between descriptive Statistics and inferential Statistics.
-17
-
Caritas Business College BUSINESS STATISTICS – OAD 221
1.10 SUGGESTED READINGS
1. Gupta, S. P. : Statistical Methods, Sultan chand and Sons, New Delhi.
2. Hooda, R. P.: Statistics for Business and Economics, Macmillan, New Delhi.
3. Hein, L. W. Quantitative Approach to Managerial Decisions, Prentice Hall,
NJ.
4. Levin, Richard I. and David S. Rubin: Statistics for Management, Prentice
Hall, New Delhi.
5. Lawrance B. Moore: Statistics for Business & Economics, Harper Collins,
NY.
6. Watsman Terry J. and Keith Parramor: Quantitative Methods in
Finance International, Thompson Business Press, London.
-18
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 2: MEASURE OF CENTRAL TENDENCY
Topic : Mean, Median, Mode, Quartiles and Percentiles
OBJECTIVE: The present lesson imparts understanding of the calculations and
main properties of measures of central tendency, including mean,
mode, median, quartiles, percentiles, etc.
STRUCTURE:
2.1 Introduction
2.2 Arithmetic Mean
2.3 Median
2.4 Mode
2.5 Relationships of the Mean, Median and Mode
2.6 The Best Measure of Central Tendency
2.7 Geometric Mean
2.8 Harmonic Mean
2.9 Quadratic Mean
2.10 Summary
2.11 Self-Test Questions
2.12 Suggested Readings
2.1 INTRODUCTION
The description of statistical data may be quite elaborate or quite brief depending on
two factors: the nature of data and the purpose for which the same data have been
collected. While describing data statistically or verbally, one must ensure that the
description is neither too brief nor too lengthy. The measures of central tendency
enable us to compare two or more distributions pertaining to the same time period or
within the same distribution over time. For example, the average consumption of
tea in two different territories for the same period or in a territory for two years, say,
2003 and 2004, can be attempted by means of an average.
-19
-
Caritas Business College BUSINESS STATISTICS – OAD 221
2.2 ARITHMETIC MEAN
Adding all the observations and dividing the sum by the number of
observations results the arithmetic mean. Suppose we have the following
observations:
10, 15,30, 7, 42, 79 and 83
These are seven observations. Symbolically, the arithmetic mean, also called simply
mean is
x = x/n, where x is simple mean.
10 15 30 7 42 79 83
= 7
266
= = 38
7
It may be noted that the Greek letter is used to denote the mean of the
population and n to denote the total number of observations in a population. Thus the
population mean = x/n. The formula given above is the basic formula that forms
the definition of arithmetic mean and is used in case of ungrouped data where
weights are not involved.
2.2.1 UNGROUPED DATA-WEIGHTED AVERAGE
In case of ungrouped data where weights are involved, our approach for calculating
arithmetic mean will be different from the one used earlier.
Example 2.1: Suppose a student has secured the following marks in three
tests: Mid-term test 30
Laboratory 25
Final 20
The simple arithmetic mean will be 30 25 20
3 25
-20
-
Caritas Business College BUSINESS STATISTICS – OAD 221
However, this will be wrong if the three tests carry different weights on the basis of
their relative importance. Assuming that the weights assigned to the three tests are:
Mid-term test 2 points
Laboratory 3 points
Final 5 points
Solution: On the basis of this information, we can now calculate a weighted mean
as shown below:
Table 2.1: Calculation of a Weighted Mean
Type of Test Relative Weight (w) Marks (x) (wx)
Mid-term 2 30 60
Laboratory 3 25 75
Final 5 20 100
Total w= 235
10
wx w1 x1 w2 x2 w3 x3
x
w w1 w2 w3
60 75 100
= 23.5 marks
235
It will be seen that weighted mean gives a more realistic picture than the simple or
unweighted mean.
Example 2.2: An investor is fond of investing in equity shares. During a period of
falling prices in the stock exchange, a stock is sold at Rs 120 per share on one day,
Rs 105 on the next and Rs 90 on the third day. The investor has purchased 50
shares on the first day, 80 shares on the second day and 100 shares on the third'
day. What average price per share did the investor pay?
-21
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution:
Table 2.2: Calculation of Weighted Average Price
Day Price per Share No of Shares Amount Paid
(Rs) (x) Purchased (w) (wx)
1 120 50 6000
2 105 80 8400
3 90 100 9000
Tota - 230 23,400
l
w1 x1 w2 x2 wx
Weighted average =
w3 x3 w1 w2 w
w3
6000 8400 9000
= marks
101.7
50 80 100
Therefore, the investor paid an average price of Rs 101.7 per share.
It will be seen that if merely prices of the shares for the three days (regardless of the
number of shares purchased) were taken into consideration, then the average price
would be
120 105 90
Rs. 3 105
This is an unweighted or simple average and as it ignores the-quantum of shares
purchased, it fails to give a correct picture. A simple average, it may be noted, is
also a weighted average where weight in each case is the same, that is, only 1.
When we use the term average alone, we always mean that it is an unweighted or
simple average.
2.2.2 GROUPED DATA-ARITHMETIC MEAN
For grouped data, arithmetic mean may be calculated by applying any of the
following methods:
(i) Direct method, (ii) Short-cut method ,(iii) Step-deviation method
-22
-
Caritas Business College BUSINESS STATISTICS – OAD 221
In the case of direct method, the formula x = fm/n is used. Here m is mid-point of
various classes, f is the frequency of each class and n is the total number of
frequencies. The calculation of arithmetic mean by the direct method is shown below.
Example 2.3: The following table gives the marks of 58 students in Statistics.
Calculate the average marks of this group.
Marks No. of Students
0-10 4
10-20 8
20-30 11
30-40 15
40-50 12
50-60 6
60-70 2
Total 58
Solution:
Table 2.3: Calculation of Arithmetic Mean by Direct Method
No. of
Mark Mid-point Stude fm
s m nts f
0-10 5 4 20
10- 15 8 120
20
20- 25 11 275
30
30- 35 15 525
40
40- 45 12 540
50
50- 55 6 330
60
60- 65 2 130
70
fm =
1940
Where,
33.45 marks or 33 marks approximately.
fm 1940
x
-23
-
Caritas Business College BUSINESS STATISTICS – OAD 221
n 58
It may be noted that the mid-point of each class is taken as a good approximation of
the true mean of the class. This is based on the assumption that the values are
distributed fairly evenly throughout the interval. When large numbers of frequency
occur, this assumption is usually accepted.
In the case of short-cut method, the concept of arbitrary mean is followed.
The formula for calculation of the arithmetic mean by the short-cut method is given
below:
fd
xA
n
Where A = arbitrary or assumed mean
f = frequency
d = deviation from the arbitrary or assumed mean
When the values are extremely large and/or in fractions, the use of the direct method
would be very cumbersome. In such cases, the short-cut method is preferable. This
is because the calculation work in the short-cut method is considerably reduced
particularly for calculation of the product of values and their respective frequencies.
However, when calculations are not made manually but by a machine calculator, it
may not be necessary to resort to the short-cut method, as the use of the direct
method may not pose any problem.
As can be seen from the formula used in the short-cut method, an arbitrary or
assumed mean is used. The second term in the formula (fd n) is the correction
factor for the difference between the actual mean and the assumed mean. If the
assumed mean turns out to be equal to the actual mean, (fd n) will be zero. The
use of the short-cut method is based on the principle that the total of deviations taken
from an actual mean is equal to zero. As such, the deviations taken from any other
figure will depend on how the assumed mean is related to the actual mean. While
one may choose any value as assumed mean, it would be proper to avoid extreme
-24
-
Caritas Business College BUSINESS STATISTICS – OAD 221
values, that is, too small or too high to simplify calculations. A value apparently close
to the arithmetic mean should be chosen.
For the figures given earlier pertaining to marks obtained by 58 students, we
calculate the average marks by using the short-cut method.
Example 2.4:
Table 2.4: Calculation of Arithmetic Mean by Short-cut Method
Mid-
Marks point f d fd
m
0-10 5 4 -30 -120
10-20 1 8 -20 -160
5
20-30 2 11 -10 -110
5
30-40 3 15 0 0
5
40-50 4 12 10 120
5
50-60 5 6 20 120
5
60-70 6 2 30 60
5
fd = -90
It may be noted that we have taken arbitrary mean as 35 and deviations from
midpoints. In other words, the arbitrary mean has been subtracted from each value of
mid-point and the resultant figure is shown in column d.
fd
xA
n
35 90
58
= 35 - 1.55 = 33.45 or 33 marks approximately.
Now we take up the calculation of arithmetic mean for the same set of data using
the step-deviation method. This is shown in Table 2.5.
Table 2.5: Calculation of Arithmetic Mean by Step-deviation Method
Mark Mid- f d d’= Fd’
s point d/10
0-10 5 4 -30 -3 -12
10- 1 8 -20 -2 -16
20 5
20- 2 11 -10 -1 -11
30 5
30- 3 15 0 0 0
-25
-
Caritas Business College BUSINESS STATISTICS – OAD 221
40 5
40- 4 12 10 1 12
50 5
50- 5 6 20 2 12
60 5
60- 6 2 30 3 6
70 5
fd’ =-
9
-26
-
Caritas Business College BUSINESS STATISTICS – OAD 221
fd '
xA
C
n = 33.45 or 33 marks approximately.
9
35
10
58
It will be seen that the answer in each of the three cases is the same. The step-
deviation method is the most convenient on account of simplified calculations. It may
also be noted that if we select a different arbitrary mean and recalculate deviations
from that figure, we would get the same answer.
Now that we have learnt how the arithmetic mean can be calculated by using
different methods, we are in a position to handle any problem where calculation of
the arithmetic mean is involved.
Example 2.6: The mean of the following frequency distribution was found to be 1.46.
No. of Accidents No. of Days (frequency)
0 4
6
1 ?
2 ?
3 2
5
4 1
0
5 5
Total 200 days
Calculate the missing frequencies.
Solution:
Here we are given the total number of frequencies and the arithmetic mean. We have
to determine the two frequencies that are missing. Let us assume that the frequency
against 1 accident is x and against 2 accidents is y. If we can establish two
simultaneous equations, then we can easily find the values of X and Y.
Mean = (0.46) (1. x) (2 . y) (3 . 25) (4 . l0)
(5 . 5)
200
-27
-
Caritas Business College BUSINESS STATISTICS – OAD 221
x 2y 140
1.46 =
200
x + 2y + 140 = (200) (1.46)
x + 2y = 152
x + y=200- {46+25 + 1O+5}
x + y = 200 - 86
x + y = 114
Now subtracting equation (ii) from equation (i), we get
x + 2y = 152
x+y = 114
- - -
y = 38
Substituting the value of y = 38 in equation (ii) above, x + 38 = 114
Therefore, x = 114 - 38 = 76
Hence, the missing frequencies are:
Against accident 1 : 76
Against accident 2 : 38
2.2.3 CHARACTERISTICS OF THE ARITHMETIC MEAN
Some of the important characteristics of the arithmetic mean are:
1. The sum of the deviations of the individual items from the arithmetic mean is
always zero. This means I: (x - x ) = 0, where x is the value of an item and x
is the arithmetic mean. Since the sum of the deviations in the positive
direction
is equal to the sum of the deviations in the negative direction, the arithmetic
mean is regarded as a measure of central tendency.
2. The sum of the squared deviations of the individual items from the arithmetic
mean is always minimum. In other words, the sum of the squared deviations
taken from any value other than the arithmetic mean will be higher.
-28
-
Caritas Business College BUSINESS STATISTICS – OAD 221
3. As the arithmetic mean is based on all the items in a series, a change in the
value of any item will lead to a change in the value of the arithmetic mean.
4. In the case of highly skewed distribution, the arithmetic mean may get
distorted on account of a few items with extreme values. In such a case,
it may cease to be the representative characteristic of the distribution.
2.3 MEDIAN
Median is defined as the value of the middle item (or the mean of the values of
the two middle items) when the data are arranged in an ascending or descending
order of magnitude. Thus, in an ungrouped frequency distribution if the n values are
arranged in ascending or descending order of magnitude, the median is the middle
value if n is odd. When n is even, the median is the mean of the two middle values.
Suppose we have the following series:
15, 19,21,7, 10,33,25,18 and 5
We have to first arrange it in either ascending or descending order. These figures are
arranged in an ascending order as follows:
5,7,10,15,18,19,21,25,33
Now as the series consists of odd number of items, to find out the value of the
middle item, we use the formula
Where n
1
n
Where n is the number of items. In this case, n is 9, as 1 = 5, that is, the size
such
2
of the 5th item is the median. This happens to be 18.
Suppose the series consists of one more items 23. We may, therefore, have to
include 23 in the above series at an appropriate place, that is, between 21 and 25.
Thus, the series is now 5, 7, 10, 15, 18, 19, and 21,23,25,33. Applying the above
-29
-
Caritas Business College BUSINESS STATISTICS – OAD 221
formula, the
-30
-
Caritas Business College BUSINESS STATISTICS – OAD 221
median is the size of 5.5th item. Here, we have to take the average of the values of
5th and 6th item. This means an average of 18 and 19, which gives the median as
18.5.
It may be noted that the formula n
itself is not the formula for the median; it
1
merely indicates the position of the median, namely, the number of items we have to
count until we arrive at the item whose value is the median. In the case of the even
number of items in the series, we identify the two items whose values have to be
averaged to obtain the median. In the case of a grouped series, the median is
calculated by linear interpolation with the help of the following formula:
l2 l1
M = l1 (m
c) f
Where M = the median
l1 = the lower limit of the class in which the median lies
12 = the upper limit of the class in which the median lies
f = the frequency of the class in which the median lies
m = the middle item or (n + 1)/2th, where n stands for total number of
items
c = the cumulative frequency of the class preceding the one in which the median lies
Example 2.7:
Monthly Wages (Rs) No. of Workers
800-1,000 18
1,000-1,200 25
1,200-1,400 30
1,400-1,600 34
1,600-1,800 26
1,800-2,000 10
Total 143
In order to calculate median in this case, we have to first provide cumulative
-31
-
Caritas Business College BUSINESS STATISTICS – OAD 221
frequency to the table. Thus, the table with the cumulative frequency is written as:
-32
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Cumulative
Monthly Frequen Frequency
Wages cy
800 -1,000 18 18
1,000 -1,200 25 43
1,200 -1,400 30 73
1,400 -1,600 34 107
1,600 -1,800 26 133
1.800 -2,000 10 143
l2 l1
M = l1 (m
c) f
n 1 143 1
M= = 72
2 2
It means median lies in the class-interval Rs 1,200 - 1,400.
Now, M = 1200 + 1400 1200 (72 43)
30
200
1200 (29)
30
= Rs 1393.3
At this stage, let us introduce two other concepts viz. quartile and decile. To
understand these, we should first know that the median belongs to a general class of
statistical descriptions called fractiles. A fractile is a value below that lays a given
fraction of a set of data. In the case of the median, this fraction is one-half (1/2).
Likewise, a quartile has a fraction one-fourth (1/4). The three quartiles Q 1, Q2 and Q3
are such that 25 percent of the data fall below Q1, 25 percent fall between Q1 and Q2,
25 percent fall between Q2 and Q3 and 25 percent fall above Q3 It will be seen that Q 2
is the median. We can use the above formula for the calculation of quartiles as well.
The only difference will be in the value of m. Let us calculate both Q 1 and Q3 in
respect of the table given in Example 2.7.
l2 l1
Q1 = l1 (m
c) f
-33
-
Caritas Business College BUSINESS STATISTICS – OAD 221
n 143 1
Here, m will be = 1 = = 36
4
1200 1000
Q 1000 (36 18)
1
25
200
1000 (18)
25
= Rs. 1,144
n1 3144
In the case of Q3, m will be 3 = 4 = 108
= 4
1800 1600
Q 1600 (108 107)
1
26
200
1600 (1)
26
Rs. 1,607.7 approx
In the same manner, we can calculate deciles (where the series is divided into
10 parts) and percentiles (where the series is divided into 100 parts). It may be noted
that unlike arithmetic mean, median is not affected at all by extreme values, as it is a
positional average. As such, median is particularly very useful when a distribution
happens to be skewed. Another point that goes in favour of median is that it can be
computed when a distribution has open-end classes. Yet, another merit of median is
that when a distribution contains qualitative data, it is the only average that can be
used. No other average is suitable in case of such a distribution. Let us take a
couple of examples to illustrate what has been said in favour of median.
-34
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 2.8:Calculate the most suitable average for the following data:
Size of the Item Below 50 50-100 100-150 150-200 200 and above
Frequency 15 20 36 40 10
Solution: Since the data have two open-end classes-one in the beginning (below 50) and
the other at the end (200 and above), median should be the right choice as a measure of
central tendency.
Table 2.6: Computation of Median
Size of Item Frequency Cumulative Frequency
Below 50 15 15
50-100 20 35
100-150 36 71
150-200 40 111
200 and above 10 121
Median is the size n1
th item
of 2
121 1
= = 61st
item 2
Now, 61st item lies in the 100-150 class
l2 l1
Median = 11 = l1 (m c)
f
150 100
= 100 + (61 35)
36
= 100 + 36.11 = 136.11 approx.
Example 2.9: The following data give the savings bank accounts balances of nine
sample households selected in a survey. The figures are in rupees.
745 2,000 1,500 68,000 461 549 3750 1800 4795
(a) Find the mean and the median for these data; (b) Do these data contain an outlier? If
so, exclude this value and recalculate the mean and median. Which of these summary
measures
-35
-
Caritas Business College BUSINESS STATISTICS – OAD 221
has a greater change when an outlier is dropped?; (c) Which of these two summary
measures is more appropriate for this series?
Solution:
Mean = 745 2,000 1,500 68,000 461 549 3,750 1,800
Rs. 4,795
Rs
=
83,600 = Rs 9,289
9
n 1
Median = Size th item
2
of
9 1
=
= 5th item
2
Arranging the data in an ascending order, we find that the median is Rs 1,800.
(b) An item of Rs 68,000 is excessively high. Such a figure is called an 'outlier'. We
exclude this figure and recalculate both the mean and the median.
Mean = 83,600
68,000
Rs.
8
=
15,600 Rs = Rs. 1,950
8
n 1
Median = Size of th item
2
8
=
1 item.
2 4.5th
1,500
= Rs.
1,800 = Rs. 1,650
2
It will be seen that the mean shows a far greater change than the median when
the outlier is dropped from the calculations.
-36
-
Caritas Business College BUSINESS STATISTICS – OAD 221
(c) As far as these data are concerned, the median will be a more appropriate
measure than the mean.
Further, we can determine the median graphically as follows:
-37
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 2.10: Suppose we are given the following series:
Class interval 0-10 10-20 20-30 30-40 40-50 50-60 60-70
Frequency 6 12 22 37 17 8 5
We are asked to draw both types of ogive from these data and to determine
the median.
Solution:
First of all, we transform the given data into two cumulative frequency distributions,
one based on ‘less than’ and another on ‘more than’ methods.
Table A
Frequenc
y
Less than 10 6
Less than 20 18
Less than 30 40
Less than 40 77
Less than 50 94
Less than 60 102
Less than 70 107
Table B
Frequency
More than 0 107
More than 10 101
More than 20 89
More than 30 67
More than 40 30
More than 50 13
More than 60 5
It may be noted that the point of
intersection of the two ogives gives the
value of the median. From this point of
intersection A, we draw a straight line to
-38
-
Caritas Business College BUSINESS STATISTICS – OAD 221
meet the X-axis at M. Thus, from the point of origin to the point at M gives the value
of the median, which comes to 34, approximately. If we calculate the median by
applying the formula, then the answer comes to 33.8, or 34, approximately. It may be
pointed out that even a single ogive can be used to determine the median. As we
have determined the median graphically, so also we can find the values of quartiles,
deciles or percentiles graphically. For example, to determine we have to take size of
{3(n + 1)} /4 = 81st item. From this point on the Y-axis, we can draw a perpendicular
to meet the 'less than' ogive from which another straight line is to be drawn to meet
the X-axis. This point will give us the value of the upper quartile. In the same manner,
other values of Q1 and deciles and percentiles can be determined.
2.3.1 CHARACTERISTICS OF THE MEDIAN
1. Unlike the arithmetic mean, the median can be computed from open-ended
distributions. This is because it is located in the median class-interval, which
would not be an open-ended class.
2. The median can also be determined graphically whereas the arithmetic mean
cannot be ascertained in this manner.
3. As it is not influenced by the extreme values, it is preferred in case of a
distribution having extreme values.
4. In case of the qualitative data where the items are not counted or measured
but are scored or ranked, it is the most appropriate measure of central
tendency.
2.4 MODE
The mode is another measure of central tendency. It is the value at the point around
which the items are most heavily concentrated. As an example, consider the
following series: 8,9, 11, 15, 16, 12, 15,3, 7, 15
-39
-
Caritas Business College BUSINESS STATISTICS – OAD 221
There are ten observations in the series wherein the figure 15 occurs maximum
number of times three. The mode is therefore 15. The series given above is a
discrete series; as such, the variable cannot be in fraction. If the series were
continuous, we could say that the mode is approximately 15, without further
computation.
In the case of grouped data, mode is determined by the following formula:
Mode= l1
f1 f 0
i
( f 1 f 0 ) ( f1 f 2 )
Where, l1 = the lower value of the class in which the mode lies
fl = the frequency of the class in which the mode lies
fo = the frequency of the class preceding the modal class
f2 = the frequency of the class succeeding the modal
class i = the class-interval of the modal class
While applying the above formula, we should ensure that the class-intervals are
uniform throughout. If the class-intervals are not uniform, then they should be made
uniform on the assumption that the frequencies are evenly distributed throughout the
class. In the case of inequal class-intervals, the application of the above formula will
give misleading results.
Example 2.11: Let us take the following frequency distribution:
Class intervals (1) Frequency (2)
30-40 4
40-50 6
50-60 8
60-70 12
70-80 9
80-90 7
90-100 4
We have to calculate the mode in respect of this series.
Solution: We can see from Column (2) of the table that the maximum frequency
of 12 lies in the class-interval of 60-70. This suggests that the mode lies in this class-
interval. Applying the formula given earlier, we get:
-40
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Mode = 60 + 12 - 8
10
12 - 8 (12 - 8) (12 -
9)
4
= 60 + 10
43
= 65.7 approx.
In several cases, just by inspection one can identify the class-interval in which the
mode lies. One should see which the highest frequency is and then identify to which
class-interval this frequency belongs. Having done this, the formula given for
calculating the mode in a grouped frequency distribution can be applied.
At times, it is not possible to identify by inspection the class where the mode lies. In
such cases, it becomes necessary to use the method of grouping. This method
consists of two parts:
(i) Preparation of a grouping table: A grouping table has six columns, the first
column showing the frequencies as given in the problem. Column 2 shows
frequencies grouped in two's, starting from the top. Leaving the first
frequency, column 3 shows frequencies grouped in two's. Column 4 shows
the frequencies of the first three items, then second to fourth item and so on.
Column 5 leaves the first frequency and groups the remaining items in
three's. Column 6 leaves the first two frequencies and then groups the
remaining in three's. Now, the maximum total in each column is marked and
shown either in a circle or in a bold type.
(ii) Preparation of an analysis table: After having prepared a grouping table, an
analysis table is prepared. On the left-hand side, provide the first column for
column numbers and on the right-hand side the different possible values of
mode. The highest values marked in the grouping table are shown here by
a bar or by simply entering 1 in the relevant cell corresponding to the
values
-41
-
Caritas Business College BUSINESS STATISTICS – OAD 221
they represent. The last row of this table will show the number of times a
particular value has occurred in the grouping table. The highest value in the
analysis table will indicate the class-interval in which the mode lies. The
procedure of preparing both the grouping and analysis tables to locate the
modal class will be clear by taking an example.
Example 2.12: The following table gives some frequency data:
Size of Item Frequency
10-20 10
20-30 18
30-40 25
40-50 26
50-60 17
60-70 4
Solution:
Grouping Table
Size of item 1 2 3 4 5 6
10-20 10
28
20-30 18 53
43
30-40 25 69
51
40-50 26 68
43
50-60 17 47
21
60-70 4
Analysis table
Size of item
Col. No. 10-20 20-30 30-40 40-50 50-60
1 1
2 1 1
3 1 1 1 1
4 1 1 1
5 1 1 1
-42
-
Caritas Business College BUSINESS STATISTICS – OAD 221
6 1 1 1
Total 1 3 5 5 2
This is a bi-modal series as is evident from the analysis table, which shows that the
two classes 30-40 and 40-50 have occurred five times each in the grouping. In such
a situation, we may have to determine mode indirectly by applying the following
formula:
Mode = 3 median - 2 mean
Median = Size of (n + l)/2th item, that is, 101/2 = 50.5th item. This lies in the class 30-
40. Applying the formula for the median, as given earlier, we get
40 - 30
= 30 + (50.5 28)
25
= 30 + 9 = 39
Now, arithmetic mean is to be calculated. This is shown in the following table.
Class- interval Frequency Mid- points d d' = d/10 fd'
10-20 10 15 -20 -2 -20
20-30 18 25 -10 -I -18
30-40 25 35 0 0 0
40-50 26 45 10 1 26
50-60 17 55 20 2 34
60-70 4 65 30 3 12
Total 100
Deviation is taken from arbitrary mean = 35 34
fd '
Mean = A+ i
n
34
= 35 + 10
100
= 38.4
Mode = 3 median - 2 mean
= (3 x 39) - (2 x 38.4)
= 117 -76.8
-43
-
Caritas Business College BUSINESS STATISTICS – OAD 221
= 40.2
This formula, Mode = 3 Median-2 Mean, is an empirical formula only. And it can
give only approximate results. As such, its frequent use should be avoided. However,
when mode is ill defined or the series is bimodal (as is the case in the
present example) it may be used.
2.5 RELATIONSHIPS OF THE MEAN, MEDIAN AND MODE
Having discussed mean, median and mode, we now turn to the relationship amongst
these three measures of central tendency. We shall discuss the relationship
assuming that there is a unimodal frequency distribution.
(i) When a distribution is symmetrical, the mean, median and mode are the
same, as is shown below in the following figure.
In case, a distribution is
skewed to the right, then
mean> median> mode.
Generally, income distri-
bution is skewed to the right where a large number of families have relatively
low income and a small number of families have extremely high income. In
such a case, the mean is pulled up by the extreme high incomes and the
relation among these three measures is as shown in Fig. 6.3. Here, we find
that mean> median> mode.
(ii) When a distribution is skewed to
the left, then mode> median>
mean. This is because here mean
is pulled down below the
median by extremely low values.
This is
-44
-
Caritas Business College BUSINESS STATISTICS – OAD 221
shown as in the figure.
(iii) Given the mean and median of a unimodal distribution, we can determine
whether it is skewed to the
right or left. When mean>
median, it is skewed to the
right; when median> mean, it
is skewed to the left. It may be noted that the median is always in the middle
between mean and mode.
2.6 THE BEST MEASURE OF CENTRAL TENDENCY
At this stage, one may ask as to which of these three measures of central tendency
the best is. There is no simple answer to this question. It is because these three
measures are based upon different concepts. The arithmetic mean is the sum of the
values divided by the total number of observations in the series. The median is the
value of the middle observation that divides the series into two equal parts. Mode is
the value around which the observations tend to concentrate. As such, the use of a
particular measure will largely depend on the purpose of the study and the nature
of the data; For example, when we are interested in knowing the consumers
preferences for different brands of television sets or different kinds of advertising, the
choice should go in favour of mode. The use of mean and median would not be
proper. However, the median can sometimes be used in the case of qualitative data
when such data can be arranged in an ascending or descending order. Let us take
another example. Suppose we invite applications for a certain vacancy in our
company. A large number of candidates apply for that post. We are now interested to
know as to which age or age group has the largest concentration of applicants.
Here, obviously the mode will be the most appropriate choice. The arithmetic mean
may not be appropriate as it may
-45
-
Caritas Business College BUSINESS STATISTICS – OAD 221
be influenced by some extreme values. However, the mean happens to be the most
commonly used measure of central tendency as will be evident from the discussion in
the subsequent chapters.
2.7 GEOMETRIC MEAN
Apart from the three measures of central tendency as discussed above, there are two
other means that are used sometimes in business and economics. These are the
geometric mean and the harmonic mean. The geometric mean is more important
than the harmonic mean. We discuss below both these means. First, we take up the
geometric mean. Geometric mean is defined at the nth root of the product of n
observations of a distribution.
Symbolically, GM = n x1....x2 .....xn ... If we have only two observations, say, 4 and
16 then GM =
4 16 Similarly, if there are three observations, then we
8. 64
have to calculate the cube root of the product of these three observations; and so on.
When the number of items is large, it becomes extremely difficult to multiply the
numbers and to calculate the root. To simplify calculations, logarithms are used.
Example 2.13: If we have to find out the geometric mean of 2, 4 and 8, then we find
Log GM = log xi n
Log 2 Log 4
= Log8 3
0.3010 0.6021
= 0.9031
3
=
1.8062
0.60206
3
GM = Antilog 0.60206
=4
-46
-
Caritas Business College BUSINESS STATISTICS – OAD 221
When the data are given in the form of a frequency distribution, then the
geometric mean can be obtained by the formula:
Log GM = f .log f 2 .log x .. f n . log x n
1
xl .
2
f1 f 2 ..........fn
f .log x
= f1 f 2 ..........fn
Then, GM = Antilog n
The geometric mean is most suitable in the following three cases:
1. Averaging rates of change.
2. The compound interest formula.
3. Discounting, capitalization.
Example 2.14: A person has invested Rs 5,000 in the stock market. At the end of the
first year the amount has grown to Rs 6,250; he has had a 25 percent profit. If at the
end of the second year his principal has grown to Rs 8,750, the rate of increase is 40
percent for the year. What is the average rate of increase of his investment during
the two years?
Solution:
GM =
1.25 1.40 = 1.323
1.75.
The average rate of increase in the value of investment is therefore 1.323 - 1 =
0.323, which if multiplied by 100, gives the rate of increase as 32.3 percent.
Example 2.15: We can also derive a compound interest formula from the above set
of data. This is shown below:
Solution: Now, 1.25 x 1.40 = 1.75. This can be written as 1.75 = (1 + 0.323)2.
Let P2 = 1.75, P0 = 1, and r = 0.323, then the above equation can be written as P2 = (1
+ r)2 or P2 = P0 (1 + r)2.
-47
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Where P2 is the value of investment at the end of the second year, P0 is the initial
investment and r is the rate of increase in the two years. This, in fact, is the familiar
compound interest formula. This can be written in a generalised form as Pn = P0(1 +
r)n. In our case Po is Rs 5,000 and the rate of increase in investment is 32.3 percent.
Let us apply this formula to ascertain the value of Pn, that is, investment at the end of
the second year.
Pn = 5,000 (1 + 0.323)2
= 5,000 x 1.75
= Rs 8,750
It may be noted that in the above example, if the arithmetic mean is used, the resultant
25 40
figure will be wrong. In this case, the average rate for the two years percent
is 2
165
per year, which comes to 32.5. Applying this rate, we get Pn = x 5,000
100
= Rs 8,250
This is obviously wrong, as the figure should have been Rs 8,750.
Example 2.16: An economy has grown at 5 percent in the first year, 6 percent in the
second year, 4.5 percent in the third year, 3 percent in the fourth year and 7.5
percent in the fifth year. What is the average rate of growth of the economy during
the five
years?
Solution:
Year Rate of Growth Value at the end of the Log x
( percent) Year x (in Rs)
1 5 105 2.02119
2 6 106 2.02531
3 4.5 104.5 2.01912
4 3 103 2.01284
5 7.5 107.5 2.03141
log X = 10.10987
-48
-
Caritas Business College BUSINESS STATISTICS – OAD 221
log x
GM = Antilog
n
10.10987
= Antilog
5
= Antilog 2.021974
= 105.19
Hence, the average rate of growth during the five-year period is 105.19 - 100 = 5.19
percent per annum. In case of a simple arithmetic average, the corresponding rate of
growth would have been 5.2 percent per annum.
2.7.1 DISCOUNTING
The compound interest formula given above was
Pn=P0(1+r)n This can be written as P0 Pn
=
(1 r)
n
This may be expressed as follows:
If the future income is Pn rupees and the present rate of interest is 100 r percent, then
the present value of P n rupees will be P0 rupees. For example, if we have a machine
that has a life of 20 years and is expected to yield a net income of Rs 50,000 per
year, and at the end of 20 years it will be obsolete and cannot be used, then the
machine's present value is
50,000 + 50,000 + 50,000 50,000
+.................
(1 r) n
(1 r) 2 (1 r)3 (1 r) 20
This process of ascertaining the present value of future income by using the interest
rate is known as discounting.
In conclusion, it may be said that when there are extreme values in a series,
geometric mean should be used as it is much less affected by such values. The
-49
-
Caritas Business College BUSINESS STATISTICS – OAD 221
arithmetic mean in such cases will give misleading results.
-50
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Before we close our discussion on the geometric mean, we should be aware of its
advantages and limitations.
2.7.2 ADVANTAGES OF G. M.
1. Geometric mean is based on each and every observation in the data set.
2. It is rigidly defined.
3. It is more suitable while averaging ratios and percentages as also in
calculating growth rates.
4. As compared to the arithmetic mean, it gives more weight to small values and
less weight to large values. As a result of this characteristic of the geometric
mean, it is generally less than the arithmetic mean. At times it may be equal
to the arithmetic mean.
5. It is capable of algebraic manipulation. If the geometric mean has two or more
series is known along with their respective frequencies. Then a combined
geometric mean can be calculated by using the logarithms.
2.7.3 LIMITATIONS OF G.M.
1. As compared to the arithmetic mean, geometric mean is difficult to
understand.
2. Both computation of the geometric mean and its interpretation are rather
difficult.
3. When there is a negative item in a series or one or more observations
have zero value, then the geometric mean cannot be calculated.
In view of the limitations mentioned above, the geometric mean is not frequently
used.
2.8 HARMONIC MEAN
-51
-
Caritas Business College BUSINESS STATISTICS – OAD 221
The harmonic mean is defined as the reciprocal of the arithmetic mean of the
reciprocals of individual observations. Symbolically,
n 1/ x
HM= Re ciprocal
1/ x1 1/ x 2 1/ x 3 . .. 1/ x n n
The calculation of harmonic mean becomes very tedious when a distribution has a
large number of observations. In the case of grouped data, the harmonic mean is
calculated by using the following formula:
n 1
HM = Reciprocal of f
i 1 i i
x
or
n n 1
f
i 1 i i
x
Where n is the total number of observations.
Here, each reciprocal of the original figure is weighted by the corresponding
frequency (f).
The main advantage of the harmonic mean is that it is based on all observations in a
distribution and is amenable to further algebraic treatment. When we desire to give
greater weight to smaller observations and less weight to the larger observations,
then the use of harmonic mean will be more suitable. As against these
advantages, there are certain limitations of the harmonic mean. First, it is difficult to
understand as well as difficult to compute. Second, it cannot be calculated if any of
the observations is zero or negative. Third, it is only a summary figure, which may not
be an actual observation in the distribution.
It is worth noting that the harmonic mean is always lower than the geometric mean,
which is lower than the arithmetic mean. This is because the harmonic mean
assigns
-52
-
Caritas Business College BUSINESS STATISTICS – OAD 221
lesser importance to higher values. Since the harmonic mean is based on
reciprocals, it becomes clear that as reciprocals of higher values are lower than those
of lower values, it is a lower average than the arithmetic mean as well as the
geometric mean.
Example 2.17: Suppose we have three observations 4, 8 and 16. We are required to
1
calculate the harmonic mean. Reciprocals of 4,8 and 16 1 1 respectively
, ,
are:
4 8 16
Since HM = n
1/ x1 1/ x 2 1/ x
= 3
3
=
1/ 4 1/ 8
1/ 16
0.25 0.125
0.0625
= 6.857 approx.
Example 2.18: Consider the following series:
Class-interval 2-4 4-6 6-8 8-10
Frequency 20 40 30 10
Solution:
Let us set up the table as follows:
Class- Mid- Frequency Reciprocal of f x 1/x
interval value MV
2-4 3 20 0.3333 6.6660
4-6 5 40 0.2000 8.0000
6-8 7 30 0.1429 4.2870
8-10 9 10 0.1111 1.1111
Total 20.064
1
n 1
f i
x
i 1 i
= n
-53
-
Caritas Business College BUSINESS STATISTICS – OAD 221
100
= = 4.984 approx.
20.0641
-54
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 2.19: In a small company, two typists are employed. Typist A types one
page in ten minutes while typist B takes twenty minutes for the same. (i) Both are
asked to type 10 pages. What is the average time taken for typing one page? (ii)
Both are asked to type for one hour. What is the average time taken by them for
typing one page?
Solution: Here Q-(i) is on arithmetic mean while Q-(ii) is on harmonic mean.
(10 10) (20 20)(min
(i) M= utes)
10 2(
pages)
= 15 minutes
60 (min utes)
HM =
60 / 10 60 / 20(
pages)
120 40
= 13 min utes and 20 seconds.
120
3
60
20
Example 2.20: It takes ship A 10 days to cross the Pacific Ocean; ship B takes
15 days and ship C takes 20 days. (i) What is the average number of days taken by a
ship to cross the Pacific Ocean? (ii) What is the average number of days taken by a
cargo to cross the Pacific Ocean when the ships are hired for 60 days?
Solution: Here again Q-(i) pertains to simple arithmetic mean while Q-(ii) is
concerned with the harmonic mean.
(i) M = 10 15 20
3 = 15 days
(ii) HM = 60 3(days) _
60 / 10 60 / 15 60 /
= 20
-55
-
Caritas Business College BUSINESS STATISTICS – OAD 221
1
8
0
3
6
0
2
4
0
1
8
0
6
0
-56
-
Caritas Business College BUSINESS STATISTICS – OAD 221
= 13.8 days approx.
2.9 QUADRATIC MEAN
We have seen earlier that the geometric mean is the antilogarithm of the arithmetic
mean of the logarithms, and the harmonic mean is the reciprocal of the arithmetic
mean of the reciprocals. Likewise, the quadratic mean (Q) is the square root of the
arithmetic mean of the squares. Symbolically,
Q= x 2 x 12n
2 2
n
Instead of using original values, the quadratic mean can be used while averaging
deviations when the standard deviation is to be calculated. This will be used in
the next chapter on dispersion.
2.9.1 Relative Position of Different Means
The relative position of different means will always be:
Q> x >G>H provided that all the individual observations in a series are positive and
all of them are not the same.
2.9.2 Composite Average or Average of Means
Sometimes, we may have to calculate an average of several averages. In such
cases, we should use the same method of averaging that was employed in
calculating the original averages. Thus, we should calculate the arithmetic mean of
several values of x, the geometric mean of several values of GM, and the harmonic
mean of several values of HM. It will be wrong if we use some other average in
averaging of means.
2.10 SUMMARY
It is the most important objective of statistical analysis is to get one single value that
describes the characteristics of the entire mass of cumbersome data. Such a value is
finding out, which is known as central value to serve our purpose.
-57
-
Caritas Business College BUSINESS STATISTICS – OAD 221
2.11 SELF-TEST QUESTIONS
1. What are the desiderata (requirements) of a good average? Compare the
mean, the median and the mode in the light of these desiderata? Why
averages are called measures of central tendency?
2. "Every average has its own peculiar characteristics. It is difficult to say which
average is the best." Explain with examples.
3. What do you understand .by 'Central Tendency'? Under what conditions is the
median more suitable than other measures of central tendency?
4. The average monthly salary paid to all employees in a company was Rs 8,000.
The average monthly salaries paid to male and female employees of the
company were Rs 10,600 and Rs 7,500 respectively. Find out the
percentages of males and females employed by the company.
5. Calculate the arithmetic mean from the following data:
Class 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89
Frequency 2 4 9 11 12 6 4 2
6. Calculate the mean, median and mode from the following data:
Height in Inches Number of
Persons
62-63 2
63-64 6
64-65 14
65-66 16
66-67 8
67-68 3
68-69 1
Total 50
7. A number of particular articles have been classified according to their weights.
After drying for two weeks, the same articles have again been weighed and
similarly classified. It is known that the median weight in the first weighing
-58
-
Caritas Business College BUSINESS STATISTICS – OAD 221
was 20.83 gm while in the second weighing it was 17.35 gm. Some
frequencies a and b in the first weighing and x and y in the second are
missing. It is known that a = 1/3x and b = 1/2 y. Find out the values of the
missing frequencies.
Class Frequencies
First Weighing Second Weighing
0- 5 a z
5-10 b y
10-15 11 40
15-20 52 50
20-25 75 30
25-30 22 28
8 Cities A, Band C are equidistant from each other. A motorist travels from A to
B at 30 km/h; from B to C at 40 km/h and from C to A at 50 km/h. Determine
his average speed for the entire trip.
9 Calculate the harmonic mean from the following data:
Class-Interval 2-4 4-6 6-8 8-10
Frequency 20 40 30 10
10 A vehicle when climbing up a gradient, consumes petrol @ 8 km per litre.
While coming down it runs 12 km per litre. Find its average consumption for
to and fro travel between two places situated at the two ends of 25 Ian long
gradient.
-59
-
Caritas Business College BUSINESS STATISTICS – OAD 221
2.12 SUGGESTED READINGS
1. Levin, Richard I. and David S. Rubin: Statistics for Management, Prentice
Hall, New Delhi.
2. Watsman Terry J. and Keith Parramor: Quantitative Methods in
Finance International, Thompson Business Press, London.
3. Hooda, R. P.: Statistics for Business and Economics, Macmillan, New Delhi.
4. Hein, L. W. Quantitative Approach to Managerial Decisions, Prentice Hall,
NJ.
-60
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 3: VARIATION
Topic : Standard Deviation and Variance
OBJECTIVE: The objective of the present lesson is to impart the knowledge of measures
of dispersion and skewness and to enable the students to distinguish
between average, dispersion, skewness, moments and kurtosis.
STRUCTURE:
3.1 Introduction
3.2 Meaning and Definition of Dispersion
3.3 Significance and Properties of Measuring Variation
3.4 Measures of Dispersion
3.5 Range
3.6 Interquartile Range or Quartile Deviation
3.7 Mean Deviation
3.8 Standard Deviation
3.9 Lorenz Curve
3.10 Skewness: Meaning and Definitions
3.11 Tests of Skewness
3.12 Measures of Skewness
3.13 Moments
3.14 Kurtosis
3.15 Summary
3.16 Self-Test Questions
3.17 Suggested Readings
3.1 INTRODUCTION
In the previous chapter, we have explained the measures of central tendency. It
may be noted that these measures do not indicate the extent of dispersion or
variability in a distribution. The dispersion or variability provides us one more step in
increasing our understanding of the pattern of the data. Further, a high degree of
uniformity (i.e. low degree of dispersion) is a desirable quality. If in a business there
is a high degree of variability in the raw material, then it could not find mass
production economical.
-61
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Suppose an investor is looking for a suitable equity share for investment. While
examining the movement of share prices, he should avoid those shares that are
highly fluctuating-having sometimes very high prices and at other times going
very low. Such extreme fluctuations mean that there is a high risk in the investment
in shares. The investor should, therefore, prefer those shares where risk is not so
high.
3.2 MEANING AND DEFINITIONS OF DISPERSION
The various measures of central value give us one single figure that represents the
entire data. But the average alone cannot adequately describe a set of
observations, unless all the observations are the same. It is necessary to describe
the variability or dispersion of the observations. In two or more distributions the
central value may be the same but still there can be wide disparities in the
formation of distribution.
Measures of dispersion help us in studying this important characteristic of a
distribution.
Some important definitions of dispersion are given below:
1. "Dispersion is the measure of the variation of the items." -A.L. Bowley
2. "The degree to which numerical data tend to spread about an average value
is called the variation of dispersion of the data." -Spiegel
3. Dispersion or spread is the degree of the scatter or variation of the variable
about a central value." -Brooks & Dick
4. "The measurement of the scatterness of the mass of figures in a series about
an average is called measure of variation or dispersion." -Simpson & Kajka
It is clear from above that dispersion (also known as scatter, spread or variation)
measures the extent to which the items vary from some central value. Since
measures of dispersion give an average of the differences of various items from
an average, they are also called averages of the second order. An average is
more meaningful when it is examined in the light of dispersion. For example, if the
average wage of the
-62
-
Caritas Business College BUSINESS STATISTICS – OAD 221
workers of factory A is Rs. 3885 and that of factory B Rs. 3900, we cannot
necessarily conclude that the workers of factory B are better off because in factory B
there may be much greater dispersion in the distribution of wages. The study of
dispersion is of great significance in practice as could well be appreciated from the
following example:
Series A Series B Series C
100 100 1
100 105 489
100 102 2
100 103 3
100 90 5
Total 500 500 500
x 100 100 100
Since arithmetic mean is the same in all three series, one is likely to conclude that
these series are alike in
nature. But a close
examination shall reveal
that distributions differ
widely from one another.
In series A, (In Box-3.1)
each and every item is
perfectly represented by
the arithmetic mean or in
other words none of the
items of series A deviates
from the
-63
-
Caritas Business College BUSINESS STATISTICS – OAD 221
arithmetic mean and hence there is no dispersion. In series B, only one item is
perfectly represented by the arithmetic mean and the other items vary but the
variation is very small as compared to series C. In series C. not a single item is
represented by the arithmetic mean and the items vary widely from one another. In
series C, dispersion is much greater compared to series B. Similarly, we may have
two groups of labourers with the same mean salary and yet their distributions may
differ widely. The mean salary may not be so important a characteristic as the
variation of the items from the mean. To the student of social affairs the mean
income is not so vitally important as to know how this income is distributed. Are a
large number receiving the mean income or are there a few with enormous incomes
and millions with incomes far below the mean? The three figures given in Box 3.1
represent frequency distributions with some of the characteristics. The two
curves in diagram (a) represent two
distractions with the same mean X , but with different dispersions. The two curves in
(b) represent two distributions with the same dispersion but with unequal means X l
and X 2, (c) represents two distributions with unequal dispersion. The measures of
central tendency are, therefore insufficient. They must be supported and
supplemented with other measures.
In the present chapter, we shall be especially concerned with the measures of
variability or spread or dispersion. A measure of variation or dispersion is one that
measures the extent to which there are differences between individual observation
and some central or average value. In measuring variation we shall be interested in
the amount of the variation or its degree but not in the direction. For example, a
measure of 6 inches below the mean has just as much dispersion as a measure of
six inches above the mean.
-64
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Literally meaning of dispersion is ‘scatteredness’. Average or the measures of central
tendency gives us an idea of the concentration of the observations about the central
part of the distribution. If we know the average alone, we cannot form a complete
idea about the distribution. But with the help of dispersion, we have an idea about
homogeneity or heterogeneity of the distribution.
3.3 SIGNIFICANCE AND PROPERTIES OF MEASURING
VARIATION
Measures of variation are needed for four basic purposes:
1. Measures of variation point out as to how far an average is representative
of the mass. When dispersion is small, the average is a typical value in the
sense that it closely represents the individual value and it is reliable in the
sense that it is a good estimate of the average in the corresponding universe.
On the other hand, when dispersion is large, the average is not so typical, and
unless the sample is very large, the average may be quite unreliable.
2. Another purpose of measuring dispersion is to determine nature and cause of
variation in order to control the variation itself. In matters of health variations
in body temperature, pulse beat and blood pressure are the basic guides to
diagnosis. Prescribed treatment is designed to control their variation. In
industrial production efficient operation requires control of quality variation
the causes of which are sought through inspection is basic to the control of
causes of variation. In social sciences a special problem requiring the
measurement of variability is the measurement of "inequality" of the
distribution of income or wealth etc.
3. Measures of dispersion enable a comparison to be made of two or more
series with regard to their variability. The study of variation may also be
looked
-65
-
Caritas Business College BUSINESS STATISTICS – OAD 221
upon as a means of determining uniformity of consistency. A high degree of
variation would mean little uniformity or consistency whereas a low degree of
variation would mean great uniformity or consistency.
4. Many powerful analytical tools in statistics such as correlation analysis. the
testing of hypothesis, analysis of variance, the statistical quality control,
regression analysis is based on measures of variation of one kind or another.
A good measure of dispersion should possess the following properties
1. It should be simple to understand.
2. It should be easy to compute.
3. It should be rigidly defined.
4. It should be based on each and every item of the distribution.
5. It should be amenable to further algebraic treatment.
6. It should have sampling stability.
7. Extreme items should not unduly affect it.
3.4 MEAURES OF DISPERSION
There are five measures of dispersion: Range, Inter-quartile range or Quartile
Deviation, Mean deviation, Standard Deviation, and Lorenz curve. Among them, the
first four are mathematical methods and the last one is the graphical method.
These are discussed in the ensuing paragraphs with suitable examples.
3.5 RANGE
The simplest measure of dispersion is the range, which is the difference between the
maximum value and the minimum value of data.
Example 3.1: Find the range for the following three sets of data:
Set 1: 05 15 15 05 15 05 15 15 15 15
Set 2: 8 7 15 11 12 5 13 11 15 9
-66
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Set 3: 5 5 5 5 5 5 5 5 5 5
Solution: In each of these three sets, the highest number is 15 and the lowest
number is 5. Since the range is the difference between the maximum value and the
minimum value of the data, it is 10 in each case. But the range fails to give any idea
about the dispersal or spread of the series between the highest and the lowest value.
This becomes evident from the above data.
In a frequency distribution, range is calculated by taking the difference between the
upper limit of the highest class and the lower limit of the lowest class.
Example 3.2: Find the range for the following frequency distribution:
Size of Frequen
Item cy
20- 40 7
40- 60 11
60- 80 30
80-100 17
100-120 5
Total 70
Solution: Here, the upper limit of the highest class is 120 and the lower limit of the
lowest class is 20. Hence, the range is 120 - 20 = 100. Note that the range is not
influenced by the frequencies. Symbolically, the range is calculated b the formula L -
S, where L is the largest value and S is the smallest value in a distribution. The
coefficient of range is calculated by the formula: (L-S)/ (L+S). This is the relative
measure. The coefficient of the range in respect of the earlier example having three
sets of data is: 0.5.The coefficient of range is more appropriate for purposes of
comparison as will be evident from the following example:
Example 3.3: Calculate the coefficient of range separately for the two sets of data
given below:
Set 1 8 10 20 9 15 10 13 28
Set 2 30 35 42 50 32 49 39 33
-67
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution: It can be seen that the range in both the sets of data is the
same: Set 1 28 - 8 = 20
Set 2 50 - 30 = 20
Coefficient of range in Set 1 is:
28 – 8 = 0.55
28+8
Coefficient of range in set 2 is:
50 – 30
= 0.25
50 +30
3.5.1 LIMITATIONS OF RANGE
There are some limitations of range, which are as follows:
1. It is based only on two items and does not cover all the items in a distribution.
2. It is subject to wide fluctuations from sample to sample based on the
same population.
3. It fails to give any idea about the pattern of distribution. This was evident from
the data given in Examples 1 and 3.
4. Finally, in the case of open-ended distributions, it is not possible to
compute the range.
Despite these limitations of the range, it is mainly used in situations where one wants
to quickly have some idea of the variability or' a set of data. When the sample size is
very small, the range is considered quite adequate measure of the variability. Thus,
it is widely used in quality control where a continuous check on the variability of raw
materials or finished products is needed. The range is also a suitable measure in
weather forecast. The meteorological department uses the range by giving the
maximum and the minimum temperatures. This information is quite useful to the
common man, as he can know the extent of possible variation in the temperature on
a particular day.
-68
-
Caritas Business College BUSINESS STATISTICS – OAD 221
3.6 INTERQUARTILE RANGE OR QUARTILE DEVIATION
The interquartile range or the quartile deviation is a better measure of variation in a
distribution than the range. Here, avoiding the 25 percent of the distribution at both
the ends uses the middle 50 percent of the distribution. In other words, the
interquartile range denotes the difference between the third quartile and the first
quartile.
Symbolically, interquartile range = Q3- Ql
Many times the interquartile range is reduced in the form of semi-interquartile range
or quartile deviation as shown below:
Semi interquartile range or Quartile deviation = (Q3 – Ql)/2
When quartile deviation is small, it means that there is a small deviation in the central
50 percent items. In contrast, if the quartile deviation is high, it shows that the
central
50 percent items have a large variation. It may be noted that in a symmetrical
distribution, the two quartiles, that is, Q3 and QI are equidistant from the median.
Symbolically,
M-QI = Q3-M
However, this is seldom the case as most of the business and economic data are
asymmetrical. But, one can assume that approximately 50 percent of the
observations are contained in the interquartile range. It may be noted that
interquartile range or the quartile deviation is an absolute measure of dispersion. It
can be changed into a relative measure of dispersion as follows:
Coefficient of QD = Q3 –Q1
Q3 +Q1
The computation of a quartile deviation is very simple, involving the computation of
upper and lower quartiles. As the computation of the two quartiles has already been
explained in the preceding chapter, it is not attempted here.
-69
-
Caritas Business College BUSINESS STATISTICS – OAD 221
3.6.1 MERITS OF QUARTILE DEVIATION
The following merits are entertained by quartile deviation:
1. As compared to range, it is considered a superior measure of dispersion.
2. In the case of open-ended distribution, it is quite suitable.
3. Since it is not influenced by the extreme values in a distribution, it is
particularly suitable in highly skewed or erratic distributions.
3.6.2 LIMITATIONS OF QUARTILE DEVIATION
1. Like the range, it fails to cover all the items in a distribution.
2. It is not amenable to mathematical manipulation.
3. It varies widely from sample to sample based on the same population.
4. Since it is a positional average, it is not considered as a measure of dispersion.
It merely shows a distance on scale and not a scatter around an average.
In view of the above-mentioned limitations, the interquartile range or the quartile
deviation has a limited practical utility.
3.7 MEAN DEVIATION
The mean deviation is also known as the average deviation. As the name implies, it
is the average of absolute amounts by which the individual items deviate from the
mean. Since the positive deviations from the mean are equal to the negative
deviations, while computing the mean deviation, we ignore positive and negative
signs.
Symbolically,
MD = | x Where MD = mean deviation, |x| = deviation of an item
|n
from the mean ignoring positive and negative signs, n = the total number of
observations.
-70
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 3.4:
Size of Item Frequency
2-4 20
4-6 40
6-8 30
8-10 10
Solution:
Size of Item Mid-points (m) Frequency (f) fm d from x f |d|
2-4 3 20 60 -2.6 52
4-6 5 40 200 -0.6 24
6-8 7 30 210 1.4 42
8-10 9 10 90 3.4 34
Total 100 560 152
fm 560
x = n 5.6
100
f |d | 152
1.52
MD ( x ) = n 100
3.7.1 MERITS OF MEAN DEVIATION
1. A major advantage of mean deviation is that it is simple to understand
and easy to calculate.
2. It takes into consideration each and every item in the distribution. As a
result, a change in the value of any item will have its effect on the magnitude
of mean deviation.
3. The values of extreme items have less effect on the value of the mean
deviation.
4. As deviations are taken from a central value, it is possible to have meaningful
comparisons of the formation of different distributions.
3.7.2 LIMITATIONS OF MEAN DEVIATION
1. It is not capable of further algebraic treatment.
-71
-
Caritas Business College BUSINESS STATISTICS – OAD 221
2. At times it may fail to give accurate results. The mean deviation gives best
results when deviations are taken from the median instead of from the mean.
But in a series, which has wide variations in the items, median is not a
satisfactory measure.
3. Strictly on mathematical considerations, the method is wrong as it ignores the
algebraic signs when the deviations are taken from the mean.
In view of these limitations, it is seldom used in business studies. A better measure
known as the standard deviation is more frequently used.
3.8 STANDARD DEVIATION
The standard deviation is similar to the mean deviation in that here too the deviations
are measured from the mean. At the same time, the standard deviation is preferred
to the mean deviation or the quartile deviation or the range because it has desirable
mathematical properties.
Before defining the concept of the standard deviation, we introduce another concept
viz. variance.
Example 3.5:
X X- (X-)2
20 20-18=12 4
15 15-18= -3 9
19 19-18 = 1 1
24 24-18 = 6 36
16 16-18 = -2 4
14 14-18 = -4 16
108 Total 70
Solution:
108
Mean = = 18
6
-72
-
Caritas Business College BUSINESS STATISTICS – OAD 221
The second column shows the deviations from the mean. The third or the last column
shows the squared deviations, the sum of which is 70. The arithmetic mean of the
squared deviations is:
2
x
= 70/6=11.67 approx.
N
This mean of the squared deviations is known as the variance. It may be noted
that this variance is described by different terms that are used interchangeably: the
variance of the distribution X; the variance of X; the variance of the distribution; and
just simply, the
variance. x
2
Symbolically, Var (X) = N
It is also written as 2 x i 2
N
Where 2 (called sigma squared) is used to denote the variance.
Although the variance is a measure of dispersion, the unit of its measurement is
(points). If a distribution relates to income of families then the variance is (Rs) 2 and
not rupees. Similarly, if another distribution pertains to marks of students, then the
unit of variance is (marks) 2. To overcome this inadequacy, the square root of
variance is taken, which yields a better measure of dispersion known as the standard
deviation. Taking our earlier example of individual observations, we take the square
root of the variance
SD or = Variance = = 3.42 points
11.67
Symbolically, = x i
2
In applied Statistics, the standard deviation is more frequently used than the
variance. This can also be written as:
-73
-
Caritas Business College BUSINESS STATISTICS – OAD 221
x
2
x 2 i
i
= N
N
We use this formula to calculate the standard deviation from the individual
observations given earlier.
Example 7.6:
X X
2
20 400
15 225
19 361
24 576
16 256
14 196
108 2014
Solution:
xi
x 2014
2
N=6
i 108
1082 2014 11664
2014
=
6 Or, = 6
6 6
12084 11664 420
6 6
= 6 Or, =6
= 70 Or, = 11.67
6
= 3.42
Example 3.7:
The following distribution relating to marks obtained by students in an examination:
Marks Number of
Students
0- 10 1
10- 20 3
20- 30 6
30- 40 10
40- 50 12
-74
-
Caritas Business College BUSINESS STATISTICS – OAD 221
50- 60 11
-75
-
Caritas Business College BUSINESS STATISTICS – OAD 221
60- 70 6
70- 80 3
80- 90 2
90-100 1
Solution:
Marks Frequency Mid- Deviations Fd’ fd'2
(f) points (d)/10=d’
0- 10 1 5 -5
-5 25
10- 20 3 15 -4
-12 48
20- 30 6 25 -3
-18 54
30- 40 10 35 -2
-20 40
40- 50 12 45 -1
-12 12
50- 60 11 55 0
0 0
60- 70 6 65 1
6 6
70- 80 3 75 2
6 12
80- 90 2 85 3
6 18
90- 1 95 4
4 16
100
Total 55 Total -45 231
In the case of frequency distribution where the individual values are not known, we use
the midpoints of the class intervals. Thus, the formula used for calculating the
standard deviation is as given below:
= K
fim
i
i1
2
N
Where mi is the mid-point of the class intervals is the mean of the distribution, fi is
the frequency of each class; N is the total number of frequency and K is the number
of classes. This formula requires that the mean be calculated and that deviations
(mi -
) be obtained for each class. To avoid this inconvenience, the above formula can be
modified as:
2
K K
= ii 1 i 1i
fid fd
N
Where C is the class interval: fi is the frequency of the ith class and di is the
deviation of the of item from an assumed origin; and N is the total number of
observations.
Applying this formula for the table given earlier,
45 2
= 10 231
55 55
-76
-
Caritas Business College BUSINESS STATISTICS – OAD 221
=104.2 0.669421
=18.8 marks
When it becomes clear that the actual mean would turn out to be in fraction, calculating
deviations from the mean would be too cumbersome. In such cases, an assumed
mean is used and the deviations from it are calculated. While mid- point of any class
can be taken as an assumed mean, it is advisable to choose the mid-point of that
class that would make calculations least cumbersome. Guided by this consideration,
in Example 3.7 we have decided to choose 55 as the mid-point and, accordingly,
deviations have been taken from it. It will be seen from the calculations that they are
considerably simplified.
3.8.1 USES OF THE STANDARD DEVIATION
The standard deviation is a frequently used measure of dispersion. It enables us to
determine as to how far individual items in a distribution deviate from its mean. In a
symmetrical, bell-shaped curve:
(i) About 68 percent of the values in the population fall within: + 1 standard
deviation from the mean.
(ii) About 95 percent of the values will fall within +2 standard deviations from the
mean.
(iii) About 99 percent of the values will fall within + 3 standard deviations from
the mean.
The standard deviation is an absolute measure of dispersion as it measures variation
in the same units as the original data. As such, it cannot be a suitable measure while
comparing two or more distributions. For this purpose, we should use a relative
measure of dispersion. One such measure of relative dispersion is the coefficient of
variation, which relates the standard deviation and the mean such that the standard
deviation is expressed as a percentage of mean. Thus, the specific unit in which the
standard deviation is measured is done away with and the new unit becomes
percent.
-77
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Symbolically, CV (coefficient of variation) = x 100
Example 3.8: In a small business firm, two typists are employed-typist A and typist
B. Typist A types out, on an average, 30 pages per day with a standard deviation of 6.
Typist B, on an average, types out 45 pages with a standard deviation of 10.
Which typist shows greater consistency in his output?
Solution: Coefficient of variation for A x 100
Or A 6
x 100
30
Or 20% and
Coefficient of variation for B x 100
10
B x 100
45
or 22.2 %
These calculations clearly indicate that although typist B types out more pages,
there is a greater variation in his output as compared to that of typist A. We can say
this in a different way: Though typist A's daily output is much less, he is more
consistent than typist B. The usefulness of the coefficient of variation becomes
clear in comparing two groups of data having different means, as has been the case
in the above example.
3.8.2 STANDARDISED VARIABLE, STANDARD SCORES
The variable Z = (x - x )/s or (x - )/, which measures the deviation from the
mean in units of the standard deviation, is called a standardised variable. Since both
the numerator and the denominator are in the same units, a standardised variable is
independent of units used. If deviations from the mean are given in units of the
standard deviation, they are said to be expressed in standard units or standard
-78
-
Caritas Business College BUSINESS STATISTICS – OAD 221
scores.
-79
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Through this concept of standardised variable, proper comparisons can be made
between individual observations belonging to two different distributions whose
compositions differ.
Example 3.9: A student has scored 68 marks in Statistics for which the
average marks were 60 and the standard deviation was 10. In the paper on
Marketing, he scored 74 marks for which the average marks were 68 and the
standard deviation was
15. In which paper, Statistics or Marketing, was his relative standing higher?
Solution: The standardised variable Z = (x - x ) s measures the deviation of x
from the mean x in terms of standard deviation s. For Statistics, Z = (68 - 60) 10 =
0.8 For Marketing, Z = (74 - 68) 15 = 0.4
Since the standard score is 0.8 in Statistics as compared to 0.4 in Marketing,
his relative standing was higher in Statistics.
Example 3.10: Convert the set of numbers 6, 7, 5, 10 and 12 into standard scores:
Solution:
X X2
6 36
7 49
5 25
10 100
12 144
2
X = 40 X =
354
x x N 40 5 8
X 2 402
NN
x2 354
5
= or, =
5
= 354 320 = 2.61 approx.
5
-80
-
Caritas Business College BUSINESS STATISTICS – OAD 221
xx 68
Z= = -0.77 (Standard score)
2.61
Applying this formula to other values:
78
(i) = -0.38
2.61
(ii)
5 8 = -1.15
(iii) 2.61
= 0.77
10
8
(iv) (
i = 1.53
v 2.61
)
12
8
2.61
Thus the standard scores for 6,7,5,10 and 12 are -0.77, -0.38, -1.15, 0.77 and 1.53,
respectively.
3.9 LORENZ CURVE
This measure of dispersion is graphical. It is known as the Lorenz curve named after
Dr. Max Lorenz. It is generally used to show the extent of concentration of income
and wealth. The steps involved in plotting the Lorenz curve are:
1. Convert a frequency distribution into a cumulative frequency table.
2. Calculate percentage for each item taking the total equal to 100.
3. Choose a suitable scale and plot the cumulative percentages of the persons
and income. Use the horizontal axis of X to depict percentages of persons
and the vertical axis of Y to depict percent ages of income.
4. Show the line of equal distribution, which will join 0 of X-axis with 100 of Y-
axis.
5. The curve obtained in (3) above can now be compared with the straight line of
equal distribution obtained in (4) above. If the Lorenz curve is close to the line
of equal distribution, then it implies that the dispersion is much less. If, on the
-81
-
Caritas Business College BUSINESS STATISTICS – OAD 221
contrary, the Lorenz curve is farther away from the line of equal distribution,
it implies that the dispersion is considerable.
The Lorenz curve is a simple graphical device to show the disparities of distribution
in any phenomenon. It is, used in business and economics to represent inequalities
in income, wealth, production, savings, and so on.
Figure 3.1 shows two Lorenz curves by way of illustration. The straight line AB is a
line of equal distribution, whereas AEB shows complete inequality. Curve ACB and
curve ADB are the Lorenz curves.
A F
Figure 3.1: Lorenz Curve
As curve ACB is nearer to the line of equal distribution, it has more equitable
distribution of income than curve ADB. Assuming that these two curves are for the
same company, this may be interpreted in a different manner. Prior to taxation, the
curve ADB showed greater inequality in the income of its employees. After the
taxation, the company’s data resulted into ACB curve, which is closer to the line of
equal distribution. In other words, as a result of taxation, the inequality has reduced.
3.10 SKEWNESS: MEANING AND DEFINITIONS
In the above paragraphs, we have discussed frequency distributions in detail. It
may be repeated here that frequency distributions differ in three ways: Average
value, Variability or dispersion, and Shape. Since the first two, that is, average
value and
-82
-
Caritas Business College BUSINESS STATISTICS – OAD 221
variability or dispersion have already been discussed in previous chapters, here our
main spotlight will be on the shape of frequency distribution. Generally, there are two
comparable characteristics called skewness and kurtosis that help us to understand
a distribution. Two distributions may have the same mean and standard deviation but
may differ widely in their overall appearance as can be seen from the following:
In both these distributions the value of
mean and standard deviation is the
same ( X = 15, = 5). But it does not
imply that the distributions are alike in
nature.
The distribution on the left-hand side is
a symmetrical one whereas the distribution on the right-hand side is symmetrical or
skewed. Measures of skewness help us to distinguish between different types of
distributions.
Some important definitions of skewness are as follows:
1. "When a series is not symmetrical it is said to be asymmetrical or skewed."
-Croxton & Cowden.
2. "Skewness refers to the asymmetry or lack of symmetry in the shape of a
frequency distribution." -Morris Hamburg.
3. "Measures of skewness tell us the direction and the extent of skewness. In
symmetrical distribution the mean, median and mode are identical. The more
the mean moves away from the mode, the larger the asymmetry or
skewness."
-Simpson & Kalka
4. "A distribution is said to be 'skewed' when the mean and the median fall at
different points in the distribution, and the balance (or centre of gravity) is
shifted to one side or the other-to left or right." -Garrett
-83
-
Caritas Business College BUSINESS STATISTICS – OAD 221
The above definitions show that the term 'skewness' refers to lack of symmetry" i.e.,
when a distribution is not symmetrical (or is asymmetrical) it is called a skewed
distribution.
The concept of skewness will be clear from the following three diagrams showing a
symmetrical distribution. a positively skewed distribution and a negatively skewed
distribution.
1. Symmetrical Distribution. It is clear from the diagram (a) that in a sym-
metrical distribution the values of mean, median and mode coincide. The
spread of the frequencies is the same on
both sides of the centre point of the curve.
2. Asymmetrical Distribution. A
distribution, which is not symmetrical, is
called a skewed distribution and such a
distribution could either be positively
skewed or negatively skewed as would be
clear from the diagrams (b) and (c).
3. Positively Skewed Distribution. In the
positively skewed distribution the value of
the mean is maximum and that of mode least-the median lies in between the
two as is clear from the diagram (b).
4. Negatively Skewed Distribution. The following is the shape of negatively
skewed distribution. In a negatively skewed distribution the value of mode is
maximum and that of mean least-the median lies in between the two. In the
positively skewed distribution the frequencies are spread out over a
greater
-84
-
Caritas Business College BUSINESS STATISTICS – OAD 221
range of values on the high-value end of the curve (the right-hand side) than
they are on the low-value end. In the negatively skewed distribution the
position is reversed, i.e. the excess tail is on the left-hand side. It should be
noted that in moderately symmetrical distributions the interval between the
mean and the median is approximately one-third of the interval between the
mean and the mode. It is this relationship, which provides a means of
measuring the degree of skewness.
3.11 TESTS OF SKEWNESS
In order to ascertain whether a distribution is skewed or not the following tests may be
applied. Skewness is present if:
1. The values of mean, median and mode do not coincide.
2. When the data are plotted on a graph they do not give the normal bell-
shaped form i.e. when cut along a vertical line through the centre the two
halves are not equal.
3. The sum of the positive deviations from the median is not equal to the
sum of the negative deviations.
4. Quartiles are not equidistant from the median.
5. Frequencies are not equally distributed at points of equal deviation from
the mode.
On the contrary, when skewness is absent, i.e. in case of a symmetrical
distribution, the following conditions are satisfied:
1. The values of mean, median and mode coincide.
2. Data when plotted on a graph give the normal bell-shaped form.
3. Sum of the positive deviations from the median is equal to the sum of
the negative deviations.
-85
-
Caritas Business College BUSINESS STATISTICS – OAD 221
4. Quartiles are equidistant from the median.
5. Frequencies are equally distributed at points of equal deviations from
the mode.
3.12 MEASURES OF SKEWNESS
There are four measures of skewness, each divided into absolute and relative
measures. The relative measure is known as the coefficient of skewness and is more
frequently used than the absolute measure of skewness. Further, when a comparison
between two or more distributions is involved, it is the relative measure of skewness,
which is used. The measures of skewness are: (i) Karl Pearson's measure, (ii)
Bowley’s measure, (iii) Kelly’s measure, and (iv) Moment’s measure. These
measures are discussed briefly below:
3.12.1 KARL PEARON’S MEASURE
The formula for measuring skewness as given by Karl Pearson is as
follows: Skewness = Mean - Mode
Coefficient of skewness =
Mean – Mode
Standard Deviation
In case the mode is indeterminate, the coefficient of skewness is:
Skp = Mean - (3 Median - 2 Mean)
Standard deviation
Skp = 3(Mean -
Median)
Standard
deviation
Now this formula is equal to the earlier one.
3(Mean - Mean - Mode
Median) Standard
Standard deviation
deviation
Or 3 Mean - 3 Median = Mean - Mode
Or Mode = Mean - 3 Mean + 3 Median
Or Mode = 3 Median - 2 Mean
The direction of skewness is determined by ascertaining whether the mean is
greater than the mode or less than the mode. If it is greater than the mode, then
skewness is
-86
-
Caritas Business College BUSINESS STATISTICS – OAD 221
positive. But when the mean is less than the mode, it is negative. The difference
between the mean and mode indicates the extent of departure from symmetry. It is
measured in standard deviation units, which provide a measure independent of
the unit of measurement. It may be recalled that this observation was made in the
preceding chapter while discussing standard deviation. The value of coefficient of
skewness is zero, when the distribution is symmetrical. Normally, this coefficient of
skewness lies between +1. If the mean is greater than the mode, then the coefficient
of skewness will be positive, otherwise negative.
Example 3.11: Given the following data, calculate the Karl Pearson's coefficient of
skewness: x = 452 x2= 24270 Mode = 43.7 and N = 10
Solution:
Pearson's coefficient of skewness is:
SkP =
Mean - Mode
Standard deviation
X 452
Mean ( x 45.2
)=
N 10 2 2
SD x x
x2
N x2
N
N
N
2 2427 (45.2) 19.59
24270 452
2
10 10
Applying the values of mean, mode and standard deviation in the above
formula, Sk = 45.2 – 43.7
p
19.59
=0.08
This shows that there is a positive skewness though the extent of skewness is
marginal.
Example 3.12: From the following data, calculate the measure of skewness using
the mean, median and standard deviation:
X 10 - 20 - 30 - 40 40 - 50- 60 - 70 -
20 30 50 60 70 80
f 18 30 40 55 38 20 16
-87
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution:
2
x MVx dx f fdx fdX cf
10 - 15 -3 18 -54 162 18
20
20 - 25 -2 30 -60 120 48
30
30 - 35 -1 40 -40 40 88
40
40-50 45= 0 55 0 0 143
a
50 - 55 1 38 38 38 181
60
60 - 65 2 20 40 80 201
70
70 - 75 3 16 48 144 217
80
Tota 217 -28 584
l
a = Assumed mean = 45, cf = Cumulative frequency, dx = Deviation from assumed
mean, and i = 10
fdx
xa i
N
45 28
10 43.71
21
7
l2 l1
Median= l1 (m c)
f1
Where m = (N + 1)/2th item
= (217 + 1)/2 = 109th item
50 40
Median 40 (109 88)
55
10
40 21
55
= 43.82
SD =
fd 2fd
x x
584 28 2
f f 10
217 217 10
-88
-
Caritas Business College BUSINESS STATISTICS – OAD 221
= 10 16.4
2.69 - 0.016
Skewness = 3 (Mean - Median)
= 3 (43.71 - 43.82)
= 3 x -0.011
-89
-
Caritas Business College BUSINESS STATISTICS – OAD 221
= -0.33
Coefficient of skewness
Skewness or
SD
= -0.33
16.4
= -0.02
The result shows that the distribution is negatively skewed, but the extent of
skewness is extremely negligible.
3.12.2 Bowley's Measure
Bowley developed a measure of skewness, which is based on quartile values. The
formula for measuring skewness is:
Skewness Q3 Q1
= 2M Q3
Q1
Where Q3 and Q1 are upper and lower quartiles and M is the median. The value of
this skewness varies between +1. In the case of open-ended distribution as well as
where extreme values are found in the series, this measure is particularly useful. In a
symmetrical distribution, skewness is zero. This means that Q3 and Q1 are positioned
equidistantly from Q2 that is, the median. In symbols, Q3 - Q2 = Q2 – Q1' In contrast,
when the distribution is skewed, then Q3 - Q2 will be different from Q2 – Q1' When Q3
- Q2 exceeds Q2 – Q1' then skewness is positive. As against this; when Q3 - Q2 is less
than Q2 – Q1' then skewness is negative. Bowley’s measure of skewness can- be
written as:
Skewness = (Q3 - Q2) - (Q2 – Q1 or Q3 - Q2 - Q2 + Q1
Or Q3 + Q1 - 2Q2 (2Q2 is 2M)
However, this is an absolute measure of skewness. As such, it cannot be used
while comparing two distributions where the units of measurement are different. In
view of this limitation, Bowley suggested a relative measure of skewness as given
below:
-90
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Relative Skewness = (Q3 Q2 )
(Q2 Q1 ) (Q3
=
Q2 ) (Q2
= Q1 ) Q3 Q2
Q2 Q1 Q3
=
Q2 Q2
Q1 Q3 Q1
2Q2
Q3 Q1
Q3 Q1
2M Q3
Q1
Example 3.13: For a distribution, Bowley’s coefficient of skewness is - 0.56,
Q1=16.4 and Median=24.2. What is the coefficient of quartile deviation?
Solution:
Q3 Q1 2M
Bowley's coefficient of skewness is: SkB =
Q3 Q1
Substituting the values in the above formula,
Q3 16.4 - (2 x
SkB = 24.2)
Q3 16.4
Q3 16.4 -
0.56 48.4
Q3 16.4
or - 0.56 (Q3-16.4) = Q3-32
or - 0.56 Q3 + 9.184 = Q3-32
or - 0.56 Q3 - Q3 = -32 - 9.184
- 1.56 Q3 = - 41.184
Q3 = 41.184
1.56 26.4
Now, we have the values of both the upper and the lower quartiles.
Q Q1 Q3 Q1
Coefficient of quartile deviation = =
3
-91
-
Caritas Business College BUSINESS STATISTICS – OAD 221
26.4 16.4
26.4 16.4 42.
8
10 0.234 Approx.
Example 3.14: Calculate an appropriate measure of skewness from the
following data:
-92
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Value in Rs Frequency
Less than 50 40
50 - 100 80
100 - 150 130
150 – 200 60
200 and above 30
Solution: It should be noted that the series given in the question is an open-ended
series. As such, Bowley's coefficient of skewness, which is based on quartiles, would
be the most appropriate measure of skewness in this case. In order to calculate the
quartiles and the median, we have to use the cumulative frequency. The table is
reproduced below with the cumulative frequency.
Value in Rs Frequenc Cumulative Frequency
y
Less than 50 40 40
50 - 100 80 120
100 - 150 130 250
150 - 200 60 310
200 and 30 340
above
l2 l1
Q1 = l1 (m
c) f1
341
n1 = 85.25, which lies in 50 - 100 class
Now m=( ) item 4
=
4
100 50
Q1 = 50 + (85.25 40) 78.28
80
n1 4
M=( ) item
=
-93
-
Caritas Business College BUSINESS STATISTICS – OAD 221
341
= 170.25, which lies in 100 -
150 class
4
-94
-
Caritas Business College BUSINESS STATISTICS – OAD 221
150 100
M= 100 + (170.5 120) 119.4
130
l2 l1
Q3 = l1 (m
c) f1
m = 3(341) 4 = 255.75
200 150
Q3 = 150 + (255.75 250) 154.79
60
Bowley's coefficient of skewness is:
Q3 + QI - 2M 154.79+ 78.28 - (2 x 119.4) -5.73
Q3 - QI = 154.79 -78.28 = 76.51
= - 0.075 approx.
This shows that there is a negative skewness, which has a very negligible magnitude.
3.12.3 Kelly's Measure
Kelly developed another measure of skewness, which is based on percentiles.
The formula for measuring skewness is as follows:
Coefficient of skewness = P90 2P50
P10 P90
Or, P10
D1 D9 2M
D9 D1
Where P and D stand for percentile and decile respectively. In order to calculate the
coefficient of skewness by this formula, we have to ascertain the values of 10th,
50th and 90th percentiles. Somehow, this measure of skewness is seldom used. All
the same, we give an example to show how it can be calculated.
Example 3.15: Use Kelly's measure to calculate skewness.
Class Intervals f cf
10 - 20 18 18
20 - 30 30 48
-95
-
Caritas Business College BUSINESS STATISTICS – OAD 221
30- 40 40 88
40- 50 55 143
50 - 60 38 181
60 – 70 20 201
70 - 80. 16 217
Solution: Now we have to calculate P10 P30 and P90.
l2 l1
PIO = l1 (m c) , where m = (n + 1)/10th item
f1
217 1
item
21.8th
10
This lies in the 20 - 30 class.
30 20 10 3.8
20 (21.8 18) 20 21.27approx.
30 30
P50 (median): where m = (n + 1)/2th item = 217 1
= 109th item
2
This lies in the class 40 - 50. Applying the above formula:
50 40 10 21
40 (109 88) 40 21 43.82approx.
55 55
P90: here m = 90 (217 + 1)/100th item = 196.2th item
This lies in the class 60 - 70. Applying the above formula:
70 60 10 15.2
60 (196.2 181) 60 67.6approx.
20 20
Kelley's skewness
P90 2P50 P10
SkK P90 P10
67.6 - (2 x 43.82) 21.27
= 67.6 - 21.27
88.87 - 87.64
= 46.63
-96
-
Caritas Business College BUSINESS STATISTICS – OAD 221
= 0.027
-97
-
Caritas Business College BUSINESS STATISTICS – OAD 221
This shows that the series is positively skewed though the extent of skewness is extremely
negligible. It may be recalled that if there is a perfectly symmetrical distribution, then
the skewness will be zero. One can see that the above answer is very close to zero.
3.13 MOMENTS
In mechanics, the term moment is used to denote the rotating effect of a force. In
Statistics, it is used to indicate peculiarities of a frequency distribution. The utility of
moments lies in the sense that they indicate different aspects of a given distribution.
Thus, by using moments, we can measure the central tendency of a series,
dispersion or variability, skewness and the peakedness of the curve. The moments
about the actual arithmetic mean are denoted by . The first four moments
about mean or
central moments are as follows:
1
First moment 1 = N x1 x
Second moment 2 = 2
1 x
Third moment 3 =
3
x
x
N 1
Fourth moment 3 =
4
1 x
1
x
N
1
1
x
N
These moments are in relation to individual items. In the case of a frequency
distribution, the first four moments will be:
1
fi x
First moment 1 = N 1
x
Second moment 2 = T moment
h =
ir
d1
-98
-
Caritas Business College BUSINESS STATISTICS – OAD 221
1
fix N 2
x
1
fix N
x
3 1
-99
-
Caritas Business College BUSINESS STATISTICS – OAD 221
Fourth moment 3 = 1 4
x
1
fix N
It may be noted that the first central moment is zero, that is, =
0. The second central moment is 2=, indicating the variance.
The third central moment 3 is used to measure skewness. The fourth central
moment gives an idea about the Kurtosis.
Karl Pearson suggested another measure of skewness, which is based on the third
and second central moments as given below:
2
1 3
3
2
Example 3.16: Find the (a) first, (b) second, (c) third and (d) fourth moments for the
set of numbers 2,3,4,5 and 6.
Solution:
(a) x 2 3 4 5 6 20
x 4
N 5 5
(b)
x 22 32 42 52
x
62
2
5
N
4 9 16 25 36
5 18
(c)
x
x 23 33 43 53
63
3
5
N
8 27 64 125 216
5 88
(d)
-
100
Caritas Business College BUSINESS STATISTICS – OAD 221
x 4
34 44 54 64
x 4
5
N 2
16 81 256 625 1296
5 454.8
-
101
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 3.17: Using the same set of five figures as given in Example 3.7, find the
(a) first, (b) second, (c) third and (d) fourth moments about the mean.
Solution:
(x x) (2 4) (3 4) (4 4) (5 4) (6 4)
m (x x)
1
N 5
- 2 -1 0 1 2
= 5 =0
m2 (x x)2
(x x)
2
(2 4)2 (3 4)2 (4 4)2 (5 4)2 (6 4)2
N 5
(-2)2 (_1) 2 02 12 22
=
5
41014
= = 2. It may be noted that m2 is the
variance 5
(x x)
3
(2 4)3 (3 4)3 (4 4)3 (5 4)3 (6 4)3
m3= (x x)3 N 5
(-2)3 (_1)3 03 13
23 - 8 -1 0 1 8
= = 5 0
5
4 4
(x x) (2 4) (3 4)4 (4 4)4 (5 4)4 (6
N 4)4
m4= (x x)4
5
(-2)4 (_1) 4 04 14 24
=
5
16 1 0 1 016
= 5 6.8
Example 3.18: Calculate the first four central moments from the following data:
Class interval 50-60 60-70 70-80 80-90 90-100
Frequency 5 12 20 7 6
Solution:
-
102
Caritas Business College BUSINESS STATISTICS – OAD 221
Class f MV d from d/10 fd fd2 fd3 fd4
Interval 75
50- 60 5 55 -20 -2 -10 20 - 80
40
-
103
Caritas Business College BUSINESS STATISTICS – OAD 221
60- 70 12 6 -10 -1 -12 12 - 12
5 12
70- 80 20 7 0 0 0 0 0 0
5
80- 90 7 8 10 1 7 7 7 7
5
90-100 6 9 20 2 12 24 48 96
5
Total 50 -3 -4 195
fd i 3 10
' 0.6
1 N 50
fd i
2
63 12.6
2' N
10
50
fd i
3
0.8
N
2' 4
10
50
fd i4
2' 195 19
N
10
50
Moments about Mean
1=1’ - 1’= -0.6-(-0.6) = 0
2=2’ - 1’2=10-( -0.6)2= 10-3.6=6.4
3=3’ - 32’’1+21’3=-0.8-3(12.6)(-0.6)+2(-0.6)3
= -0.8 + 22.68 + 0.432 = 22.312
’2 ’4
4=4’ - 43’’1+621 -31
= 19 + 4(-0.8)(-0.6) + 6(10)(-0.6)2- 3(-0.6)4
= 19 + 1.92 + 21.60 - 0.3888
= 42.1312
3.14 KURTOSIS
Kurtosis is another measure of the shape of a frequency curve. It is a Greek word,
which means bulginess. While skewness signifies the extent of asymmetry, kurtosis
measures the degree of peakedness of a frequency distribution. Karl Pearson
classified curves into three types on the basis of the shape of their peaks. These are
mesokurtic, leptokurtic and platykurtic. These three types of curves are shown in
figure below:
-
104
Caritas Business College BUSINESS STATISTICS – OAD 221
It will be seen from Fig.
3.2 that mesokurtic curve is
neither too much flattened
nor too much peaked. In
fact, this is the frequency
curve of a normal
distribution. Leptokurtic
curve is a more peaked than the normal curve. In contrast, platykurtic is a relatively
flat curve. The coefficient of kurtosis as given by Karl Pearson is 2=4/22. In case
of a normal distribution, that is, mesokurtic curve, the value of 2=3. If 2 turn out to
be
> 3, the curve is called a leptokurtic curve and is more peaked than the normal curve.
Again, when 2 < 3, the curve is called a platykurtic curve and is less peaked than
the normal curve. The measure of kurtosis is very helpful in the selection of an
appropriate average. For example, for normal distribution, mean is most appropriate;
for a leptokurtic distribution, median is most appropriate; and for platykurtic
distribution, the quartile range is most appropriate.
Example 3.19: From the data given in Example 3.18, calculate the kurtosis.
Solution: For this, we have to calculate 2 This can be done by using the formula
2=4/ 2. In the preceding example, values of 4 and 2 are given. Hence, 2 =
2
42.1312 (6.4)2 = 1.03.
As 2. < 3, the distribution is platykurtic.
Another measure of kurtosis is based on both quartiles and percentiles and is given
by the following formula:
Q
K
P90
P10
-
105
Caritas Business College BUSINESS STATISTICS – OAD 221
Where K = kurtosis, Q = ½ (Q3 – Q1) is the semi-interquartile range; P90 is 90th
percentile and P10 is the 10th percentile. This is also known as the percentile
coefficient of kurtosis. In case of the normal distribution, the value of K is 0.263.
Example 3.20: From the data given below, calculate the percentile coefficient of
kurtosis.
Daily Wages in Rs. Number of Workers cf
50- 60 10 10
60-70 14 24
70-80 18 42
80 - 90 24 66
90-100 16 82
100 -110 12 94
110 - 120 6 10
0
Total 100
Solution: It may be noted that the question involved first two columns and in order to
calculate quartiles and percentiles, cumulative frequencies have been shown in
column three of the above table.
l2 l1
Q1 = l1 (m c) , where m = (n + 1)/4th item, which is = 25.25th item
f1
This falls in 70 - 80 class interval.
80 70
= 70 + (25.25 = 70.69
24)
18
l2 l1
Q3 = l1 + (m c) , where m = 75.75
f1
This falls in 90 - 100 class interval.
100
=
90 90 + (75.75 - 66) = 96.09
16
-
106
Caritas Business College BUSINESS STATISTICS – OAD 221
l2 l1
PI0 = l1 + (m c) , where m = 10.1
f1
This falls in 60 - 70 class interval.
70
= 60 + 60 (10.01 -10) = 60.07
14
l2 l1
P90 = l1 + (m c) , where m = 90.9
f1
This falls in 100 - 110 class interval.
110
= 100 +
100 (90.9 - 82) = 107.41
12
Q
K
P90
P10
1/ 2(Q3 Q1 )
= P90 P10
½ (96.09 - 70.69)
= 107.41 - 60.07
= 0.268
It will be seen that the above distribution is very close to normal distribution as the
value of K is 0.268, which is extremely close to 0.263.
3.15 SUMMARY
The average value cannot adequately describe a set of observations, unless all the
observations are the same. It is necessary to describe the variability or dispersion of
the observations. In two or more distributions the central value may be the same but
still there can be wide disparities in the formation of distribution. Therefore, we have
to use the measures of dispersion.
Further, two distributions may have the same mean and standard deviation but may
differ widely in their overall appearance in terms of symmetry and skewness. To
-
107
Caritas Business College BUSINESS STATISTICS – OAD 221
distinguish between different types of distributions, we may use the measures of
skewness.
3.16 SELF TEST QUESTIONS
1. What do you mean by dispersion? What are the different measures of dispersion?
2. “Variability is not an important factor because even though the outcome is more
certain, you still have an equal chance of falling either above or below the median.
Therefore, on an average, the outcome will be the same.” Do you agree with this
statement? Give reasons for your answer.
3. Why is the standard deviation the most widely used measure of dispersion? Explain.
4. Define skewness and Dispersion.
5. Define Kurtosis and Moments.
6. What are the different measures of skewness? Which one is repeatedly used?
7. Measures of dispersion and skewness are complimentary to one another in
understanding a frequency distribution." Elucidate the statement.
8. Calculate Karl Pearson's coefficient of skewness from the following data:
Weekly Sales (Rs lakh) Number of
Companies
10-12 1
2
12 – 14 1
8
14 – 16 3
5
16 - 18 4
2
18 – 20 5
0
22-24 3
0
24-26 8
9. For a distribution, the first four moments about zero are 1,7,38 and 155 respectively.
(i) Compute the moment coefficients of skewness and kurtosis. (ii) Is the distribution
mesokurtic? Give reason.
10. The first four moments of a distribution about the value 4 are 1,4, 10 and 45. Obtain
various characteristics of the distribution on the basis of the information given.
Comment upon the nature of the distribution.
-
108
Caritas Business College BUSINESS STATISTICS – OAD 221
11. Define kurtosis. If β1=1 and β2 =4 and variance = 9, find the values of β3 and β4 and
comment upon the nature of the distribution.
12. Calculate the first four moments about the mean from the following data. Also
calculate the values of β1 and β2
Marks 0-10 10 – 20 20-3030 – 40 40 – 50 50 - 60 60 - 70
No. of students5 12 18 40 15 7 3
3.17 SUGGESTED READINGS
1. Levin, Richard I. and David S. Rubin: Statistics for Management, Prentice
Hall, New Delhi.
2. Watsman terry J. and Keith Parramor: Quantitative Methods in
Finance International, Thompson Business Press, London.
3. Hooda, R. P.: Statistics for Business and Economics, Macmillan,
New Delhi.
4. Hein, L. W. Quantitative Approach to Managerial Decisions, Prentice
Hall, NJ.
-
109
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 5: CORRELATION ANALYSIS
Topic : Correlation
Objectives : The overall objective of this lesson is to give you an understanding
of bivariate linear correlation, there by enabling you to understand the
importance as well as the limitations of correlation analysis.
Structure
4.1 Introduction
4.2 What is Correlation?
4.3 Correlation Analysis
4.3.1 Scatter Diagram
4.3.2 Correlation Graph
4.3.3 Pearson’s Coefficient of Correlation
4.3.4 Spearman’s Rank Correlation
4.3.5 Concurrent Deviation Method
4.4 Limitations of Correlation Analysis
4.5 Self-Assessment Questions
4.6 Suggested Readings
...if we have information on more than one variables, we might be
interested in seeing if there is any connection - any association -
between them.
4.1 INTRODUCTION
Statistical methods of measures of central tendency, dispersion, skewness and kurtosis are
helpful for the purpose of comparison and analysis of distributions involving only one
variable i.e. univariate distributions. However, describing the relationship between two or
more variables, is another important part of statistics.
In many business research situations, the key to decision making lies in understanding the
relationships between two or more variables. For example, in an effort to predict the
behavior of the bond market, a broker might find it useful to know whether the interest rate of
bonds is related to the prime interest rate. While studying the effect of advertising on sales,
an account executive may find it useful to know whether there is a strong relationship
between advertising dollars and sales dollars for a company.
The statistical methods of Correlation (discussed in the present lesson) and Regression (to
be discussed in the next lesson) are helpful in knowing the relationship between two or more
Caritas Business College BUSINESS STATISTICS – OAD 221
variables which may be related in same way, like interest rate of bonds and prime
interest rate; advertising expenditure and sales; income and consumption; crop-yield and
fertilizer used; height and weights and so on.
In all these cases involving two or more variables, we may be interested in seeing:
if there is any association between the variables;
if there is an association, is it strong enough to be useful;
if so, what form the relationship between the two variables takes;
how we can make use of that relationship for predictive purposes, that is,
forecasting; and
how good such predictions will be.
Since these issues are inter related, correlation and regression analysis, as two sides of
a single process, consists of methods of examining the relationship between two or more
variables. If two (or more) variables are correlated, we can use information about one (or
more) variable(s) to predict the value of the other variable(s), and can measure the
error of estimations - a job of regression analysis.
4.2 WHAT IS CORRELATION?
Correlation is a measure of association between two or more variables. When two or more
variables very in sympathy so that movement in one tends to be accompanied by
corresponding movements in the other variable(s), they are said to be correlated.
“The correlation between variables is a measure of the nature and degree of
association between the variables”.
As a measure of the degree of relatedness of two variables, correlation is widely used in
exploratory research when the objective is to locate variables that might be related in some
way to the variable of interest.
4.2.1 TYPES OF CORRELATION
Correlation can be classified in several ways. The important ways of classifying correlation
are:
(i) Positive and negative,
(ii) Linear and non-linear (curvilinear) and
Caritas Business College BUSINESS STATISTICS – OAD 221
(iii) Simple, partial and multiple.
Positive and Negative Correlation
If both the variables move in the same direction, we say that there is a positive correlation,
i.e., if one variable increases, the other variable also increases on an average or if one
variable decreases, the other variable also decreases on an average.
On the other hand, if the variables are varying in opposite direction, we say that it is a case
of negative correlation; e.g., movements of demand and supply.
Linear and Non-linear (Curvilinear) Correlation
If the change in one variable is accompanied by change in another variable in a constant
ratio, it is a case of linear correlation. Observe the following data:
X : 10 20 30 40 50
Y : 25 50 75 100 125
The ratio of change in the above example is the same. It is, thus, a case of linear
correlation. If we plot these variables on graph paper, all the points will fall on the same
straight line.
On the other hand, if the amount of change in one variable does not follow a constant ratio
with the change in another variable, it is a case of non-linear or curvilinear correlation. If a
couple of figures in either series X or series Y are changed, it would give a non-linear
correlation.
Simple, Partial and Multiple Correlation
The distinction amongst these three types of correlation depends upon the number of
variables involved in a study. If only two variables are involved in a study, then the
correlation is said to be simple correlation. When three or more variables are involved in a
study, then it is a problem of either partial or multiple correlation. In multiple correlation, three
or more variables are studied simultaneously. But in partial correlation we consider only two
variables influencing each other while the effect of other variable(s) is held constant.
Suppose we have a problem comprising three variables X, Y and Z. X is the number of
hours studied, Y is I.Q. and Z is the number of marks obtained in the examination. In a
Caritas Business College BUSINESS STATISTICS – OAD 221
multiple correlation, we will study the relationship between the marks obtained (Z) and the
two variables, number of hours studied (X) and I.Q. (Y). In contrast, when we study the
relationship between X and Z, keeping an average I.Q. (Y) as constant, it is said to be a
study involving partial correlation.
In this lesson, we will study linear correlation between two variables.
4.2.2 CORRELATION DOES NOT NECESSARILY MEAN CAUSATION
The correlation analysis, in discovering the nature and degree of relationship between
variables, does not necessarily imply any cause and effect relationship between the
variables. Two variables may be related to each other but this does not mean that one
variable causes the other. For example, we may find that logical reasoning and creativity are
correlated, but that does not mean if we could increase peoples’ logical reasoning ability, we
would produce greater creativity. We need to conduct an actual experiment to unequivocally
demonstrate a causal relationship. But if it is true that influencing someones’ logical
reasoning ability does influence their creativity, then the two variables must be correlated
with each other. In other words, causation always implies correlation, however
converse is not true.
Let us see some situations-
1. The correlation may be due to chance particularly when the data pertain to a
small sample. A small sample bivariate series may show the relationship but such
a relationship may not exist in the universe.
2. It is possible that both the variables are influenced by one or more other variables.
For example, expenditure on food and entertainment for a given number of
households show a positive relationship because both have increased over time.
But, this is due to rise in family incomes over the same period. In other words, the
two variables have been influenced by another variable - increase in family
incomes.
3. There may be another situation where both the variables may be influencing each
other so that we cannot say which is the cause and which is the effect. For
example, take the case of price and demand. The rise in price of a commodity
Caritas Business College BUSINESS STATISTICS – OAD 221
may lead to a decline in the demand for it. Here, price is the cause and the
demand is the effect. In yet another situation, an increase in demand may lead to
a rise in price. Here, the demand is the cause while price is the effect, which is
just the reverse of the earlier situation. In such situations, it is difficult to identify
which variable is causing the effect on which variable, as both are
influencing each other.
The foregoing discussion clearly shows that correlation does not indicate any causation or
functional relationship. Correlation coefficient is merely a mathematical relationship and
this has nothing to do with cause and effect relation. It only reveals co-variation between
two variables. Even when there is no cause-and-effect relationship in bivariate series and
one interprets the relationship as causal, such a correlation is called spurious or non-sense
correlation. Obviously, this will be misleading. As such, one has to be very careful in
correlation exercises and look into other relevant factors before concluding a cause-and-
effect relationship.
4.3 CORRELATION ANALYSIS
Correlation Analysis is a statistical technique used to indicate the nature and degree of
relationship existing between one variable and the other(s). It is also used along with
regression analysis to measure how well the regression line explains the variations of the
dependent variable with the independent variable.
The commonly used methods for studying linear relationship between two variables involve
both graphic and algebraic methods. Some of the widely used methods include:
1. Scatter Diagram
2. Correlation Graph
3. Pearson’s Coefficient of Correlation
4. Spearman’s Rank Correlation
5. Concurrent Deviation Method
4.3.1 SCATTER DIAGRAM
This method is also known as Dotogram or Dot diagram. Scatter diagram is one of the
simplest methods of diagrammatic representation of a bivariate distribution. Under this
1
method, both the variables are plotted on the graph paper by putting dots. The diagram so
Caritas Business College BUSINESS STATISTICS – OAD 221
obtained is called "Scatter Diagram". By studying diagram, we can have rough idea about
the nature and degree of relationship between two variables. The term scatter refers to the
spreading of dots on the graph. We should keep the following points in mind while
interpreting correlation:
if the plotted points are very close to each other, it indicates high degree of
correlation. If the plotted points are away from each other, it indicates low degree of
correlation.
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 4-1 Scatter Diagrams
if the points on the diagram reveal any trend (either upward or downward), the
variables are said to be correlated and if no trend is revealed, the variables are
uncorrelated.
if there is an upward trend rising from lower left hand corner and going upward to the
upper right hand corner, the correlation is positive since this reveals that the values
of the two variables move in the same direction. If, on the other hand, the points
depict a downward trend from the upper left hand corner to the lower right hand
corner, the correlation is negative since in this case the values of the two variables
move in the opposite directions.
in particular, if all the points lie on a straight line starting from the left bottom and
going up towards the right top, the correlation is perfect and positive, and if all the
points like on a straight line starting from left top and coming down to right bottom,
the correlation is perfect and negative.
The various diagrams of the scattered data in Figure 4-1 depict different forms of correlation.
Example 4-1
102
Caritas Business College BUSINESS STATISTICS – OAD 221
Given the following data on sales (in thousand units) and expenses (in thousand rupees) of a
firm for 10 month:
Month : J F M A M J J A S O
Sales: 50 50 55 60 62 65 68 60 60 50
Expenses: 11 13 14 16 16 15 15 14 13 13
a) Make a Scatter Diagram
b) Do you think that there is a correlation between sales and expenses of
the firm? Is it positive or negative? Is it high or low?
Solution:(a) The Scatter Diagram of the given data is shown in Figure 4-2
20
Expens
15
10
0
0 20 40 60 80
Sales
Figure 4.2 Scatter Diagram
(b) Figure 4-2 shows that the plotted points are close to each other and reveal an upward
trend. So there is a high degree of positive correlation between sales and expenses of the
firm.
4.3.2 CORRELATION GRAPH
This method, also known as Correlogram is very simple. The data pertaining to two series
are plotted on a graph sheet. We can find out the correlation by examining the direction and
closeness of two curves. If both the curves drawn on the graph are moving in the same
direction, it is a case of positive correlation. On the other hand, if both the curves are moving
in opposite direction, correlation is said to be negative. If the graph does not show
any definite pattern on account of erratic fluctuations in the curves, then it shows an absence
of correlation.
103
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 4-2
Find out graphically, if there is any correlation between price yield per plot (qtls); denoted by
Y and quantity of fertilizer used (kg); denote by X.
Plot No.: 1 2 3 4 5 6 7 8 9 10
Y: 3.5 4.3 5.2 5.8 6.4 7.3 7.2 7.5 7.8 8.3
X: 6 8 9 12 10 15 17 20 18 24
Solution: The Correlogram of the given data is shown in Figure 4-3
30
25
20
15
10
X and
5
0
12345678910
Plot Number
Figure 4-3 Correlation Graph
Figure 4-3 shows that the two curves move in the same direction and, moreover, they are
very close to each other, suggesting a close relationship between price yield per plot (qtls)
and quantity of fertilizer used (kg)
Remark: Both the Graphic methods - scatter diagram and correlation graph provide a
‘feel for’ of the data – by providing visual representation of the association between the
variables. These are readily comprehensible and enable us to form a fairly good,
though rough idea of the nature and degree of the relationship between the two variables.
However, these methods are unable to quantify the relationship between them. To quantify
the extent of correlation, we make use of algebraic methods - which calculate correlation
coefficient.
4.3.3 PEARSON’S COEFFICIENT OF CORRELATION
A mathematical method for measuring the intensity or the magnitude of linear relationship
104
Caritas Business College BUSINESS STATISTICS – OAD 221
between two variables was suggested by Karl Pearson (1867-1936), a great British
Biometrician and Statistician and, it is by far the most widely used method in practice.
Karl Pearson’s measure, known as Pearsonian correlation coefficient between two variables
X and Y, usually denoted by r(X,Y) or rxy or simply r is a numerical measure of linear
relationship between them and is defined as the ratio of the covariance between X and Y,
to the product of the standard deviations of X and Y.
Symbolically
Cov( X ,Y
)r …………(4.1)
Sx .S y
x
when, ( X 1 ,Y1 );( X 2 ,Y2 );..........( X n
,Yn ) are N pairs of observations of the variables X and
Y in a bivariate distribution,
( X X )(Y
Cov(
Y ) X ,Y ) …………(4.2a)
N
Sx ( X X )2 …………(4.2b)
N
and Sy …………(4.2c)
(Y Y )2
N
Thus by substituting Eqs. (4.2) in Eq. (4.1), we can write the Pearsonian
correlation coefficient as
1
rxy ( X X )(Y Y )
N
1 ( X X )21 (Y Y )2
N N
rxy …………(4.3)
( X X )(Y Y )
( X X )2 (Y Y )2
105
Caritas Business College BUSINESS STATISTICS – OAD 221
If we denote, dNx X an
X d dy YY
The rxy dxd …………(4.3a)
n y
dd x
2
y
2
We can further simply the calculations of Eqs. (4.2)
We have
1
Cov( X ,Y ) ( X X )(Y Y )
N
1
XY
XY N
1 X Y
XY
N N N
1 …………(4.4)
N XY X Y
and 1
S2 ( X X )
2
x
N
1
X 2 ( X ) 2
N
1 2
2
X
N
N
1 2
N X 2 X …………(4.5a)
N
Similarly, we have
1 2 …………(4.5b)
S2 N Y 2 Y
y 2
N
So Pearsonian correlation coefficient may be found as
1
N XY X Y
N 2
rxy
1 N X 2
X 1 N Y 2
Y
2
2
106
Caritas Business College BUSINESS STATISTICS – OAD 221
or rxy N XY X Y …………(4.6)
N X 2 X N Y 2 Y
2 2
107
Caritas Business College BUSINESS STATISTICS – OAD 221
Remark: Eq. (4.3) or Eq. (4.3a) is quite convenient to apply if the means X and
Y come out to be integers. If X or/and Y is (are) fractional then the Eq. (4.3) or Eq. (4.3a) is
2 2
quite cumbersome to apply, since the computations of ( X X ) , (Y Y ) and
( X X )(Y Y
are quite time consuming and tedious. In such a case Eq. (4.6) may be
)
used provided the values of X or/ and Y are small. But if X and Y assume large values, the
calculation of X 2 , Y 2 and XY is again quite time consuming.
Thus if (i) X and Y are fractional and (ii) X and Y assume large values, the Eq. (4.3) and Eq.
(4.6) are not generally used for numerical problems. In such cases, the step deviation
method where we take the deviations of the variables X and Y from any arbitrary points is
used. We will discuss this method in the properties of correlation coefficient.
4.3.3.1 Properties of Pearsonian Correlation Coefficient
The following are important properties of Pearsonian correlation coefficient:
1. Pearsonian correlation coefficient cannot exceed 1 numerically. In other words it lies
between –1 and +1. Symbolically,
-1 ≤ r ≤1
Remarks: (i) This property provides us a check on our calculations. If in any problem,
the obtained value of r lies outside the limits + 1, this implies that there is some mistake in
our calculations.
(ii) The sign of r indicate the nature of the correlation. Positive value of r indicates
positive correlation, whereas negative value indicates negative correlation. r = 0 indicate
absence of correlation.
(iii) The following table sums up the degrees of correlation corresponding to
various values of r:
108
Caritas Business College BUSINESS STATISTICS – OAD 221
Value of r Degree of correlation
±1 perfect correlation
±0.90 or more very high degree of correlation
sufficiently high degree of
±0.75 to ±0.90
correlation
±0.60 to ±0.75 moderate degree of correlation
only the possibility of a
±0.30 to ±0.60
correlation
less than ±0.30 possibly no correlation
0 absence of correlation
2. Pearsonian Correlation coefficient is independent of the change of origin and scale.
Mathematically, if given variables X and Y are transformed to new variables U and V
by change of origin and scale, i. e.
XA
U= an YB
h V
d k
Where A, B, h and k are constants and h > 0, k > 0; then the correlation
coefficient between X and Y is same as the correlation coefficient between U and V
i.e.,
r(X,Y) = r(U, V) => rxy = ruv
Remark: This is one of the very important properties of the correlation coefficient and
is extremely helpful in numerical computation of r. We had already stated that Eq. (4.3) and
Eq.(4.6) become quite tedious to use in numerical problems if X and/or Y are in fractions or if
X and Y are large. In such cases we can conveniently change the origin and scale (if
possible) in X or/and Y to get new variables U and V and compute the correlation between U
and V by the Eq. (4.7)
rxy r
uv N UV U V …………(4.7)
N U 2 U N V 2 V
2 2
109
Caritas Business College BUSINESS STATISTICS – OAD 221
3. Two independent variables are uncorrelated but the converse is not true
110
Caritas Business College BUSINESS STATISTICS – OAD 221
If X and Y are independent variables then
rxy = 0
However, the converse of the theorem is not true i.e., uncorrelated variables need
not necessarily be independent. As an illustration consider the following bivariate
distribution.
X : 1 2 3 -3 -2 -1
Y : 1 4 9 9 4 1
For this distribution, value of r will be 0.
Hence in the above example the variable X and Y are uncorrelated. But if we
examine the data carefully we find that X and Y are not independent but are
connected by the relation Y = X2. The above example illustrates that uncorrelated
variables need not be independent.
Remarks: One should not be confused with the words uncorrelation and
independence. rxy = 0 i.e., uncorrelation between the variables X and Y simply implies the
absence of any linear (straight line) relationship between them. They may, however, be
related in some other form other than straight line e.g., quadratic (as we have seen in the
above example), logarithmic or trigonometric form.
4. Pearsonian coefficient of correlation is the geometric mean of the two regression
coefficients, i.e.
rxy = bxy .byx
The signs of both the regression coefficients are the same, and so the value of r will
also have the same sign.
This property will be dealt with in detail in the next lesson on Regression Analysis.
111
Caritas Business College BUSINESS STATISTICS – OAD 221
5. The square of Pearsonian correlation coefficient is known as the coefficient of
determination.
Coefficient of determination, which measures the percentage variation in the
dependent variable that is accounted for by the independent variable, is a much
better and useful measure for interpreting the value of r. This property will also
be dealt with in detail in the next lesson.
4.3.3.2 Probable Error of Correlation Coefficient
The correlation coefficient establishes the relationship of the two variables. After ascertaining
this level of relationship, we may be interested to find the extent upto which this coefficient is
dependable. Probable error of the correlation coefficient is such a measure of testing the
reliability of the observed value of the correlation coefficient, when we consider it as
satisfying the conditions of the random sampling.
If r is the observed value of the correlation coefficient in a sample of N pairs of observations
for the two variables under consideration, then the Probable Error, denoted by PE (r) is
expressed as
PE(r) 0.6745
SE(r)
or PE(r) 0.6745 1 r 2
N
There are two main functions of probable error:
1. Determination of limits: The limits of population correlation coefficient are r ±
PE(r), implying that if we take another random sample of the size N from the same
population, then the observed value of the correlation coefficient in the second
sample can be expected to lie within the limits given above, with 0.5 probability.
When sample size N is small, the concept or value of PE may lead to wrong
112
Caritas Business College BUSINESS STATISTICS – OAD 221
conclusions. Hence to use the concept of PE effectively, sample size N it should
be fairly large.
2. Interpretation of 'r': The interpretation of 'r' based on PE is as under:
If r < PE(r), there is no evidence of correlation, i.e. a case of insignificant
correlation.
If r > 6 PE(r), correlation is significant. If r < 6 PE(r), it is insignificant.
If the probable error is small, correlation exist where r > 0.5
Example 4-3
Find the Pearsonian correlation coefficient between sales (in thousand units) and expenses (in
thousand rupees) of the following 10 firms:
Firm: 1 2 3 4 5 6 7 8 9 10
Sales: 50 50 55 60 65 65 65 60 60 50
Expenses: 11 13 14 16 16 15 15 14 13 13
Solution: Let sales of a firm be denoted by X and expenses be denoted by Y
Calculations for Coefficient of Correlation
{Using Eq. (4.3) or (4.3a)}
Firm X Y 2 2 dx.dy
dx dy
dx dy
XX YY
1 50 11 -8 -3 64 9 24
2 50 13 -8 -1 64 1 8
3 55 14 -3 0 9 0 0
4 60 16 2 2 4 4 4
5 65 16 7 2 49 4 14
6 65 15 7 1 49 1 7
7 65 15 7 1 49 1 7
8 60 14 2 0 4 0 0
9 60 13 2 -1 4 1 -2
10 50 13 -8 -1 64 1 8
X Y 2 2 dxdy
d x d y
113
Caritas Business College BUSINESS STATISTICS – OAD 221
= = =360 =22 =70
580 140
X 580 Y 140
X= = = 58 and Y= = = 14
N 10 N 10
Applying the Eq. (4.3a), we have, Pearsonian coefficient of correlation
rxy dx d y
rxy dd x
2
y
2
70
rxy 360x22
70
7920
rxy 0.78
The value of rxy 0.78 , indicate a high degree of positive correlation between sales and expenses.
Example 4-4
The data on price and quantity purchased relating to a commodity for 5 months is
given below:
Month : January February March April May
Prices(Rs): 10 10 11 12 12
Quantity(Kg): 5 6 4 3 3
Find the Pearsonian correlation coefficient between prices and quantity and comment on
its sign and magnitude.
Solution: Let price of the commodity be denoted by X and quantity be denoted by Y
Calculations for Coefficient of Correlation
{Using Eq. (4.6)}
Month X Y 2 2 XY
X Y
1 10 5 100 25 50
2 10 6 100 36 60
3 11 4 121 16 44
4 12 3 144 9 36
5 12 3 144 9 36
114
Caritas Business College BUSINESS STATISTICS – OAD 221
2 2
X =55 Y =21 X 609 Y 95 XY 226
Applying the Eq. (4.6), we have, Pearsonian coefficient of correlation
rxy N XY X Y
N X 2 X N Y 2 Y
2 2
rxy
5x226 55x21
(5x609 55x55)(5x95 21x21)
rxy
1130 1155
20x34
rxy
25
680
rxy 0.98
The negative sign of r indicate negative correlation and its large magnitude indicate a very
high degree of correlation. So there is a high degree of negative correlation between prices
and quantity demanded.
Example 4-5
Find the Pearsonian correlation coefficient from the following series of marks obtained by 10
students in a class test in mathematics (X) and in Statistics (Y):
X: 45 70 65 30 90 40 50 75 85 60
Y: 35 90 70 40 95 40 60 80 80 50
Also calculate the Probable Error.
Solution:
Calculations for Coefficient of Correlation
{Using Eq. (4.7)}
X Y U V 2 2 UV
U V
45 35 -3 -6 9 36 18
70 90 2 5 4 25 10
65 70 1 1 1 1 1
115
Caritas Business College BUSINESS STATISTICS – OAD 221
30 40 -6 -5 36 25 30
90 95 6 6 36 36 36
40 40 -4 -5 16 25 20
50 60 -2 -1 4 1 2
75 80 3 3 9 9 9
85 80 5 3 25 9 15
60 50 0 -3 0 9 0
U V U
2
V
2
176 UV
141
2 2 140
We have, defined variables U and V as
X
U
60 and Y 65
V 5
5
Applying the Eq. (4.7)
N UV U V
rxy ruv
N U 2 U 2N V 2 V 2
10x141 2x(2)
=
10x140 2x2 10x176 (2)x(2)
1410 4
=
1400 4 1760 4
1414
=
2451376
= 0.9
So there is a high degree of positive correlation between marks obtained in Mathematics
and in Statistics.
Probable Error, denoted by PE (r) is given as
PE(r) 0.6745 1 r 2
116
Caritas Business College BUSINESS STATISTICS – OAD 221
1 0.9 2
PE(r) 0.6745 10
PE(r) 0.0405
So the value of r is highly significant.
4.3.4 SPEARMAN’S RANK CORRELATION
Sometimes we come across statistical series in which the variables under consideration
are not capable of quantitative measurement but can be arranged in serial order. This
happens when we are dealing with qualitative characteristics (attributes) such as honesty,
beauty, character, morality, etc., which cannot be measured quantitatively but can be
arranged serially. In such situations Karl Pearson’s coefficient of correlation cannot be used
as such. Charles Edward Spearman, a British Psychologist, developed a formula in 1904,
which consists in obtaining the correlation coefficient between the ranks of N individuals in
the two attributes under study.
Suppose we want to find if two characteristics A, say, intelligence and B, say, beauty are
related or not. Both the characteristics are incapable of quantitative measurements but we
can arrange a group of N individuals in order of merit (ranks) w.r.t. proficiency in the two
characteristics. Let the random variables X and Y denote the ranks of the individuals in the
characteristics A and B respectively. If we assume that there is no tie, i.e., if no two
individuals get the same rank in a characteristic then, obviously, X and Y assume numerical
values ranging from 1 to N.
The Pearsonian correlation coefficient between the ranks X and Y is called the rank
correlation coefficient between the characteristics A and B for the group of individuals.
Spearman’s rank correlation coefficient, usually denoted by ρ(Rho) is given by the equation
6 d 2
ρ =1 …………(4.8)
N (N 2
1)
117
Caritas Business College BUSINESS STATISTICS – OAD 221
Where d is the difference between the pair of ranks of the same individual in the two
characteristics and N is the number of pairs.
Example 4-6
Ten entries are submitted for a competition. Three judges study each entry and list the ten
in rank order. Their rankings are as follows:
Entry: A B C D E F G H I J
Judge J1: 9 3 7 5 1 6 2 4 10 8
Judge J2: 9 1 10 4 3 8 5 2 7 6
Judge J3: 6 3 8 7 2 4 1 5 9 10
Calculate the appropriate rank correlation to help you answer the following questions:
(i) Which pair of judges agrees the most?
(ii) Which pair of judges disagrees the most?
Solution:
Calculations for Coefficient of Rank Correlation
{Using Eq.(4.8)}
Entry Rank by
Difference in Ranks
Judges
J1 J2 J3 d(J1&J2) 2 d(J1&J3) 2 d(J2&J3) 2
d d d
A 9 9 6 0 0 +3 9 +3 9
B 3 1 3 +2 4 0 0 -2 4
C 7 10 8 -3 9 -1 1 +2 4
D 5 4 7 +1 1 -2 4 -3 9
E 1 3 2 -2 4 -1 1 +1 1
F 6 8 4 -2 4 +2 4 +4 16
G 2 5 1 -3 9 +1 1 +4 16
H 4 2 5 +2 4 -1 1 -3 9
I 10 7 9 +3 9 +1 1 -2 4
J 8 6 10 +2 4 -2 4 -4 16
d2 =48 d2 =26 d2 =88
6 d 2
(J1 & J2) = 1
N (N 2 1)
118
Caritas Business College BUSINESS STATISTICS – OAD 221
=1 6 x 48
10(102
1)
288
=1 990
=1 – 0.29
= +0.71
6 d 2
(J1 & J3) =1
N (N 2
1)
=1
6 x 26
10(102
1)
156
=1 990
=1 – 0.1575
= +0.8425
6 d 2
(J2 & J3) =1
N (N 2
1)
=1
6 x 88
10(102
1)
528
=1 990
=1 – 0.53
= +0.47
So (i) Judges J1 and J3 agree the most
(ii) Judges J2 and J3 disagree the most
Spearman’s rank correlation Eq.(4.8) can also be used even if we are dealing with variables,
which are measured quantitatively, i.e. when the actual data but not the ranks relating to two
variables are given. In such a case we shall have to convert the data into ranks. The
119
Caritas Business College BUSINESS STATISTICS – OAD 221
highest (or the smallest) observation is given the rank 1. The next highest (or the next
lowest) observation is given rank 2 and so on. It is immaterial in which way (descending or
ascending) the ranks are assigned. However, the same approach should be followed for all
the variables under consideration.
120
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 4-7
Calculate the rank coefficient of correlation from the following data:
X: 75 88 95 70 60 80 81 50
Y: 120 134 150 115 110 140 142 100
Solution:
Calculations for Coefficient of Rank Correlation
{Using Eq.(4.8)}
X Ranks RX Y Ranks RY d = RX -RY d2
75 5 120 5 0 0
88 2 134 4 -2 4
95 1 150 1 0 0
70 6 115 6 0 0
60 7 110 7 0 0
80 4 140 3 +1 1
81 3 142 2 +1 1
50 8 100 8 0 0
d2 = 6
6 d 2
=1
N (N 2 1)
6x6
=1
8(82 1)
=1 36
504
= 1 – 0.07
= + 0.93
Hence, there is a high degree of positive correlation between X and Y
Repeated Ranks
In case of attributes if there is a tie i.e., if any two or more individuals are placed together in
any classification w.r.t. an attribute or if in case of variable data there is more than one item
with the same value in either or both the series then Spearman’s Eq.(4.8) for calculating the
rank correlation coefficient breaks down, since in this case the variables X [the ranks
of
121
Caritas Business College BUSINESS STATISTICS – OAD 221
individuals in characteristic A (1st series)] and Y [the ranks of individuals in characteristic B
(2nd series)] do not take the values from 1 to N.
In this case common ranks are assigned to the repeated items. These common ranks are
the arithmetic mean of the ranks, which these items would have got if they were different
from each other and the next item will get the rank next to the rank used in computing the
common rank. For example, suppose an item is repeated at rank 4. Then the common rank
to be assigned to each item is (4+5)/2, i.e., 4.5 which is the average of 4 and 5, the ranks
which these observations would have assumed if they were different. The next item will be
assigned the rank 6. If an item is repeated thrice at rank 7, then the common rank to be
assigned to each value will be (7+8+9)/3, i.e., 8 which is the arithmetic mean of 7,8 and 9
viz., the ranks these observations would have got if they were different from each other. The
next rank to be assigned will be 10.
If only a small proportion of the ranks are tied, this technique may be applied together with
Eq.(4.8). If a large proportion of ranks are tied, it is advisable to apply an adjustment or a
correction factor to Eq.(4.8)as explained below:
“In the Eq.(4.8) add the factor
m(m 2
1) …………(4.8a)
12
2
to d ; where m is the number of times an item is repeated. This correction factor is to be
added for each repeated value in both the series”.
Example 4-8
For a certain joint stock company, the prices of preference shares (X) and debentures (Y)
are given below:
X: 73.2 85.8 78.9 75.8 77.2 81.2 83.8
Y: 97.8 99.2 98.8 98.3 98.3 96.7 97.1
122
Caritas Business College BUSINESS STATISTICS – OAD 221
Use the method of rank correlation to determine the relationship between preference
prices and debentures prices.
Solution:
Calculations for Coefficient of Rank Correlation
{Using Eq. (4.8) and (4.8a)}
X Y Rank of X (XR) Rank of Y (YR) d = XR – YR 2
d
73.2 97.8 7 5 2 4
85.8 99.2 1 1 0 0
78.9 98.8 4 2 2 4
75.8 98.3 6 3.5 2.5 6.25
77.2 98.3 5 3.5 1.5 2.25
81.2 96.7 3 7 -4 16
83.8 97.1 2 6 -4 16
2
d 0 d
48.50
In this case, due to repeated values of Y, we have to apply ranking as average of 2 ranks,
which could have been allotted, if they were different values. Thus ranks 3 and 4 have been
allotted as 3.5 to both the values of Y = 98.3. Now we also have to apply correction
factor
m(m 2
2
1) to d , where m in the number of times the value is repeated, here m = 2.
12
2 mm 2 1
6 d
2
=
N (N 2 1)
24 1
6 48.5
12
=
7(72 1)
6 x 49
= 1-
7 x 48
= 0.125
Hence, there is a very low degree of positive correlation, probably no correlation,
123
Caritas Business College BUSINESS STATISTICS – OAD 221
between preference share prices and debenture prices.
124
Caritas Business College BUSINESS STATISTICS – OAD 221
Remarks on Spearman’s Rank Correlation Coefficient
1. We always have d 0 , which provides a check for numerical calculations.
2. Since Spearman’s rank correlation coefficient, , is nothing but Karl Pearson’s
correlation coefficient, r, between the ranks, it can be interpreted in the same way
as the Karl Pearson’s correlation coefficient.
3. Karl Pearson’s correlation coefficient assumes that the parent population from
which sample observations are drawn is normal. If this assumption is violated
then we need a measure, which is distribution free (or non-parametric).
Spearman’s is such a distribution free measure, since no strict assumption are
made about the from of the population from which sample observations are
drawn.
4. Spearman’s formula is easy to understand and apply as compared to Karl
Pearson’s formula. The values obtained by the two formulae, viz Pearsonian r
and Spearman’s are generally different. The difference arises due to the fact
that when ranking is used instead of full set of observations, there is always some
loss of information. Unless many ties exist, the coefficient of rank correlation
should be only slightly lower than the Pearsonian coefficient.
5. Spearman’s formula is the only formula to be used for finding correlation
coefficient if we are dealing with qualitative characteristics, which cannot be
measured quantitatively but can be arranged serially. It can also be used where
actual data are given. In case of extreme observations, Spearman’s formula is
preferred to Pearson’s formula.
6. Spearman’s formula has its limitations also. It is not practicable in the case of
bivariate frequency distribution. For N >30, this formula should not be used
unless the ranks are given.
125
Caritas Business College BUSINESS STATISTICS – OAD 221
4.3.5 CONCURRENT DEVIATION METHOD
This is a casual method of determining the correlation between two series when we are not
very serious about its precision. This is based on the signs of the deviations (i.e. the
direction of the change) of the values of the variable from its preceding value and does not
take into account the exact magnitude of the values of the variables. Thus we put a plus (+)
sign, minus (-) sign or equality (=) sign for the deviation if the value of the variable is greater
than, less than or equal to the preceding value respectively. The deviations in the values of
two variables are said to be concurrent if they have the same sign (either both deviations are
positive or both are negative or both are equal). The formula used for computing correlation
coefficient rc by this method is given by
rc 2c N
…………(4.9)
N
Where c is the number of pairs of concurrent deviations and N is the number of pairs of
deviations. If (2c-N) is positive, we take positive sign in and outside the square root in Eq.
(4.9) and if (2c-N) is negative, we take negative sign in and outside the square root in Eq.
(4.9).
Remarks: (i) It should be clearly noted that here N is not the number of pairs of
observations but it is the number of pairs of deviations and as such it is one less than the
number of pairs of observations.
(ii) Coefficient of concurrent deviations is primarily based on the following principle:
“If the short time fluctuations of the time series are positively correlated or in other words,
if their deviations are concurrent, their curves would move in the same direction and would
indicate positive correlation between them”
Example 4-9
126
Caritas Business College BUSINESS STATISTICS – OAD 221
Calculate coefficient of correlation by the concurrent deviation method
Supply: 112 125 126 118 118 121 125 125 131 135
Price: 106 102 102 104 98 96 97 97 95 90
Solution:
Calculations for Coefficient of Concurrent Deviations
{Using Eq. (4.9)}
Supply Sign of deviation from Price Sign of deviation Concurrent
(X) preceding value (X) (Y) preceding value (Y) deviations
112 106
125 + 102 -
126 + 102 =
118 - 104 +
118 = 98 -
121 + 96 -
125 + 97 + +(c)
125 = 97 = = (c)
131 + 95 -
135 + 90 -
We have
Number of pairs of deviations, N =10 – 1 = 9
c = Number of concurrent deviations
= Number of deviations having like signs
=2
Coefficient of correlation by the method of concurrent deviations is given by:
rc
2c N
N
2x2 9
rc
9
rc 0.5556
Since 2c – N = -5 (negative), we take negative sign inside and outside the square root
127
Caritas Business College BUSINESS STATISTICS – OAD 221
0.5556
rc
rc
0.5556
rc 0.7
Hence there is a fairly good degree of negative correlation between supply and price.
4.4 LIMITATIONS OF CORRELATION ANALYSIS
As mentioned earlier, correlation analysis is a statistical tool, which should be properly used
so that correct results can be obtained. Sometimes, it is indiscriminately used by
management, resulting in misleading conclusions. We give below some errors frequently
made in the use of correlation analysis:
1. Correlation analysis cannot determine cause-and-effect relationship. One should not
assume that a change in Y variable is caused by a change in X variable unless one is
reasonably sure that one variable is the cause while the other is the effect. Let us
take an example. .
Suppose that we study the performance of students in their graduate examination
and their earnings after, say, three years of their graduation. We may find that these
two variables are highly and positively related. At the same time, we must not forget
that both the variables might have been influenced by some other factors such as
quality of teachers, economic and social status of parents, effectiveness of the
interviewing process and so forth. If the data on these factors are available, then it is
worthwhile to use multiple correlation analysis instead of bivariate one.
2. Another mistake that occurs frequently is on account of misinterpretation of the
coefficient of correlation. Suppose in one case r = 0.7, it will be wrong to interpret
that correlation explains 70 percent of the total variation in Y. The error can be seen
easily when we calculate the coefficient of determination. Here, the coefficient of
128
Caritas Business College BUSINESS STATISTICS – OAD 221
determination r2 will be 0.49. This means that only 49 percent of the total variation in
Y is explained.
Similarly, the coefficient of determination is misinterpreted if it is also used to
indicate causal relationship, that is, the percentage of the change in one variable is
due to the change in another variable.
3. Another mistake in the interpretation of the coefficient of correlation occurs when one
concludes a positive or negative relationship even though the two variables are
actually unrelated. For example, the age of students and their score in the
examination have no relation with each other. The two variables may show similar
movements but there does not seem to be a common link between them.
To sum up, one has to be extremely careful while interpreting coefficient of correlation. Be-
fore one concludes a causal relationship, one has to consider other relevant factors that
might have any influence on the dependent variable or on both the variables. Such an
approach will avoid many of the pitfalls in the interpretation of the coefficient of correlation. It
has been rightly said that the coefficient of correlation is not only one of the most
widely used, but also one of the widely abused statistical measures.
4.5 SELF-ASSESSMENT QUESTIONS
1. “Correlation and Regression are two sides of the same coin”. Explain.
2. Explain the meaning and significance of the concept of correlation. Does correlation
always signify casual relationships between two variables? Explain with illustration
on what basis can the following correlation be criticized?
(a) Over a period of time there has been an increased financial aid to under
developed countries and also an increase in comedy act television shows. The
correlation is almost perfect.
129
Caritas Business College BUSINESS STATISTICS – OAD 221
(b) The correlation between salaries of school teachers and amount of liquor
sold during the period 1940 – 1980 was found to be 0.96
3. Write short not on the following
(a) Spurious correlation
(b) Positive and negative correlation
(c) Linear and non-linear correlation
(d) Simple, multiple and partial correlation
4. What is a scatter diagram? How does it help in studying correlation between two
variables, in respect of both its nature and extent?
5. Write short note on the following
(a) Karl Pearson’s coefficient of correlation
(b) Probable Error
(c) Spearman’s Rank Correlation Coefficient
(d) Coefficient of Concurrent Deviation
6. Draw a scatter diagram from the data given below and interpret it.
X: 10 20 30 40 50 60 70 80
Y: 32 20 24 36 40 28 38 44
7. Calculate Karl Pearson’s coefficient of correlation between expenditure on
advertising (X) and sales (Y) from the data given below:
X: 39 65 62 90 82 75 25 98 36 78
Y: 47 53 58 86 62 68 60 91 51 84
8. To study the effectiveness of an advertisement a survey is conducted by
calling people at random by asking the number of advertisements read or seen in a
week (X) and the number of items purchased (Y) in that week.
X: 5 10 4 0 2 7 3 6
Y: 10 12 5 2 1 3 4 8
130
Caritas Business College BUSINESS STATISTICS – OAD 221
Calculate the correlation coefficient and comment on the result.
9. Calculate coefficient of correlation between X and Y series from the following data
and calculate its probable error also.
X: 78 89 96 69 59 79 68 61
Y: 125 137 156 112 107 136 123 108
10. In two set of variables X and Y, with 50 observations each, the following data
are observed:
X = 10, SD of X = 3
rxy 0.3
Y = 6, SD of Y = 2
However, on subsequent verification, it was found that one value of X (=10) and one
value of Y (= 6) were inaccurate and hence weeded out with the remaining 49 pairs
of
values. How the original value of is rxy 0.3 affected?
11. Calculate coefficient of correlation r between the marks in statistics (X) and
Accountancy (Y) of 10 students from the following:
X: 52 74 93 55 41 23 92 64 40 71
Y: 45 80 63 60 35 40 70 58 43 64
Also determine the probable error or r.
12. The coefficient of correlation between two variables X and Y is 0.48. The covariance
is 36. The variance of X is 16. Find the standard deviation of Y.
13. Twelve entries in painting competition were ranked by two judges as shown below:
Entry: A B C D E F G H I J
Judge I: 5 2 3 4 1 6 8 7 10 9
Judge II: 4 5 2 1 6 7 10 9 3 8
Find the coefficient of rank correlation.
14. Calculate Spearman’s rank correlation coefficient between advertisement cost (X)
and sales (Y) from the following data:
131
Caritas Business College BUSINESS STATISTICS – OAD 221
X: 39 65 62 90 82 75 25 98 36 78
Y: 47 53 58 86 62 68 60 91 51 84
15. An examination of eight applicants for a clerical post was taken by a firm. From the
marks obtained by the applicants in the Accountancy (X) and Statistics (Y)
paper,
compute rank coefficient of correlation.
Applicant: A B C D E F G H
X: 15 20 28 12 40 60 20 80
Y: 40 30 50 30 20 10 30 60
16. Calculate the coefficient of concurrent deviation from the following data:
Year: 1993 1994 1995 1996 1997 1998 1999 2000
Supply: 160 164 172 182 166 170 178 192
Price: 222 280 260 224 266 254 230 190
17. Obtain a suitable measure of correlation from the following data regarding changes in
price index of the shares A and B during nine months of a year:
Month: A M J J A S O N D
A: +4 +3 +2 -1 -3 +4 -5 +1 +2
B: -2 +5 +3 -2 -1 -3 +4 -1 -3
18. The cross-classification table shows the marks obtained by 105 students in the
subjects of Statistics and Finance:
Marks in Statistics
50-54 55-59 60-64 65-74 Total
50-59 4 6 8 7 25
60-69 - 10 12 13 35
Marks in
70-79 16 9 20 - 45
80-89 - - - - -
Total 20 25 40 20 105
Find the coefficient of correlation between marks obtained in two subjects.
132
Caritas Business College BUSINESS STATISTICS – OAD 221
4.6 SUGGESTED READINGS
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons.,
New Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
133
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 6: REGRESSION ANALYSIS
Topic : Regression Analysis
Objectives: The overall objective of this lesson is to give you an understanding of linear
regression, there by enabling you to understand the importance and also
the limitations of regression analysis.
Structure
5.1 Introduction
5.2 What is Regression?
5.3 Linear Regression
5.3.1 Regression Line of Y on X
5.3.1.1 Scatter Diagram
5.3.1.2 Fitting a Straight Line
5.3.1.3 Predicting an Estimate and its Preciseness
5.3.1.4 Error of Estimate
5.3.2 Regression Line of X on Y
5.4 Properties of Regression Coefficients
5.5 Regression Lines and Coefficient of Correlation
5.6 Coefficient of Determination
5.7 Correlation Analysis Versus Regression Analysis
5.8 Solved Problems
5.9 Self-Assessment Questions
5.10 Suggested Readings
134
Caritas Business College BUSINESS STATISTICS – OAD 221
...if we find any association between two or more variables, we might be
interested in estimating the value of one variable for known value(s) of
another variable(s)
5.1 INTRODUCTION
In business, several times it becomes necessary to have some forecast so that the
management can take a decision regarding a product or a particular course of action. In
order to make a forecast, one has to ascertain some relationship between two or more
variables relevant to a particular situation. For example, a company is interested to know
how far the demand for television sets will increase in the next five years, keeping in mind
the growth of population in a certain town. Here, it clearly assumes that the increase in
population will lead to an increased demand for television sets. Thus, to determine the
nature and extent of relationship between these two variables becomes important for the
company.
In the preceding lesson, we studied in some depth linear correlation between two variables.
Here we have a similar concern, the association between variables, except that we develop
it further in two respects. First, we learn how to build statistical models of relationships
between the variables to have a better understanding of their features. Second, we extend
the models to consider their use in forecasting.
For this purpose, we have to use the technique - regression analysis - which forms the
subject-matter of this lesson.
5.2 WHAT IS REGRESSION?
In 1889, Sir Francis Galton, a cousin of Charles Darwin published a paper on heredity,
“Natural Inheritance”. He reported his discovery that sizes of seeds of sweet pea plants
appeared to “revert” or “regress”, to the mean size in successive generations. He also
reported results of a study of the relationship between heights of fathers and heights of their
sons. A straight line was fit to the data pairs: height of father versus height of son. Here, too,
he found a “regression to mediocrity” The heights of the sons represented a movement away
from their
135
Caritas Business College BUSINESS STATISTICS – OAD 221
fathers, towards the average height. We credit Sir Galton with the idea of statistical
regression.
While most applications of regression analysis may have little to do with the “regression to
the mean” discovered by Galton, the term “regression” remains. It now refers to the
statistical technique of modeling the relationship between two or more variables. In
general sense, regression analysis means the estimation or prediction of the unknown value
of one variable from the known value(s) of the other variable(s). It is one of the most
important and widely used statistical techniques in almost all sciences - natural, social or
physical.
In this lesson we will focus only on simple regression –linear regression involving only two
variables: a dependent variable and an independent variable. Regression analysis for
studying more than two variables at a time is known as multiple regressions.
5.2.1 INDEPENDENT AND DEPENDENT VARIABLES
Simple regression involves only two variables; one variable is predicted by another variable.
The variable to be predicted is called the dependent variable. The predictor is called the
independent variable, or explanatory variable. For example, when we are trying to predict
the demand for television sets on the basis of population growth, we are using the demand
for television sets as the dependent variable and the population growth as the independent
or predictor variable.
The decision, as to which variable is which sometimes, causes problems. Often the choice is
obvious, as in case of demand for television sets and population growth because it would
make no sense to suggest that population growth could be dependent on TV demand! The
population growth has to be the independent variable and the TV demand the dependent
variable.
136
Caritas Business College BUSINESS STATISTICS – OAD 221
If we are unsure, here are some points that might be of use:
if we have control over one of the variables then that is the independent. For
example, a manufacturer can decide how much to spend on advertising and expect
his sales to be dependent upon how much he spends
it there is any lapse of time between the two variables being measured, then the
latter must depend upon the former, it cannot be the other way round
if we want to predict the values of one variable from your knowledge of the other
variable, the variable to be predicted must be dependent on the known one
5.3 LINEAR REGRESSION
The task of bringing out linear relationship consists of developing methods of fitting a
straight line, or a regression line as is often called, to the data on two variables.
The line of Regression is the graphical or relationship representation of the best estimate of
one variable for any given value of the other variable. The nomenclature of the line depends
on the independent and dependent variables. If X and Y are two variables of which
relationship is to be indicated, a line that gives best estimate of Y for any value of X, it is
called Regression line of Y on X. If the dependent variable changes to X, then best
estimate of X by any value of Y is called Regression line of X on Y.
5.3.1 REGRESSION LINE OF Y ON X
For purposes of illustration as to how a straight line relationship is obtained, consider the
sample paired data on sales of each of the N = 5 months of a year and the marketing
expenditure incurred in each month, as shown in Table 5-1
Table 5-1
Month Sales Marketing Expenditure
(Rs (Rs thousands)
lac)
137
Caritas Business College BUSINESS STATISTICS – OAD 221
Y X
April 14 10
May 17 12
June 23 15
July 21 20
August 25 23
Let Y, the sales, be the dependent variable and X, the marketing expenditure, the
independent variable. We note that for each value of independent variable X, there is a
specific value of the dependent variable Y, so that each value of X and Y can be seen as
paired observations.
5.3.1.1 Scatter Diagram
Before obtaining a straight-line relationship, it is necessary to discover whether the
relationship between the two variables is linear, that is, the one which is best explained by a
straight line. A good way of doing this is to plot the data on X and Y on a graph so as to
yield a scatter diagram, as may be seen in Figure 5-1. A careful reading of the scatter
diagram reveals that:
the overall tendency of the points is to move upward, so the relationship is positive
the general course of movement of the various points on the diagram can be
best explained by a straight line
there is a high degree of correlation between the variables, as the points are very
close to each other
1
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 5-1 Scatter Diagram with Line of Best Fit
5.3.1.2 Fitting a Straight Line on the Scatter Diagram
If the movement of various points on the scatter diagram is best described by a straight line,
the next step is to fit a straight line on the scatter diagram. It has to be so fitted that on the
whole it lies as close as possible to every point on the scatter diagram. The
necessary requirement for meeting this condition being that the sum of the squares of the
vertical deviations of the observed Y values from the straight line is minimum.
As shown in Figure 5-1, if dl, d2,..., dN are the vertical deviations' of observed Y values from
the straight line, fitting a straight line requires that
d 2
d 2 ..................... d
2 Nd 2
1 2 j
N j 1
is the minimum. The deviations dj have to be squared to avoid negative deviations canceling
out the positive deviations. Since a straight line so fitted best approximates all the points on
the scatter diagram, it is better known as the best approximating line or the line of best fit. A
line of best fit can be fitted by means of:
1. Free hand drawing method, and
2. Least square method
Free Hand Drawing:
Free hand drawing is the simplest method of fitting a straight line. After a careful inspection
of the movement and spread of various points on the scatter diagram, a straight line is
drawn through these points by using a transparent ruler such that on the
135
Caritas Business College BUSINESS STATISTICS – OAD 221
whole it is closest to every point. A straight line so drawn is particularly useful when future
approximations of the dependent variable are promptly required.
Whereas the use of free hand drawing may yield a line nearest to the line of best fit, the
major drawback is that the slope of the line so drawn varies from person to person because
of the influence of subjectivity. Consequently, the values of the dependent variable estimated
on the basis of such a line may not be as accurate and precise as those based on the line of
best fit.
Least Square Method:
The least square method of fitting a line of best fit requires minimizing the sum of the
squares of vertical deviations of each observed Y value from the fitted line. These
deviations, such as d1 and d3, are shown in Figure 5-1 and are given by Y - Yc, where Y is
the observed value and Yc the corresponding computed value given by the fitted line
Yc a …………(5.1)
bX i
for the ith value of X.
The straight line relationship in Eq.(5.1), is stated in terms of two constants a and b
The constant a is the Y-intercept; it indicates the height on the vertical axis
from where the straight line originates, representing the value of Y when X is zero.
Constant b is a measure of the slope of the straight line; it shows the absolute
change in Y for a unit change in X. As the slope may be positive or negative, it
indicates the nature of relationship between Y and X. Accordingly, b is also known
as the regression coefficient of Y on X.
Since a straight line is completely defined by its intercept a and slope b, the task of fitting
the same reduces only to the computation of the values of these two constants. Once these
two values are known, the computed Yc values against each value of X can be easily
obtained by substituting X values in the linear equation.
136
Caritas Business College BUSINESS STATISTICS – OAD 221
In the method of least squares the values of a and b are obtained by solving simultaneously
the following pair of normal equations
Y aN b X …………(5.2)
XY a X b X
2 …………(5.2)
The value of the expressions - X , Y ,
XY and X can be obtained from the given
2
observations and then can be substituted in the above equations to obtain the value of a and
b. Since simultaneous solving the two normal equations for a and b may quite often be
cumbersome and time consuming, the two values can be directly obtained as
…………(5.3)
a= Y b
X
and
N XY X Y
b N X 2 X 2 …………(5.4)
Note: Eq. (5.3) is obtained simply by dividing both sides of the first of Eqs. (5.2) by N and
Eq.(5.4) is obtained by substituting (Y b X ) in place of a in the second of Eqs. (5.2)
Instead of directly computing b, we may first compute value of a as
2
Y X X XY
a …………(5.5)
2
N X X
2
and
b= …………(5.6)
Y
aX
Note: Eq. (5.5) is obtained by substituting N XY X Y 2 for b in Eq. (5.3) and Eq.
N X 2 X
(5.6) is obtained simply by rearranging Eq. (5.3)
Table 5-2
Computation of a and b
Y X XY X2 Y2
137
Caritas Business College BUSINESS STATISTICS – OAD 221
14 10 140 100 196
17 12 204 144 289
23 15 345 225 529
21 20 420 400 441
25 23 575 529 625
Y 100 X 80 XY 1684
2
X 1398
2
Y 2080
So using Eqs. (5.5) and (5.4)
100x1398 80x1684
a 2
5x1398 80
139800 134720
6990 6400
5080
= 590
= 8.6101695
and
5x1684
b 80x100 5x1398
2
80
8420 8000
6990 6400
420
= 590
= 0.7118644
Now given a = 8.61 and b=
0.71 The regression Eq.(5.1) takes the form
Yc = 8.61 + 0.71X..................................................................(5.1a)
138
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 5-2 Regression Line of Y on X
Then, to fit the line of best fit on the scatter diagram, only two computed Yc values are
needed. These can be easily obtained by substituting any two values of X in Eq. (5.1a).
When these are plotted on the diagram against their corresponding values of X, we get
two points, by joining which (by means of a straight line) gives us the required line of best
fit, as shown in Figure 5-2
Some Important Relationships
We can have some important relationships for data analysis, involving other measures such as
X , Y , Sx, Sy and the correlation coefficient rxy.
Substituting Y b [from Eq.(5.3)] for a in Eq.(5.1)
X
Yc = ( Y b X ) +bX
or Yc - Y = b(X- X )........................................................(5.7)
Dividing the numerator and denominator of Eq.(5.4) by N2, we get
XY X Y
N N N
2
b X
X
2
N N
XY
XY
N
or b
Sx 2
…………(5.8)
Cov( X ,Y )
or b
Sx 2
We know, coefficient of correlation, rxy is given by
Cov( X , Y )
r xy
Sx Sy
139
Caritas Business College BUSINESS STATISTICS – OAD 221
or Cov( X , Y ) rxy S x S y
So Eq. (5.8) becomes
b r Sx S y
xy 2
Sx
S
br …………(5.9)
y
xy
Sx
for b in Eq.(5.7), we get
S
Substituting r y
xy
Sx
(X- X )...........................................................(5.10)
Y-Y = r Sy
c xy
Sx
These are important relationships for data analysis.
5.3.1.3 Predicting an Estimate and its Preciseness
The main objective of regression analysis is to know the nature of relationship between two
variables and to use it for predicting the most likely value of the dependent variable
corresponding to a given, known value of the independent variable. This can be done by
substituting in Eq.(5.1a) any known value of X corresponding to which the most likely
estimate of Y is to be found.
For example, the estimate of Y (i.e. Yc), corresponding to X = 15 is
Yc = 8.61 + 0.71(15)
= 8.61 + 10.65
= 19.26
It may be appreciated that an estimate of Y derived from a regression equation will not be
exactly the same as the Y value which may actually be observed. The difference between
estimated Yc values and the corresponding observed Y values will depend on the extent of
scatter of various points around the line of best fit.
140
Caritas Business College BUSINESS STATISTICS – OAD 221
The closer the various paired sample points (Y, X) clustered around the line of best fit, the
smaller the difference between the estimated Yc and observed Y values, and vice-versa. On
the whole, the lesser the scatter of the various points around, and the lesser the vertical
distance by which these deviate from the line of best fit, the more likely it is that an estimated
Yc value is close to the corresponding observed Y value.
The estimated Yc values will coincide the observed Y values only when all the points on the
scatter diagram fall in a straight line. If this were to be so, the sales for a given marketing
expenditure could have been estimated with l00 percent accuracy. But such a situation is too
rare to obtain. Since some of the points must lie above and some below the straight line,
perfect prediction is practically non-existent in the case of most business and economic
situations.
This means that the estimated values of one variable based on the known values of the
other variable are always bound to differ. The smaller the difference, the greater the
precision of the estimate, and vice-versa. Accordingly, the preciseness of an estimate can
be obtained only through a measure of the magnitude of error in the estimates, called the
error of estimate.
5.3.1.4 Error of Estimate
A measure of the error of estimate is given by the standard error of estimate of Y on X,
denoted as Syx and defined as
Syx Y Y c
2
…………(5.11)
N
Syx measures the average absolute amount by which observed Y values depart from the
corresponding computed Yc values.
Computation of Syx becomes little cumbersome where the number of observations N is
large. In such cases Syx may be computed directly by using the equation:
141
Caritas Business College BUSINESS STATISTICS – OAD 221
Syx = Y 2 aY b XY N …………(5.12)
By substituting the values of Y 2 ,
Y , and XY from the Table 5-2, and the calculated
values of a and b
We
have
2080 8.61x100 0.71x1684
Syx =
5
2080 861 1195.64
=
5
23.36
=
5
= 4.67
= 2.16
Interpretations of Syx
A careful observation of how the standard error of estimate is computed reveals the
following:
1. Syx is a concept statistically parallel to the standard deviation Sy . The only difference
between the two being that the standard deviation measures the dispersion around
the mean; the standard error of estimate measures the dispersion around the
regression line. Similar to the property of arithmetic mean, the sum of the deviations
of different Y values from their corresponding estimated Yc values is equal to zero.
That is
( Yi - Y ) = ( Yi - Yc) = 0 where i = 1, 2, ..., N.
2. Syx tells us the amount by which the estimated Yc values will, on an average, deviate
from the observed Y values. Hence it is an estimate of the average amount of error in
the estimated Yc values. The actual error (the residual of Y and Yc) may, however, be
smaller or larger than the average error. Theoretically, these errors follow a normal
distribution. Thus, assuming that n ≥ 30, Yc ± 1.Syx means that 68.27% of the
estimates
142
Caritas Business College BUSINESS STATISTICS – OAD 221
based on the regression equation will be within 1.Syx Similarly, Yc ± 2.Syx means that
95.45% of the estimates will fall within 2.Syx
Further, for the estimated value of sales against marketing expenditure of Rs 15
thousand being Rs 19.26 lac, one may like to know how good this estimate is. Since
Syx is estimated to be Rs 2.16 lac, it means there are about 68 chances (68.27) out of
100 that this estimate is in error by not more than Rs 2.16 lac above or below
Rs
19.26 lac. That is, there are 68% chances that actual sales would fall between (19.26
- 2.16) = Rs 17.10 lac and (19.26 + 2.16) = Rs 21.42 lac.
3. Since Syx measures the closeness of the observed Y values and the estimated Yc
values, it also serves as a measure of the reliability of the estimate. Greater the
closeness between the observed and estimated values of Y, the lesser the error
and, consequently, the more reliable the estimate. And vice-versa.
4. Standard error of estimate Syx can also be seen as a measure of correlation insofar
as it expresses the degree of closeness of scatter of observed Y values about the
regression line. The closer the observed Y values scattered around the regression
line, the higher the correlation between the two variables.
A major difficulty in using Syx as a measure of correlation is that it is expressed in the
same units of measurement as the data on the dependent variable. This creates
problems in situations requiring comparison of two or more sets of data in terms of
correlation. It is mainly due to this limitation that the standard error of estimate is not
generally used as a measure of correlation. However, it does serve as the basis of
evolving the coefficient of determination, denoted as r2, which provides an alternate
method of obtaining a measure of correlation.
5.3.2 REGRESSION LINE OF X ON Y
143
Caritas Business College BUSINESS STATISTICS – OAD 221
So far we have considered the regression of Y on X, in the sense that Y was in the role of
dependent and X in the role of an independent variable. In their reverse position, such that
X is now the dependent and Y the independent variable, we fit a line of regression of X
on Y. The regression equation in this case will be
Xc = a’ + b’Y...........................................................................(5.13)
Where Xc denotes the computed values of X against the corresponding values of Y. a’ is the
X-intercept and b’ is the slope of the straight line.
Two normal equations to solve a’and b’ are
…………(5.14)
X a' N b'Y
XY a'Y b'Y
2
…………(5.14)
The value of a’ and b’ can also be obtained directly
a’ = X - b’ Y...........................................................................(5.15)
and
N XY X Y
b' N Y 2 Y 2 …………(5.16)
or
2
X Y Y XY N
a' 2 …………(5.17)
Y Y
2
and
b' …………(5.18)
X a'
Y
Cov(Y , X
b' …………(5.19)
) 2
S
y
b' r
S
…………(5.20)
x
yx
Sy
So, Regression equation of X on Y may also be written as
144
Caritas Business College BUSINESS STATISTICS – OAD 221
Xc - X = b’ (Y- Y )................................................................(5.21)
S
Xc - X = r (Y - Y )........................................................(5.22)
x
yx
S
y
As before, once the values of a’ and b’ have been found, their substitution in Eq.(5.13) will
enable us to get an estimate of X corresponding to a known value of Y
Standard Error of estimate of X on Y i.e. Sxy will be
X X 2
Sxy =.......................................................................................(5.23)
c
N
or
Sxy = X 2 a' X b' XY N …………(5.24)
For example, if we want to estimate the marketing expenditure to achieve a sale target of
Rs 40 lac, we have to obtain regression line of X on Y i. e.
Xc = a’ + b’Y
So using Eqs. (5.17) and (5.16), and substituting the values of X , Y 2 , Y and XY
from Table 5-2, we have
80x2080 100x1684
a' 2
5x2080 100
166400 168400
10400 10000
2000
= 400
= -5.00
and
5x1684 80x100
b' 2
5x2080 100
8420 8000
10400 10000
145
Caritas Business College BUSINESS STATISTICS – OAD 221
420
= 400
= 1.05
Now given that a’= -5.00 and b’=1.05, Regression equation (5.13) takes the form
Xc = -5.00 +1.05Y
So when Y = 40(Rs lac), the corresponding X value is
Xc = -5.00+1.05x40
= -5 + 42
= 37
That is to achieve a sale target of Rs 40 lac, there is a need to spend Rs 37 thousand on
marketing.
5.4 PROPERTIES OF REGRESSION COEFFICIENTS
As explained earlier, the slope of regression line is called the regression coefficient. It tells
the effect on dependent variable if there is a unit change in the independent variable.
Since for a paired data on X and Y variables, there are two regression lines: regression line
of Y on X and regression line of X on Y, so we have two regression coefficients:
a. Regression coefficient of Y on X, denoted by byx [b in Eq.(5.1)]
b. Regression coefficient of X on Y, denoted by bxy [b’ in Eq.(5.13)]
The following are the important properties of regression coefficients that are helpful in data
analysis
1. The value of both the regression coefficients cannot be greater than 1. However,
value of both the coefficients can be below 1 or at least one of them must be
below 1, so that the square root of the product of two regression coefficients must
lie in the limit
±1.
2. Coefficient of correlation is the geometric mean of the regression coefficients, i.e.
146
Caritas Business College BUSINESS STATISTICS – OAD 221
b. b'
r = ±............................................................................(5.25)
The signs of both the regression coefficients are the same, and so the value of r
will also have the same sign.
3. The mean of both the regression coefficients is either equal to or greater than
the coefficient of correlation, i.e.
b b'
2 r
3. Regression coefficients are independent of change of origin but not of change
of scale. Mathematically, if given variables X and Y are transformed to new variables
U and V by change of origin and scale, i. e.
XA YB
U= an V
h k
d
Where A, B, h and k are constants, h > 0, k > 0 then
Regression coefficient of Y on X = k/h (Regression coefficient of V on U)
k
b b
yx vu
h
and
Regression coefficient of X on Y = h/k (Regression coefficient of U on V)
h
b b
xy uv
k
5. Coefficient of determination is the product of both the regression coefficients i.e.
r2 = b.b’
5.5 REGRESSION LINES AND COEFFICIENT OF CORRELATION
The two regression lines indicate the nature and extent of correlation between the
variables. The two regression lines can be represented as
S
Y- Y = r
y (X - X ) and X - X = rSx (Y - Y )
S
Sx
y
147
Caritas Business College BUSINESS STATISTICS – OAD 221
We can write the slope of these lines, as
S
b= r Sx
and b’ =
y r S
y
Sx
If is the angle between these lines, then
tan = b
b' 1
bb'
S S
= x y r 2 1
S2 S2 r
x y
r 2 1
S
–1
Sx
2y
or = S S 2 r …………(5.26)
tan
x y
1
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 5-3 Regression Lines and Coefficient of Correlation
Eq. (5.26) reveals the following:
In case of perfect positive correlation (r = +1) and in case of perfect negative
correlation (r = -1), = 0, so the two regression lines will coincide, i.e. we have only
one line, see (a) and (b) in Figure 5-3.
The farther the two regression lines from each other, lesser will be the degree of
correlation and nearer the two regression lines, more will be the degree of
correlation, see (c) and (d) in Figure 5-3.
If the variables are independent i.e. r = 0, the lines of regression will cut each other at
right angle. See (g) in Figure 5-3.
Note : Both the regression lines cut each other at mean value of X and mean value of Y i.e. at
X and Y .
5.6 COEFFICIENT OF DETERMINATION
Coefficient of determination gives the percentage variation in the dependent variable that is
accounted for by the independent variable. In other words, the coefficient of determination
gives the ratio of the explained variance to the total variance. The coefficient of
determination is given by the square of the correlation coefficient, i.e. r2. Thus,
Coefficient of determination
2 Explained Variance
r = Total Variance
r2 = Y Y …………(5.27)
2
c
2
YY
149
Caritas Business College BUSINESS STATISTICS – OAD 221
We can calculate another coefficient K2, known as coefficient of Non-Determination, which
is the ratio of unexplained variance to the total variance.
2 Un exp lained Variance
K = Total Variance
2
Y Y
K2 = 2 c …………(5.28)
YY
K2 = 1- Explained Variance
Total Variance
...............................................................................
= 1 - r2 (5.29)
The square root of the coefficient of non-determination, i.e. K gives the coefficient of
alienation
1r2
K = ±...........................................................................(5.30)
Relation Between Syx and r:
A simple algebraic operation on Eq. (5.30) brings out some interesting points about the
relation between Syx and r. Thus, since
Y
2 and 2
Yc
N Y Y NS2
S2
y y
So we have coefficient of Non-determination
2
Y Y
K2 c
2
Y Y
K2 = N 2
yx
N S2
y
S2
yx
S
2
y
150
Caritas Business College BUSINESS STATISTICS – OAD 221
Syx2
So 1 – r2
Sy2
S yx
or
1 r 2 …………(5.31)
=
Sy
151
Caritas Business College BUSINESS STATISTICS – OAD 221
If coefficient of correlation, r, is defined as the under root of the coefficient of determination
r= r2
Syx2
r2 = 1
Sy2
1 S yx
r .............................................................................(5.32)
S y2
On carefully observing Eq. (5.32), it will be noticed that the ratio Syx/Sy will be large if the
coefficient of determination is small, and it will be small when the coefficient of determination
is large. Thus
if r2 = r = 0, Syx/Sy =1, which means that Syx = Sy.
if r2 = r = 1, Syx/Sy =0, which means that Syx = 0.
when r = 0.865, Syx = 0.427 Sy means that Syx is 42.7% of Sy.
Eq. (5.32) also implies that Syx is generally less than Sy. The two can at the most be equal,
but only in the extreme situation when r = 0.
Interpretations of r2:
1. Even though the coefficient of determination, whose under root measures the
degree of correlation, is based on Syx,; it is expressed as 1 - ( Syx/Sy ). As it is a
dimensionless pure number, the unit in which Syx is measured becomes irrelevant.
This facilitates comparison between the two sets of data in terms of their coefficient
of determination r2 (or the coefficient of correlation r). This was not possible in
terms of Sy x as the units of measurement could be different.
2. The value of r2 can range between 0 and 1. When r2 = 1, all the points on the scatter
diagram fall on the regression line and the entire variations are explained by the
straight line. On the other hand, when r2 = 0, none of the points on the scatter
diagram falls on the regression line, meaning thereby that there is no relationship
between the two variables. However, being always non-negative coefficient of
determination does
152
Caritas Business College BUSINESS STATISTICS – OAD 221
not tell us about the direction of the relationship (whether it is positive or negative)
between the two variables.
3. When r2 = 0.7455 (or any other value), 74.55% of the total variations in sales are
explained by the marketing expenditure used. What remains is the coefficient of non-
determination K2 (= 1 - r2) = 0.2545. It means 25.45% of the total variations remain
unexplained, which are due to factors other than the changes in the marketing
expenditure.
4. r2 provides the necessary link between regression and correlation which are the two
related aspects of a single problem of the analysis of relationship between two
variables. Unlike regression, correlation quantifies the degrees of relationship
between the variables under study, without making a distinction between the
dependent and independent ones. Nor does it, therefore, help in predicting the value
of one variable for a given value of the other.
5. The coefficient of correlation overstates the degree of relationship and it’s meaning is
not as explicit as that of the coefficient of determination. The coefficient of
correlation r = 0.865, as compared to r2 = 0.7455, indicates a higher degree of
correlation between sales and marketing expenditure. Therefore, the coefficient of'
determination is a more objective measure of the degree of relationship.
6. The sum of r and K never adds to one, unless one of the two is zero. That is, r + K
can be unity either when there is no correlation or when there is perfect
correlation. Except in these two extreme situations, (r + K) > 1.
5.7 CORRELATION ANALYSIS VERSUS REGRESSION ANALYSIS
Correlation and Regression are the two related aspects of a single problem of the analysis of
the relationship between the variables. If we have information on more than one variable, we
might be interested in seeing if there is any connection - any association - between them.
If
153
Caritas Business College BUSINESS STATISTICS – OAD 221
we found such a association, we might again be interested in predicting the value of one
variable for the given and known values of other variable(s).
1. Correlation literally means the relationship between two or more variables that vary in
sympathy so that the movements in one tend to be accompanied by the
corresponding movements in the other(s). On the other hand, regression means
stepping back or returning to the average value and is a mathematical measure
expressing the average relationship between the two variables.
2. Correlation coefficient rxy between two variables X and Y is a measure of the direction
and degree of the linear relationship between two variables that is mutual. It is
symmetric, i.e., ryx = rxy and it is immaterial which of X and Y is dependent variable
and which is independent variable.
Regression analysis aims at establishing the functional relationship between the two(
or more) variables under study and then using this relationship to predict or estimate
the value of the dependent variable for any given value of the independent
variable(s). It also reflects upon the nature of the variable, i.e., which is dependent
variable and which is independent variable. Regression coefficient are not symmetric
in X and Y, i.e., byx bxy.
3. Correlation need not imply cause and effect relationship between the variable under
study. However, regression analysis clearly indicates the cause and effect
relationship between the variables. The variable corresponding to cause is taken as
independent variable and the variable corresponding to effect is taken as dependent
variable.
4. Correlation coefficient rxy is a relative measure of the linear relationship between X
and Y and is independent of the units of measurement. It is a pure number lying
between ±1.
154
Caritas Business College BUSINESS STATISTICS – OAD 221
On the other hand, the regression coefficients, byx and bxy are absolute measures
representing the change in the value of the variable Y (or X), for a unit change in the
value of the variable X (or Y). Once the functional form of regression curve is known;
by substituting the value of the independent variable we can obtain the value of the
dependent variable and this value will be in the units of measurement of the
dependent variable.
5. There may be non-sense correlation between two variables that is due to pure
chance and has no practical relevance, e.g., the correlation, between the size of
shoe and the intelligence of a group of individuals. There is no such thing like non-
sense regression.
5.8 SOLVED PROBLEMS
Example 5-1
The following table shows the number of motor registrations in a certain territory for a term
of 5 years and the sale of motor tyres by a firm in that territory for the same period.
Year Motor Registrations No. of Tyres
Sold
1 600 1,250
2 630 1,100
3 720 1,300
4 750 1,350
5 800 1,500
Find the regression equation to estimate the sale of tyres when the motor registration is
known. Estimate sale of tyres when registration is 850.
Solution: Here the dependent variable is number of tyres; dependent on motor registrations.
Hence we put motor registrations as X and sales of tyres as Y and we have to establish the
regression line of Y on X.
Calculations of values for the regression equation are given below:
155
Caritas Business College BUSINESS STATISTICS – OAD 221
X Y 2 dx dy
dx
dx = X- X dy = Y- Y
600 1,250 -100 -50 10,000 5,000
630 1,100 -70 -200 4,900 14,000
720 1,300 20 0 400 0
750 1,350 50 50 2,500 2,500
800 1,500 100 200 10,000 20,000
2
X = 3,500 Y = 6,500 dx =0 dy=0 d x = 27,800 dx d y =
41,500
X
X= = 3,50 =700 6,500
and Y Y = 1,300
N 0 = N = 5
5
byx = Regression coefficient of Y on X
byx X X Y Y d x d 4,1500 1.4928
= 2 = 2,7800
X X y
2
d x
Now we can use these values for the regression line
Y- Y = byx (X- X )
or Y – 1300 = 1.4928 (X - 700)
Y = 1.4928 X + 255.04
When X = 850, the value of Y can be calculated from the above equation, by putting X =
850 in the equation.
Y = 1.4928 x 850 + 255. 04
= 1523.92
= 1,524 Tyres
Example 5-2
A panel of Judges A and B graded seven debators and independently awarded the
following marks:
Debator Marks by A Marks by B
1 40 32
156
Caritas Business College BUSINESS STATISTICS – OAD 221
2 34 39
157
Caritas Business College BUSINESS STATISTICS – OAD 221
3 28 26
4 30 30
5 44 38
6 38 34
7 31 28
An eighth debator was awarded 36 marks by judge A, while Judge B was not present.
If Judge B were also present, how many marks would you expect him to award to the eighth
debator, assuming that the same degree of relationship exists in their judgement?
Solution: Let us use marks from Judge A as X and those from Judge B as Y. Now we have
to work out the regression line of Y on X from the calculation below:
Debtor X Y U = X-35 V = Y-30 2 2 UV
U V
1 40 32 5 2 25 4 10
2 34 39 -1 9 1 81 -9
3 28 26 -7 -4 49 16 28
4 30 30 -5 0 25 0 0
5 44 38 9 8 81 64 72
6 38 34 3 4 9 16 12
7 31 28 -4 -2 16 4 8
N=7 U
2
206 V
2
185 UV 121
U 0 V 17
U 0 V 17
X= A = 35 + = 35 and Y= A = 30 + = 32.43
N 7 N 7
N UV U V
byx = bvu = 2
N U 2 U
7x121 - 0x17
= 7x206 - 0 0.587
Hence regression equation can be written as
Y- Y = byx (X- X )
Y – 32.43 = 0.587 (X-35)
158
Caritas Business College BUSINESS STATISTICS – OAD 221
or Y = 0.587X + 11.87
When X = 36 (awarded by Judge A)
Y = 0.587 x 36 + 11.87
= 33
Thus if Judge B were present, he would have awarded 33 marks to the eighth debator.
Example 5-3
For some bivariate data, the following results were obtained.
Mean value of variable X = 53.2
Mean value of variable Y = 27.9
Regression coefficient of Y on X = - 1.5
Regression coefficient of X on Y = - 0.2
What is the most likely value of Y, when X = 60?
What is the coefficient of correlation between X and Y?
Solution: Given data indicate
X = 53.2 Y = 27.9
byx = -1.5 bxy = -0.2
To obtain value of Y for X = 60, we establish the regression line of Y on
X, Y- Y = byx (X- X )
Y – 27.9 = -1.5 (X-53.2)
or Y = -1.5X + 107.7
Putting value of X = 60, we obtain
Y = -1.5 x 60 + 107.7
= 17.7
Coefficient of correlation between X and Y is given by G.M. of byx and bxy
r2 = byx bxy
159
Caritas Business College BUSINESS STATISTICS – OAD 221
= (-1.5) x (–0.2)
= 0.3
So r = ± 0.3 = ± 0.5477
Since both the regression coefficients are negative, we assign negative value to the
correlation coefficient
r = - 0.5477
Example 5-4
Write regression equations of X on Y and of Y on X for the following data
X: 45 48 50 55 65 70 75 72 80 85
Y: 25 30 35 30 40 50 45 55 60 65
Solution: We prepare the table for working out the values for the regression lines.
X Y U = X-65 V = Y-45 2 UV 2
U V
45 25 -20 -20 400 400 400
48 30 -17 -15 289 255 225
50 35 -15 -10 225 150 100
55 30 -10 -15 100 150 225
65 40 0 -5 0 0 25
70 50 5 5 25 25 25
75 45 10 0 100 0 0
72 55 7 5 49 35 25
80 60 15 15 225 225 225
85 65 20 20 400 400 400
U 5 V 20 U 1813 V 2 1415 UV 1675
2
X = 645 Y = 435
We have,
X 645
X= 435
= = 64.5 and Y Y = 43.5
N = 10 N = 10
N UV U V
byx = 2
N U 2 U
160
Caritas Business College BUSINESS STATISTICS – OAD 221
(10) x 1415 - (5) x (-20)
= (10) x 1813 - (5) 2
14150 100 14250
= 18130 - 25 18105 0.787
Regression equation of Y on X is
Y- Y = by (X- X )
x
Y – 43.5 = 0.787 (X-64.5)
or Y = 0.787X + 7.26
Similarly bxy can be calculated as
N UV U V
bxy = 2
N V 2 V
= (10) x 1415 - (5) x (-
20)
(10) x 1675 - (-20)2
14150 100 14250
= 16750 - 400 16350 0.87
Regression equation of X on Y will be
X-X = bx (Y- Y )
y
X – 64.5 = 0.87 (Y-43.5)
or X = 0.87Y + 26.65
Example 5-5
The lines of regression of a bivariate population
are 8X – 10Y + 66 = 0
40X – 18Y = 214
The variance of X is 9. Find
(i) The mean value of X and Y
(ii) Correlation coefficient between X and Y
(iii) Standard deviation of Y
161
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution: The regression lines given are
8X – 10Y + 66 = 0
40X – 18Y = 214
Since both the lines of regression pass through the mean values, the point ( X , Y ) will
satisfy both the equations.
Hence these equations can be written
as 8 X - 10 Y + 66 = 0
40 X - 18 Y - 214 = 0
Solving these two equations for X and Y , we obtain
X = 13 and Y = 17
(ii) For correlation coefficient between X and Y, we have to calculate the values of byx and
bxy
Rewriting the equations
10Y = 8X + 66
byx = + 8/10 = + 4/5
Similarly, 40X = 18Y + 214
bxy = 18/40 = 9/20
By these values, we can now work out the correlation coefficient.
2
r = byx . bxy
= 4/5 x 9/20 = 9/25
So r = + 9 / 25
= + 0.6
Both the values of the regression coefficients being positive, we have to consider only
the positive value of the correlation coefficient. Hence r = 0.6
(iii) We have been given variance of X i.e Sx2 = 9
162
Caritas Business College BUSINESS STATISTICS – OAD 221
Sx = ± 3
We consider Sx = 3 as SD is always
positive Since byx = r Sy /Sx
Substituting the values of byx, r and Sx we obtain,
Sy = 4/5 x 3/0.6
= 4
Example 5-6
The height of a child increases at a rate given in the table below. Fit the straight line using
the method of least-square and calculate the average increase and the standard error of
estimate.
Month: 1 2 3 4 5 6 7 8 9 10
Height: 52.5 58.7 65 70.2 75.4 81.1 87.2 95.5 102.2 108.4
Solution: For Regression calculations, we draw the following table
Month (X) Height (Y) 2 XY
X
1 52.5 1 52.5
2 58.7 4 117.4
3 65.0 9 195.0
4 70.2 16 280.8
5 75.4 25 377.0
6 81.1 36 486.6
7 87.2 49 610.4
8 95.5 64 764.0
9 102.2 81 919.8
10 108.4 100 1084.0
2
X =55 Y =796.2 X 385 XY 4887.5
Considering the regression line as Y = a + bX, we can obtain the values of a and b from
the above values.
163
Caritas Business College BUSINESS STATISTICS – OAD 221
a
2
Y X X XY
2
N X 2 X
796.2 x 385 - 55 x 4887.5
= 10 x 385 - 55 x 55
= 45.73
N XY X Y N
b 2
X X
2
10 x 4887.5 - 55 x 796.2
= 10 x 385 - 55 x 55
= 6.16
Hence the regression line can be written as
Y = 45.73 + 6.16X
For standard error of estimation, we note the calculated values of the variable against
the observed values,
When X = 1, Y1 = 45.73 + 6.16 = 51.89
for X = 2, Y2 = 45.73 + 616 x 2 = 58.05
Other values for X = 3 to X = 10 are calculated and are tabulated as follows:
Month (X) Height (Y) Yi Y-Yi (Y-Yi) 2
1 52.5 51.89 0.61 0.372
2 58.7 58.05 0.65 0.423
3 65.0 64.21 0.79 0.624
4 70.2 70.37 -0.17 0.029
5 75.4 76.53 -1.13 1.277
6 81.1 82.69 -1.59 2.528
7 87.2 88.85 -1.65 2.723
8 95.5 95.01 0.49 0.240
9 102.2 101.17 1.03 1.061
10 108.4 107.33 1.07 1.145
164
Caritas Business College BUSINESS STATISTICS – OAD 221
(Y 2
Yi ) 10.421
Standard error of estimation
1
S yx =
N (Y Y )
i
2
10.421
=
10
= 1.02
Example 5-7
Given X = 4Y+5 and Y = kX + 4 are the lines of regression of X on Y and of Y on X
respectively. If k is positive, prove that it cannot exceed ¼.
If k = 1/16, find the means of the two variables and coefficient of correlation between them.
Solution: Line X = 4Y + 5 is regression line of X on Y
So bxy = 4
Similarly from regression line of Y on X , Y = kX + 4,
We get byx = k
Now
2
r = bxy. byx
= 4k
Since 0 r 2 1, we obtain 0 4k 1,
1
Or 0k
,
4
1
Now for k = ,
16
1 1
r 2 4x
16 4
r=+½
= ½ since byx and byx are positive
165
Caritas Business College BUSINESS STATISTICS – OAD 221
When k = 1
, the regression line of Y on X becomes
16
1
Y= X+4
16
Or X – 16Y + 64 = 0
Since line of regression pass through the mean values of the variables, we obtain revised
equations as
X -4Y-5=0
X - 16 Y + 64 = 0
Solving these two equations, we get
X = 28 and Y = 5.75
Example 5-8
A firm knows from its past experience that its monthly average expenses (X) on
advertisement are Rs 25,000 with standard deviation of Rs 25.25. Similarly, its average
monthly product sales (Y) have been Rs 45,000 with standard deviation of Rs 50.50. Given
this information and also the coefficient of correlation between sales and advertisement
expenditure as 0.75, estimate
(i) the most appropriate value of sales against an advertisement expenditure of
Rs 50,000
(ii) the most appropriate advertisement expenditure for achieving a sales target
of Rs 80,000
Solution: Given the following
X = Rs 25,000 Sx = Rs 25.25
Y = Rs 45,000 Sy = Rs 50.50
r = 0.75
166
Caritas Business College BUSINESS STATISTICS – OAD 221
S
(i) Using equation y (X- X ), the most appropriate value of sales for an
Yc -Y = r Yc
S
x
advertisement expenditure X = Rs 50,000 is
Yc – 45,000 = 50.50
0.75 (50,000 – 25,000)
25.25
Yc = 45,000 + 37,500
= Rs 82,500
Sx
(ii) Using equation Xc - X =r (Y - Y ), the most appropriate value of advertisement
Sy
expenditure Xc for achieving a sales target Y= Rs 80,000 is
Xc – 25,000 = 25.25
0.75 (80,000 – 45,000)
50.50
Xc = 13,125 + 25,000
= Rs 38,125
1.8 SELF-ASSESSMENT QUESTIONS
1. Explain clearly the concept of Regression. Explain with suitable examples its role in
dealing with business problems.
2. What do you understand by linear regression?
3. What is meant by ‘regression’? Why should there be in general, two lines of
regression for each bivariate distribution? How the two regression lines are useful in
studying correlation between two variables?
4. Why is the regression line known as line of best fit?
5. Write short note on
(i) Regression Coefficients
(ii) Regression Equations
(iii) Standard Error of Estimate
167
Caritas Business College BUSINESS STATISTICS – OAD 221
(iv) Coefficient of Determination
168
Caritas Business College BUSINESS STATISTICS – OAD 221
(v) Coefficient of Non-determination
6. Distinguish clearly between correlation and regression as concept used in
statistical analysis.
7. Fit a least-square line to the following data:
(i) Using X as independent variable
(ii) Using X as dependent variable
X : 1 3 4 8 9 11 14
Y : 1 2 4 5 7 8 9
Hence obtain
c) The regression coefficients of Y on X and of X on Y
d) X and Y
e) Coefficient of correlation between and X and Y
f) What is the estimated value of Y when X = 10 and of X when Y = 5?
8. What are regression coefficients? Show that r2 = byx. bxy where the symbols have their
usual meanings. What can you say about the angle between the regression lines
when
(i) r = 0, (ii) r = 1 (iii) r increases from 0 to 1?
9. Obtain the equations of the lines of regression of Y on X from the following data.
X : 12 18 24 30 36 42 48
Y : 5.27 5.68 6.25 7.21 8.02 8.71 8.42
Estimate the most probable value of Y, when X = 40.
10. The following table gives the ages and blood pressure of 9 women.
Age (X) : 56 42 36 47 49 42 60 72 63
Blood Pressure(Y) 147 125 118 128 145 140 155 160 149
Find the correlation coefficient between X and Y.
(i) Determine the least square regression equation of Y on X.
169
Caritas Business College BUSINESS STATISTICS – OAD 221
(ii) Estimate the blood pressure of a woman whose age is 45 years.
11. Given the following results for the height (X) and weight (Y) in appropriate units of
1,000 students:
S x = 2.5, S y = 20 and r = 0.6.
X = 68, Y =
150,
Obtain the equations of the two lines of regression. Estimate the height of a student A
who weighs 200 units and also estimate the weight of the student B whose height
is 60 units.
12. From the following data, find out the probable yield when the rainfall is 29”.
Rainfall Yield
Mean 25” 40 units per
hectare
Standard Deviation 3” 6 units per hectare
Correlation coefficient between rainfall and production = 0.8.
13. A study of wheat prices at two cities yielded the following data:
City A City B
Average Price Rs 2,463 Rs 2,797
Standard Deviation Rs 0.326 Rs 0.207
Coefficient of correlation r is 0.774. Estimate from the above data the most likely
price of wheat
(i) at City A corresponding to the price of Rs 2,334 at City B
(ii) at city B corresponding to the price of Rs 3.052 at City A
14. Find out the regression equation showing the regression of capacity utilisation on
production from the following data:
Average Standard Deviation
Production (in lakh units) 35.6 10.5
Capacity Utilisation (in percentage) 84.8 8.5
r = 0.62
170
Caritas Business College BUSINESS STATISTICS – OAD 221
Estimate the production, when capacity utilisation is 70%.
15. The following table shows the mean and standard deviation of the prices of two
shares in a stock exchange.
Share Mean (in Rs) Standard Deviation (in Rs)
A Ltd. 39.5 10.8
B Ltd. 47.5 16.0
If the coefficient of correlation between the prices of two shares is 0.42, find the most
likely price of share A corresponding to a price of Rs 55, observed in the case of
share B.
16. Find out the regression coefficients of Y on X and of X on Y on the basis of following
data:
X = 50, X = 5, Y = 60, Y = 6, XY = 350
Variance of X = 4, Variance of Y = 9
17. Find the regression equation of X and Y and the coefficient of correlation from the
following data:
X = 60, Y = 40, XY = 1150, X = 4160, Y = 1720 and N = 10.
2 2
18. By using the following data, find out the two lines of regression and from them
compute the Karl Pearson’s coefficient of correlation.
X = 250, Y = 300, XY = 7900, X 2 6500, Y 2 = 10000, N = 10
19. The equations of two regression lines between two variables are expressed
as 2X – 3Y = 0 and 4Y – 5X-8 = 0.
(i) Identify which of the two can be called regression line of Y on X and of X on Y.
(ii) Find X and Y and correlation coefficient r from the equations
20. If the two lines of regression are
4X - 5Y + 30 = 0 and 20X – 9Y – 107 = 0
Which of these is the lines of regression of X and Y. Find rxy and Sy when Sx = 3
171
Caritas Business College BUSINESS STATISTICS – OAD 221
21. The regression equation of profits (X) on sales (Y) of a certain firm is 3Y – 5X +108 =
0. The average sales of the firm were Rs 44,000 and the variance of profits is 9/16 th
of the variance of sales. Find the average profits and the coefficient of correlation
between the sales and profits.
5.10 SUGGESTED READINGS
9. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
10. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
11. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
12. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
13. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons.,
New Delhi.
14. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
15. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
16. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
172
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 7: INDEX NUMBERS
Topic : Index Numbers
Objectives : The overall objective of this lesson is to give an understanding of Index
Numbers. After successful completion of the lesson, the students will
be able to understand the concepts, techniques and the problems
involved in constructing and using index numbers.
Structure
6.1 Introduction
6.2 What are Index Numbers?
6.3 Uses of Index Numbers
6.4 Types of Index Numbers
6.5 Simple Index Numbers
6.6 Composite Index Numbers
6.6.1 Simple Aggregative Price/Quantity Index
6.6.2 Index of Average of Price/Quantity Relatives
6.6.3 Weighted Aggregative Price/Quantity Index
6.6.4 Index of Weighted Average of Price/Quantity Relatives
6.6 Test of Adequacy of Index Numbers
6.7 Special Issues in the Construction of Index Numbers
6.9 Problems of Constructing Index Numbers
6.10 Self-Assessment Question
6.11 Suggested Readings
6.1 INTRODUCTION
In business, managers and other decision makers may be concerned with the way in
which the values of variables change over time like prices paid for raw materials,
numbers of
173
Caritas Business College BUSINESS STATISTICS – OAD 221
employees and customers, annual income and profits, and so on. Index numbers are one
way of describing such changes.
If we turn to any journal devoted to economic and financial matters, we are very likely to
come across an index number of one or the other type. It may be an index number of share
prices or a wholesale price index or a consumer price index or an index of industrial
production. The objective of these index numbers is to measure the changes that have
occurred in prices, cost of living, production, and so forth. For example, if a wholesale price
index number for the year 2000 with base year 1990 was 170; it shows that wholesale
prices, in general, increased by 70 percent in 2000 as compared to those in 1990. Now, if
the same index number moves to 180 in 2001, it shows that there has been 80 percent
increase in wholesale prices in 2001 as compared to those in 1990.
With the help of various index numbers, economists and businessmen are able to
describe and appreciate business and economic situations quantitatively. Index numbers
were originally developed by economists for monitoring and comparing different groups of
goods. It is necessary in business to understand and manipulate the different published
index serieses, and to construct index series of your own. Having constructed your own
index, it can then be compared to a national one such as the RPI, a similar index for your
industry as a whole and also to indexes for your competitors. These comparisons are a very
useful tool for decision making in business.
174
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 6-1 The Indexes of the Volume of Sales
For example, an accountant of a supermarket chain could construct an index of the
company's own sales and compare it to the index of the volume of sales for the general
supermarket industry. A graph of the two indexes will illustrate the company's performance
within the sector. It is immediately clear from Figure 6-1 that, after initially lagging behind the
general market, the supermarket company caught up and then overtook it. In the later
stages, the company was having better results than the general market but that, as with the
whole industry, those had levelled out.
Our focus in this lesson will be on the discussion related to the methodology of index number
construction. The scope of the lesson is rather limited and as such, it does not discuss a
large number of index numbers that are presently compiled and published by different
departments of the Government of India.
6.2 WHAT ARE INDEX NUMBERS?
“Index numbers are statistical devices designed to measure the relative changes in
the level of a certain phenomenon in two or more situations”. The phenomenon under
consideration may be any field of quantitative measurements. It may refer to a single
variable or a group of distinct but related variables. In Business and Economics, the
phenomenon under consideration may be:
the prices of a particular commodity like steel, gold, leather, etc. or a group of
commodities like consumer goods, cereals, milk and milk products, cosmetics, etc.
volume of trade, factory production, industrial or agricultural production, imports or
exports, stocks and shares, sales and profits of a business house and so on.
175
Caritas Business College BUSINESS STATISTICS – OAD 221
the national income of a country, wage structure of workers in various sectors, bank
deposits, foreign exchange reserves, cost of living of persons of a particular
community, class or profession and so on.
The various situations requiring comparison may refer to either
the changes occurring over a time, or
the difference(s) between two or more places, or
the variations between similar categories of objects/subjects, such as persons,
groups of persons, organisations etc. or other characteristics such as income,
profession, etc.
The utility of index numbers in facilitating comparison may be seen when, for example we
are interested in studying the general change in the price level of consumer goods, i.e.
good or commodities consumed by the people belonging to a particular section of society,
say, low income group or middle income group or labour class and so on. Obviously these
changes are not directly measurable as the price quotations of the various commodities are
available in different units, e.g., cereals (wheat, rice, pulses, etc) are quoted in Rs per quintal
or kg; water in Rs per gallon; milk, petrol, kerosene, etc. in Rs per liter; cloth in Rs per miter
and so on.
Further, the prices of some of the commodities may be increasing while those of others may
be decreasing during the two periods and the rates of increase or decrease may be
different for different commodities. Index number is a statistical device, which enables us to
arrive at a single representative figure that gives the general level of the price of the
phenomenon (commodities) in an extensive group. According to Wheldon:
“Index number is a statistical device for indicating the relative movements of
the data where measurement of actual movements is difficult or incapable of
being made.”
FY Edgeworth gave the classical definition of index numbers as follows:
176
Caritas Business College BUSINESS STATISTICS – OAD 221
“Index number shows by its variations the changes in a magnitude which is
not susceptible either of accurate measurement in itself or of direct valuation
in practice.”
On the basis of above discussion, the following characteristics of index numbers are apparent:
1. Index Numbers are specialized averages: An average is a summary figure
measuring the central tendency of the data, representing a group of figures. Index
number has all these functions to perform. L R Connor states, "in its simplest form, it
(index number) represents a special case of an average, generally a weighted
average compiled from a sample of items judged to be representative of the whole".
It is a special type of average – it averages variables having different units of
measurement.
2. Index Numbers are expressed in percentages: Index numbers are expressed in
terms of percentages so as to show the extent of change. However, percentage sign
(%) is never used.
3. Index Numbers measure changes not capable of direct measurement: The
technique of index numbers is utilized in measuring changes in magnitude, which are
not capable of direct measurement. Such magnitudes do not exist in themselves.
Examples of such magnitudes are 'price level', 'cost of living', 'business or economic
activity' etc. The statistical methods used in the construction of index numbers are
largely methods for combining a number of phenomena representing a particular
magnitude in such a manner that the changes in that magnitude may be measured in
a meaningful way without introduction of serious bias.
4. Index Numbers are for comparison: The index numbers by their nature are
comparative. They compare changes taking place over time or between places or
between like categories.
177
Caritas Business College BUSINESS STATISTICS – OAD 221
In brief, index number is a statistical technique used in measuring the composite change in
several similar economic variables over time. It measures only the composite change,
because some of the variables included may be showing an increase, while some others
may be showing a decrease. It synthesizes the changes taking place in different directions
and by varying extents into the one composite change. Thus, an index number is a device to
simplify comparison to show relative movements of the data concerned and to replace what
may be complicated figures by simple ones calculated on a percentage basis.
6.3 USES OF INDEX NUMBER
The first index number was constructed by an Italian, Mr G R Carli, in 1764 to compare the
changes in price for the year 1750 (current year) with the price level in 1500 (base year) in
order to study the effect of discovery of America on the price level in Italy. Though originally
designed to study the general level of prices or accordingly purchasing power of
money, today index numbers are extensively used for a variety of purposes in economics,
business, management, etc., and for quantitative data relating to production, consumption,
profits, personnel and financial matters etc., for comparing changes in the level of
phenomenon for two periods, places, etc. In fact there is hardly any field or quantitative
measurements where index numbers are not constructed. They are used in almost all
sciences – natural, social and physical. The main uses of index numbers can be
summarized as follows:
1. Index Numbers as Economic Barometers
Index numbers are indispensable tools for the management personnel of any
government organisation or individual business concern and in business planning
and formulation of executive decisions. The indices of prices (wholesale & retail),
output (volume of trade, import and export, industrial and agricultural production) and
bank deposits, foreign exchange and reserves etc., throw light on the nature of, and
178
Caritas Business College BUSINESS STATISTICS – OAD 221
variation in the general economic and business activity of the country. They are the
indicators of business environment. A careful study of these indices gives us a fairly
good appraisal of the general trade, economic development and business activity
of the country. In the world of G Simpson and F Kafka:
“Index numbers are today one of the most widely used statistical devices.
They are used to take the pulse of the economy and they have come to be
used as indicators of inflationary or deflationary tendencies.”
Like barometers, which are used in Physics and Chemistry to measure atmospheric
pressure, index numbers are rightly termed as “economic barometers”, which
measure the pressure of economic and business behaviour.
2. Index Numbers Help in Studying Trends and Tendencies
Since the index numbers study the relative change in the level of a phenomenon at
different periods of time, they are especially useful for the study of the general trend
for a group phenomenon in time series data. The indices of output (industrial and
agricultural production), volume of trade, import and export, etc., are extremely useful
for studying the changes in the level of phenomenon due to the various components
of a time series, viz. secular trend, seasonal and cyclical variations and irregular
components and reflect upon the general trend of production and business activity.
As a measure of average change in extensive group, the index numbers can be used
to forecast future events. For instance, if a businessman is interested in
establishing a new undertaking, the study of the trend of changes in the prices,
wages and incomes in different industries is extremely helpful to him to frame a
general idea of the comparative courses, which the future holds for different
undertakings.
3. Index Numbers Help in Formulating Decisions and Policies
179
Caritas Business College BUSINESS STATISTICS – OAD 221
Index numbers of the data relating to various business and economic variables
serve an important guide to the formulation of appropriate policy. For example, the
cost of living index numbers are used by the government and, the industrial and
business concerns for the regulation of dearness allowance (D.A.) or grant of bonus
to the workers so as to enable them to meet the increased cost of living from time to
time. The excise duty on the production or sales of a commodity is regulated
according to the index numbers of the consumption of the commodity from time to
time. Similarly, the indices of consumption of various commodities help in the
planning of their future production. Although index numbers are now widely used to
study the general economic and business conditions of the society, they are also
applied with advantage by sociologists (population indices), psychologists (IQs’),
health and educational authorities etc., for formulating and revising their policies from
time to time.
4. Price Indices Measure the Purchasing Power of Money
A traditional use of index numbers is in measuring the purchasing power of money.
Since the changes in prices and purchasing power of money are inversely related, an
increase in the general price index indicates that the purchasing power of money has
gone down.
In general, the purchasing power of money may be computed as
Purchasing Power 1
x100
General Price
Index
Accordingly, if the consumer price index for a given year is 150, the purchasing
power of a rupee is (1/150) 100 = 0.67. That is, the purchasing power of a rupee in
the given year is 67 paise as compared to the base year.
180
Caritas Business College BUSINESS STATISTICS – OAD 221
With the increase in prices, the amount of goods and services which money wages
can buy (or the real wages) goes on decreasing. Index numbers tell us the change
in real
wages, which are obtained
as
Money Wage
x100
Real Wage
Consumer Price Index
A real wage index equal to, say, 120 corresponding to money wage index of 160 will
indicate an increase in real wages by only 20 per cent as against 60 per cent
increase in money wages.
Index numbers also serve as the basis of determining the terms of exchange.
The terms of exchange are the parity rate at which one set of commodities is
exchanged for another set of commodities. It is determined by taking the ratio of the
price index for the two groups of commodities and expressing it in percentage.
For example, if A and B are the two groups of commodities with 120 and 150 as their
price index in a particular year, respectively, the ratio 120/150 multiplied by 100 is 80
per cent. It means that prices of A group of commodities in terms of those in group B
are lower by 20 per cent.
5. Index Numbers are Used for Deflation
Consumer price indices or cost of living index numbers are used for deflation of net
national product, income value series in national accounts. The technique of
obtaining real wages from the given nominal wages (as explained in use 4 above)
can be used to find real income from inflated money income, real sales from nominal
sales and so on by taking into account appropriate index numbers.
181
Caritas Business College BUSINESS STATISTICS – OAD 221
5.4 TYPES OF INDEX NUMBERS
Index numbers may be broadly classified into various categories depending upon the type of
the phenomenon they study. Although index numbers can be constructed for measuring
relative changes in any field of quantitative measurement, we shall primarily confine the
discussion to the data relating to economics and business i.e., data relating to prices,
production (output) and consumption. In this context index numbers may be broadly
classified into the following three categories:
1. Price Index Numbers: The price index numbers measure the general changes in the
prices. They are further sub-divided into the following classes:
(i) Wholesale Price Index Numbers: The wholesale price index numbers reflect
the changes in the general price level of a country.
(ii) Retail Price Index Numbers: These indices reflect the general changes in
the retail prices of various commodities such as consumption goods, stocks and
shares, bank deposits, government bonds, etc.
(iii) Consumer Price Index: Commonly known as the Cost of living Index, CPI is
a specialized kind of retail price index and enables us to study the effect of changes
in the price of a basket of goods or commodities on the purchasing power or cost of
living of a particular class or section of the people like labour class, industrial or
agricultural worker, low income or middle income class etc.
2. Quantity Index Numbers: Quantity index numbers study the changes in the volume
of goods produced (manufactured), consumed or distributed, like: the indices of
agricultural production, industrial production, imports and exports, etc. They are
extremely helpful in studying the level of physical output in an economy.
182
Caritas Business College BUSINESS STATISTICS – OAD 221
3. Value Index Numbers: These are intended to study the change in the total value
(price multiplied by quantity) of output such as indices of retail sales or profits or
inventories. However, these indices are not as common as price and quantity indices.
Notations Used
Since index numbers are computed for prices, quantities, and values, these are denoted by the lower
case letters:
p, q, and v represent respectively the price, the quantity, and the value of an individual
commodity.
Subscripts 0, 1, 2,… i, ... are attached to these lower case letters to distinguish price, quantity, or value
in any one period from those in the other. Thus,
p0 denotes the price of a commodity in the base period,
p1 denotes the price of a commodity in period 1, or the current period, and
pi denotes the price of a commodity in the ith period, where i = 1,2,3, ...
Similar meanings are assigned to q0, q1, ... qi, ... and v0, v1, … vi, …
Capital letters P, Q and V are used to represent the price, quantity, and value index numbers,
respectively. Subscripts attached to P, Q, and V indicates the years compared. Thus,
POI means the price index for period 1 relative to period 0,
P02 means the price index for period 2 relative to period 0,
PI2 means the price index for period 2 relative to period 1, and so on.
Similar meanings are assigned to quantity Q and value V indices. It may be noted that all indices are
expressed in percent with 100 as the index for the base period, the period with which comparison is to be
made.
Various indices can also be distinguished on the basis of the number of commodities that go
into the construction of an index. Indices constructed for individual commodities or variable
are termed as simple index numbers. Those constructed for a group of commodities or
variables are known as aggregative (or composite) index numbers.
183
Caritas Business College BUSINESS STATISTICS – OAD 221
Here, in this lesson, we will develop methods of constructing simple as well as composite
indices.
6.5 SIMPLE INDEX NUMBERS
A simple price index number is based on the price or quantity of a single commodity. To
construct a simple index, we first have to decide on the base period and then find ratio of the
value at any subsequent period to the value in that base period - the price/quantity relative.
This ratio is then finally converted to a percentage
Value in Period i
Index for any Period i x100
Value in Base
Year
i.e. Simple Price Index for period i = 1,2,3 ... will be
P0i
pp x10 …………(6-1)
0
0
Similarly, Simple Quantity Index for period i = 1,2,3 ... will be
Q0i x10 …………(6-2)
qi 0
q
0
Example 6-1
Given are the following price-quantity data of fish, with price quoted in Rs per kg and production in
qtls.
Year : 1980 1981 1982 1983 1984 1985
Price : 15 17 16 18 22 20
Production : 500 550 480 610 650 600
Construct:
(a) the price index for each year taking price of 1980 as base,
(b) the quantity index for each year taking quantity of 1980 as base.
Solution:
Simple Price and Quantity Indices of
Fish (Base Year = 1980)
Year Price Quantity Price Index Quantity Index
pi qi
(pi) (qi) P x100 Q x100
0i 0i
p0 q0
184
Caritas Business College BUSINESS STATISTICS – OAD 221
1980 15 500 100.00 100.00
1981 17 550 113.33 110.00
1982 16 480 106.66 96.00
1983 18 610 120.00 122.00
1984 22 650 146.66 130.00
1985 20 600 133.33 120.00
These simple indices facilitate comparison by transforming absolute quantities/prices into
percentages. Given such an index, it is easy to find the percent by which the price/quantity
may have changed in a given period as compared to the base period. For example,
observing the index computed in Example 6-1, one can firmly say that the output of fish was
30 per cent more in 1984 as compared to 1980.
It may also be noted that given the simple price/quantity for the base year and the index for
the period i = 1, 2, 3, …; the actual price/quantity for the period i = 1, 2, 3, … may easily be
obtained
as: p …………(6-3)
P0i
i 0
100
Q0i
and q q …………(6-4)
i
100
0
For example, with i = 1983, Q0i = 122.00, and q0 = 500,
122.00
q
i
500 100
= 610
6.6 COMPOSITE INDEX NUMBERS
The preceding discussion was confined to only one commodity. What about
price/quantity changes in several commodities? In such cases, composite index
185
Caritas Business College BUSINESS STATISTICS – OAD 221
numbers are used. Depending upon the method used for constructing an index, composite
indices may be:
1. Simple Aggregative Price/ Quantity Index
2. Index of Average of Price/Quantity Relatives
3. Weighted Aggregative Price/ Quantity Index
4. Index of Weighted Average of Price/Quantity Relatives
6.6.1 SIMPLE AGGREGATIVE PRICE/ QUANTITY INDEX
Irrespective of the units in which prices/quantities are quoted, this index for given
prices/quantities, of a group of commodities is constructed in the following three steps:
(i) Find the aggregate of prices/quantities of all commodities for each period (or
place).
(ii) Selecting one period as the base, divide the aggregate prices/quantities
corresponding to each period (or place) by the aggregate of prices/
quantities in the base period.
(iii) Express the result in percent by multiplying by 100.
The computation procedure contained in the above steps can be expressed as:
pi
P x10 …………(6-5)
0
0i
p 0
qi
and Q x100 …………(6-6)
0i
q 0
Example 6-2
Given are the following price-quantity data, with price quoted in Rs per kg and
production in qtls.
1980 1985
Item Price Production Price Production
Fish 15 500 20 600
Mutton 18 590 23 640
186
Caritas Business College BUSINESS STATISTICS – OAD 221
Chicken 22 450 24 500
Find (a) Simple Aggregative Price Index with 1980 as the base.
(b) Simple Aggregative Quantity Index with 1980 as the base.
Solution:
Calculations for
Simple Aggregative Price and Quantity Indices
(Base Year = 1980)
Item Prices Quantities
1980(p0) 1985(qi)
1985(pi) 1980(q0)
Fish 15 20 500 600
Mutton 18 23 590 640
Chicken 22 24 450 500
Sum → 55 67 1540 1740
(a) Simple Aggregative Price Index with 1980 as the base
pi
P x100
0i
p 0
P0i 67
x100
55
P0i 121.82
(b) Simple Aggregative Quantity Index with 1980 as the base
qi
Q x100
0i
q 0
Q0i 1740
x100
1540
Q0i 112.98
Although Simple Aggregative Index is simple to calculate, it has two important limitations:
First, equal weights get assigned to every item entering into the construction of this index
irrespective of relative importance of each individual item being different. For example,
187
Caritas Business College BUSINESS STATISTICS – OAD 221
items like pencil and milk are assigned equal importance in the construction of this index.
This limitation renders the index of no practical utility.
Second, different units in which the prices are quoted also sometimes unduly affect
this index. Prices quoted in higher weights, such as price of wheat per bushel as compared
to a price per kg, will have unduly large influence on this index. Consequently, the prices of
only a few commodities may dominate the index. This problem no longer exists when the
units in which the prices of various commodities are quoted have a common base.
Even the condition of common base will provide no real solution because commodities with
relatively high prices such as gold, which is not as important as milk, will continue to
dominate this index excessively. For example, in the Example 6-2 given above chicken
prices are relatively higher than those of fish, and hence chicken prices tend to influence this
index relatively more than the prices of fish.
6.6.2 INDEX OF AVERAGE OF PRICE/QUANTITY RELATIVES
This index makes an improvement over the index of simple aggregative prices/quantities as
it is not affected by the difference in the units of measurement in which prices/quantities are
expressed. However, this also suffers from the problem of equal importance getting
assigned to all the commodities.
Given the prices/quantities of a number of commodities that enter into the construction of this
index, it is computed in the following two steps:
(i) After selecting the base year, find the price relative/quantity relative of each
commodity for each year with respect to the base year price/quantity. As
defined earlier, the price relative/quantity relative of a commodity for a given
period is the ratio of the price/quantity of that commodity in the given period
to its price/quantity in the base period.
188
Caritas Business College BUSINESS STATISTICS – OAD 221
(ii) Multiply the result for each commodity by 100, to get simple price/quantity
indices for each commodity.
(iii) Take the average of the simple price/quantity indices by using arithmetic
mean, geometric mean or median.
Thus it is computed as:
P0i Average of p x100
p
0
and Q0i Average of q x100
0
Using arithmetic
mean
x100
pi
p
P0i 0 …………(6-7)
N
qi x100
q
and Q0i 0 …………(6-8)
N
Using geometric mean
pi
1
…………(6-9)
P0i Anti log log
x100 N p
0
1 qi
and …………(6-10)
Anti log log
Q0i
x100 N q
0
Example 6-3
From the data in Example 6.2 find:
(a) Index of Average of Price Relatives (base year 1980); using mean, median and
geometric mean.
(b) Index of Average of Quantity Relatives (base year 1980); using mean, median
189
Caritas Business College BUSINESS STATISTICS – OAD 221
and geometric mean.
Solution:
190
Caritas Business College BUSINESS STATISTICS – OAD 221
Calculations for
Index of Average of Price Relatives and Quantity Relatives
(Base Year = 1980)
Price Relative Quantity Relative
pi pi qi qi
Item x100 log x100 x100 log
x100
p0 p0 q0
q0
Fish 133.33 2.1248 120.00 2.0792
Mutton 127.77 2.1063 108.47 2.0354
Chicken 109.09 2.0378 111.11 2.0457
Sum → 370.19 6.2689 339.58 6.1603
(a) Index of Average of Price Relatives (base year 1980)
pi x100
p
Using arithmetic mean P 0
0i N
370.19
3
123.39
Using Median P Size of N 1
0i
th item
2
3 1
Size of
th item
2
Size of 2nd item
127.77
1 pi
Using geometric mean
P0i Anti log log x100
N p
0
1
Anti log 6.2689
3
Anti log2.08963
122.92
(b) Index of Average of Quantity Relatives (base year 1980)
191
Caritas Business College BUSINESS STATISTICS – OAD 221
qi x100
q
Using arithmetic mean Q 0
0i N
339.58
3
113.19
Using Median Q Size of N 1
0i
th item
2
3 1
Size of
th item
2
Size of 2nd item
111.11
1 qi
Using geometric mean
Q0i Anti log log x100
N 0q
1
Anti log 6.1603
3
Anti log2.05343
113.09
Apart from the inherent drawback that this index accords equal importance to all items
entering into its construction, a simple arithmetic mean and median are not appropriate
average to be applied to ratios. Because it is generally believed that a simple average
injects an upward bias in the index. So geometric mean is considered a more appropriate
average for ratios and percentages.
6.6.3 WEIGHTED AGGREGATIVE PRICE/QUANTITY INDICES
We have noted that the simple aggregative price/quantity indices do not take care of the
differences in the weights to be assigned to different commodities that enter into their
construction. It is primarily because of this limitation that the simple aggregative indices are
of very limited use. Weighted aggregative Indices make up this deficiency by assigning
proper weights to individual items.
192
Caritas Business College BUSINESS STATISTICS – OAD 221
Among several ways of assigning weights, two widely used ways are:
(i) to use base period quantities/prices as weights, popularly known as
Laspeyre's Index, and
(ii) to use the given (current) period quantities/prices as weights, popularly
known as Paasche's Index.
6.6.3.1 Laspeyre’s Index
Laspeyre’s Price Index, using base period quantities as weights is obtained as
La pi q0 x10 …………(6-11)
P 0
0i
p q0 0
Laspeyre’s Quantity Index, using base period prices as weights is obtained as
La qi p0 x10 …………(6-12)
Q 0
0i
q p0 0
6.6.3.2 Paasche’s Index
Paasche’s Price Index, using base period quantities as weights is obtained as
Pa pi
P x10 …………(6-13)
qi 0
0i
p q0 i
Paasche’s Quantity Index, using base period prices as weights is obtained as
Pa qi pi x10 …………(6-14)
Q 0
0i
q p0 i
Example 6-4
From the data in Example 6.2 find:
(a) Laspeyre’s Price Index for 1985, using 1980 as the base
(b) Laspeyre’s Quantity Index for 1985, using 1980 as the base
(c) Paasche’s Price Index for 1985, using 1980 as the base
193
Caritas Business College BUSINESS STATISTICS – OAD 221
(d) Paasche’s Quantity Index for 1985, using 1980 as the base
Solution:
Calculations for
Laspeyre’s and Paasche’s
Indices
(Base Year = 1980)
Item p0 q0 p1 q0 p0 q1 p1 q1
Fish 7500 10000 9000 12000
Mutton 10620 13570 11520 14720
Chicken 9900 10800 11000 12000
Sum → 28020 34370 31520 38720
(a) Laspeyre’s Price Index for 1985, using 1980 as the base
La pi q0 x100
P
0i
p q0 0
34370
28020 x100
122.66
(b) Laspeyre’s Quantity Index for 1985, using 1980 as the base
La qi p0 x100
Q
0i
q 0p 0
31520
28020 x100
112.49
(c) Paasche’s Price Index for 1985, using 1980 as the base
Pa pi
P x100
qi
0i
p q0 i
38720
31520 x100
194
Caritas Business College BUSINESS STATISTICS – OAD 221
122.84
(d) Paasche’s Quantity Index for 1985, using 1980 as the base
195
Caritas Business College BUSINESS STATISTICS – OAD 221
Pa qi pi x100
Q
0i
q 0p i
38720
34370 x100
112.66
Interpretations of Laspeyre's Index
On close examination it will be clear that the Laspeyre's Price Index offers the following
precise interpretations:
1. It compares the cost of collection of a fixed basket of goods selected in the base
period with the cost of collecting the same basket of goods in the given (current)
period.
Accordingly, the cost of collection of 500 qtls of fish, 590 qtls of mutton and 450 qtls
of chicken has increased by 22.66 per cent in 1985 as compared to what it was in
1980. Viewed differently, it indicates that a fixed amount of goods sold at 1985
prices yield 22.66 per cent more revenue than what it did at 1980 prices.
2. It also implies that a fixed amount of goods when purchased at 1985 prices would
cost 22.66 per cent more than what it did at 1980 prices. In this interpretation, the
Laspeyre's Price Index serves as the basis of constructing the cost of living
index, for it tells how much more does it cost to maintain the base period standard
of living at the current period prices.
Laspeyre's Quantity Index, too, has precise interpretations. It reveals the percentage change
in total expenditure in the given (current) period as compared to the base period if varying
amounts of the same basket of goods are sold at the base period prices. When viewed in
this manner, we will be required to spend 12.49 per cent more in 1985 as compared to 1980
if the quantities of fish, mutton and chicken for 1965 are sold at the base period (1980)
prices.
196
Caritas Business College BUSINESS STATISTICS – OAD 221
Interpretations of Paasche's Index
A careful examination of the Paasche's Price Index will show that this too is amenable to the
following precise interpretations:
1. It compares the cost of collection of a fixed basket of goods selected in the given
period with the cost of collection of the same basket of goods in the base period.
Accordingly, the cost of collection of a fixed basket of goods containing 600 qtls of
fish, 640 qtls of mutton and 500 qtls of chicken in 1985 is about 22.84 per cent
more than the cost of collecting the same basket of goods in 1980. Viewed a little
differently, it indicates that a fixed basket of goods sold at 1985 prices yields 22.84
per cent more revenue than what it would have earned had it been sold at the base
period (1980) prices.
2. It also tells that a fixed amount of goods purchased at 1985 prices will cost
22.84 per cent more than what it would have cost if this fixed amount of goods had
been sold at base period (1980) prices.
Analogously, Paasche's Quantity Index, too, has its own precise meaning. It tells the per
cent change in total expenditure in the given period as compared to the base period if
varying amounts of the same basket of goods are to be sold at given period prices. When so
viewed, we will be required to spend 12.66 per cent more in 1985 as compared to
1980 if the quantities of fish, mutton and chicken for 1980 are sold at the given period
(1985) prices.
Relationship Between Laspeyre’s and Paasche’s Indices
In order to understand the relationship between Laspeyre’s and Paasche’s Indices, the
assumptions on which the two indices are based be borne in mind:
Laspeyre's index is based on the assumption that unless there is a change in tastes and
preferences, people continue to buy a fixed basket of goods irrespective of how high or low
197
Caritas Business College BUSINESS STATISTICS – OAD 221
the prices are likely to be in the future. Paasche's index, on the other hand, assumes that
people would have bought the same amount of a given basket of goods in the past
irrespective of how high or low were the past prices.
However, the basic contention implied in the assumptions on which the two indices are
based is not true. For, people do make shifts in their purchase pattern and preferences by
buying more of goods that tend to become cheaper and less of those that tend to become
costlier. In view of this, the following two situations that are likely to emerge need
consideration:
1. When the prices of goods that enter into the construction of these indices show a
general tendency to rise, those whose prices increase more than the average
increase in prices will have smaller quantities in the given period than the
corresponding quantities in the base period. That is, qi’s will be smaller than q0’s
when prices in general are rising. Consequently, Paasche's index will have relatively
smaller weights
than those in the Laspeyre's index and, therefore, the former ( P Pa ) will be smaller
0
La
than the latter ( P ). In other words, Paasche's index will show a relatively smaller
0
increase when the prices in general tend to rise.
2. On the contrary, when prices in general are falling, goods whose prices show a
relatively smaller fall than the average fall in prices, will have smaller quantities in
the given period than the corresponding quantities in the base period. This means
that qi’s will be smaller than q0’s when prices in general are falling. Consequently,
Paasche's index will have smaller weights than those in the Laspeyre's index
and,
therefore, the former ( P Pa ) will be smaller than the latter ( P La ). In other words,
0i 0i
Paasche's index will show a relatively greater fall when the prices in general tend
to fall.
An important inference based on the above discussion is that the Paasche's index has a
downward bias and the Laspeyre's index an upward bias. This directly follows from the fact
198
Caritas Business College BUSINESS STATISTICS – OAD 221
that the Paasche's index, relative to the Laspeyre's index, shows a smaller rise when the
prices in general are rising, and a greater fall when the prices in general are falling.
It may, however, be noted that when the quantity demanded increases because of change in
real income, tastes and preferences, advertising, etc., the prices remaining unchanged, the
Paasche's index will show a higher value than the Laspeyre's index. In such situations, the
Paasche's index will overstate, and the Laspeyre's will understate, the changes in prices.
The former now represents the upper limit, and the latter the lower limit, of the range of price
changes.
The relationship between the two indices can be derived more precisely by making use of
the coefficient of linear correlation computed as:
fXY fX fY
N N
N
…………(6.15)
r
xy
Sx S y
pi
in which X and Y denote the relative price ) and relative quantity
movements( p0
qi
movements( ) respectively. Sx and Sy are the standard deviations of price and quantity
q0
movements, respectively. While rxy represents the coefficient of correlation between the
relative price and quantity movements; f represents the weights assigned, that is, p0 q0. N
is
the sum of frequencies i. e. N = p0 q0 .
Substituting the values of X, Y, f and N in Eq. (6-15), and then rearranging the expression, we
have
rxy Sx pi qi pi q0 p0 qi
x
Sy p q0 0 p q0 0 p q0 0
199
Caritas Business College BUSINESS STATISTICS – OAD 221
pi qi
If , is the index of value expanded between the base period and the ith period,
p q0 0 V0i
then dividing both sides by pi qi or , we get
V0i
p q0 0
rxy Sx
pi
Sy 1 p0 qi
q0 x
V0i pi qi
p0
q0
rxy Sx
1
S y 1 P La x
0
P0iP
V0i
La
P0i 1 rxy S x
…………(6.16)
P P Sy
0i
V0i
The relationship in Eq. (6.16) offers the following useful results:
La
1 P when either rxy , Sx and Sy is equal to zero. That is, the two indices will
Pa
0i 0i
give the same result either when there is no correlation between the price and
quantity movements, or when the price or quantity movements are in the same ratio
so that Sx or Sy is equal to zero.
2. Since in actual practice rxy will have a negative value between 0 and -1, and as
neither Sx = 0 nor Sy = 0, the right hand side of Eq. (6-16) will be less than 1.
This
means that P La is normally greater than PPa
0 0
3. Given the overall movement in the index of value ( V0i ) expanded, the greater the
coefficient of correlation (rxy) between price and quantity movements and/or the
200
P
Caritas Business College BUSINESS STATISTICS – OAD 221
greater the degree of dispersion (Sx and Sy) in the price and quantity movements,
the
greater the discrepancy La Pa
P and
between
4. The longer the time interval between the two periods to be compared, the more the
chances for price and quantity movements leading to higher values of Sx and Sy.
The assumption of tastes, habits, and preferences remaining unchanged
breaking down
201
Caritas Business College BUSINESS STATISTICS – OAD 221
over a longer period, people do find enough time to make shifts in their
consumption pattern, buying more of goods that may have become relatively
cheaper and less of those that may have become relatively dearer. All this will end
up with a higher degree of correlation between the price and quantity
movement.
0 P0
Consequently P La will diverge from Pa
more in the long run than in the short run.
,
So long as the periods to be compared are not much apart, P La will be quite close to
0
PPa
0i
Laspeyre’s and Paasche’s Indices Further Considered
The use of different system of weights in these two indices may give an impression as if they
are opposite to each other. Such an impression is not sound because both serve the same
purpose, although they may give different results when applied to the same data.
This raises an important question. Which one of them gives more accurate results and which
one should be preferred over the other? The answer to this question is rather difficult since
both the indices are amenable to precise and useful results.
Despite a very useful and precise difference in interpretation, in actual practice the
Laspeyre's index is used more frequently than the Paasche's index for the simple reason
that the latter requires frequent revision to take into account the yearly changes in weights.
No such revision is required in the case of the Laspeyre's index where once the weights
have been determined, these do not require any change in any subsequent period. It is on
this count that the Laspeyre's index is preferred over the Paasche's index.
However, this does not render the Paasche's index altogether useless. In fact, it
supplements the practical utility of the Laspeyre's index. The fact that the Laspeyre's index
has an upward bias and the Paasche's index downward bias, the two provide the range
between which the index can vary between the base period and the given period.
Interestingly, thus, the former represents the upper limit, and the latter the lower limit.
202
Caritas Business College BUSINESS STATISTICS – OAD 221
6.6.3.3 Improvements over the Laspeyre’s and Paasche’s Indices
To overcome the difficulty of overstatement of changes in prices by the Laspeyre's index and
understatement by the Paasche's index, different indices have been developed to
compromise and improve upon them. These are particularly useful when the given period
and the base period fall quite apart and result in a greater divergence between Laspeyre's
and Paasche's indices.
Other important Weighted Aggregative Indices are:
1. Marshall-Edgeworth Index
The Marshall-Edgeworth Index uses the average of the base period and given
period quantities/prices as the weights, and is expressed as
q0 qi
p
i 2
ME
P0i …………(6-17)
q x100
q
0 i
p 0
2
p0 pi
q
i 2
ME
Q0i …………(6-18)
p x100
p
0 i
q 0
2
2. Dorbish and Bowley Index
The Dorbish and Bowley Index is defined as the arithmetic mean of the
Laspeyre’s and Paasche’s indices.
La Pa
DBP P
P 0i 0i …………(6-19)
0i
2
La Pa
DB Q i Q0i
0 …………(6-20)
Q
0i
2
3. Fisher’s Ideal Index
203
Caritas Business College BUSINESS STATISTICS – OAD 221
The Fisher’s Ideal Index is defined as the geometric mean of the Laspeyre’s and
Paasche’s indices.
F
P0 ..........................................................................(6-21)
P0i0i
La .PPa
Q0 ..........................................................................(6-22)
F
Q0i0i
La .QPa
6.6.4 INDEX OF WEIGHTED AVERAGE OF PRICE/QUANTITY RELATIVES
An alternative system of assigning weights lies in using value weights. The value weight v for
any single commodity is the product of its price and quantity, that is, v = pq.
If the index of weighted average of price relatives is defined as
pi
v x100
p
0
P0i …………(6-23)
v
then v can be obtained either as
(i) the product of the base period prices and the base period quantities denoted as
v0 that is, v0 = p0 q0 , or
(ii) the product of the base period prices and the given period quantities
denoted as vi that is, vi = p0 qi
When v is v0 = p0 q0 , the index of weighted average of price relatives, is expressed as
pi
p0 q0
p x100
0
0 P0i …………(6-24)
p q0 0
It may be seen that 0 P0i is the same as the Laspeyre’s aggregative price index.
Similarly, When v is vi = p0 qi , the index of weighted average of price relatives, is
expressed as
204
Caritas Business College BUSINESS STATISTICS – OAD 221
pi
p0 qi
p x100
0
i P0i …………(6-25)
p q0 i
It may be seen that i P0i is the same as the Paasche’s aggregative price
index. If the index of weighted average of quantity relatives is defined as
qi
v x100
Q0i q 0
…………(6-26)
v
then v can be obtained either as
(i) the product of the base period quantities and the base period prices denoted as
v0 that is, v0 = q0 p0 , or
(ii) the product of the base period quantities and the given period prices
denoted as vi that is, vi = q0 pi
When v is v0 = q0 p0 , the index of weighted average of quantity relatives, is expressed as
qi
q 0 p 0
q x100
0 …………(6-27)
Q
0 0i q0 p0
It may be seen that 0 Q0i is the same as the Laspeyre’s aggregative quantity index.
Similarly, When v is vi = vi = q0 pi , the index of weighted average of quantity relatives, is
expressed as
qi
q 0 pi x100
q 0 …………(6-28)
Q
i 0i q0 pi
205
Caritas Business College BUSINESS STATISTICS – OAD 221
It may be seen that i Q0i is the same as the Paasche’s aggregative quantity index.
Example 6-5
From the data in Example 6.2 find the:
206
Caritas Business College BUSINESS STATISTICS – OAD 221
(a) Index of Weighted Average of Price Relatives, using
(i) v0 = p0 q0 as the value weights
(ii) vi = p0 qi as the value weights
(b) Index of Weighted Average of Quantity Relatives, using
(i) v0 = q0 p0 as the value weights
(ii) vi = q0 pi as the value weights
Solution:
Calculations for
Index of Weighted Average of Price
Relatives (Base Year = 1980)
pi p1
p q p q
Item v0 = p0 q0 v1 = p0 q1
x100 x100
0 0 0 1
0p 0p
Fish 7500 9000 1000000 1200000
Mutton 10620 11520 1357000 1472000
Chicken 9900 11000 1080000 1200000
Sum → 28020 31520 3437000 3872000
(a) Index of Weighted Average of Price Relatives, using
(i) v0 = p0 q0 as the value weights
pi
p0 q0
p x100
0
0 P0i
p 0q 0
3437000
28020
122.66
(ii) vi = p0 qi as the value weights
pi
p0 qi
p x100
207
Caritas Business College BUSINESS STATISTICS – OAD 221
0
i P0i
p q0 i
208
Caritas Business College BUSINESS STATISTICS – OAD 221
3872000
31520
122.84
Calculations for
Index of Weighted Average of Quantity Relatives
(Base Year = 1980)
q1 q1
q p q p
Item v0 = q0 p0 v1 = q0 p1
x100 x100
0 0 0 1
0q 0q
Fish 7500 10000 900000 1200000
Mutton 10620 13570 1152000 1472000
Chicken 9900 10800 1100000 1200000
Sum → 28020 34370 3152000 3872000
(b) Index of Weighted Average of Quantity Relatives, using
(i) v0 = q0 p0 as the value weights
qi
q 0 p 0
q x100
0
Q
0 0i q0 p0
3152000
28020
112.49
(ii) vi = q0 pi as the value weights
qi
q 0 pi
q x100
0
Q
i 0i q0 pi
3872000
34370
112.66
Although the indices of weighted average of price/quantity relatives yield the same results
as the Laspeyre's or Paasche's price/quantity indices, we do construct these indices
209
Caritas Business College BUSINESS STATISTICS – OAD 221
also in
210
Caritas Business College BUSINESS STATISTICS – OAD 221
situations when it is necessary and advantageous to do so. Some such situations are
as follows:
(i) When a group of commodities is to be represented by a single commodity in
the group, the price relative of the latter is weighted by the group as a whole.
(ii) Where the price/quantity relatives of individual commodities have been
computed, these can be more conveniently utilised in constructing the index.
(iii) Price/quantity relatives serve a useful purpose in splicing two index series
having different base periods.
(iv) Depersonalizing a time series requires construction of a seasonal index,
which also requires the use of relatives.
6.7 TESTS OF ADEQUACY OF INDEX NUMBERS
We have discussed various formulae for the construction of index numbers. None of the
formulae measures the price changes or quantity changes with perfection and has some
bias. The problem is to choose the most appropriate formula in a given situation. As a
measure of the formula error a number of mathematical tests, known as the tests of
consistency or tests of adequacy of index number formulae have been suggested. In this
section we will discuss these tests, which are also sometimes termed as the criteria for a
good index number.
1. Unit Test: This test requires that the index number formula should be independent of
the units in which the prices or quantities of various commodities are quoted. All the
formulae discussed in the lesson except the index number based on Simple
Aggregate of Prices/Quantities satisfy this test.
2. Time Reversal Test: The time reversal test, proposed by Prof Irving Fisher requires
the index number formula to possess time consistency by working both forward and
backward w.r.t. time. In his (Fisher’s) words:
211
Caritas Business College BUSINESS STATISTICS – OAD 221
“The formula for calculating an index number should be such that it
gives the same ratio between one point of comparison and the other, no
matter which of the two is taken as the base or putting it another way,
the index number reckoned forward should be reciprocal of the one
reckoned backward.”
In other words, if the index numbers are computed for the same data relating to two
periods by the same formula but with the bases reversed, then the two index
numbers so obtained should be the reciprocals of each other. Mathematically, we
should have
(omitting the factor 100),
Pab xPba …………(6-29)
1 or more generally
P01 xP10 …………(6-29a)
1
Time reversal test is satisfied by the following index number formulae:
(i) Marshall-Edgeworth formula
(ii) Fisher’s Ideal formula
(iii) Kelly’s fixed weight formula
(iv) Simple Aggregate index
(v) Simple Geometric Mean of Price Relatives formula
(vi) Weighted Geometric Mean of Price Relatives formula with fixed weights
Lespeyre’s and Pasche’s index numbers do not satisfy the time reversal test.
3. Factor Reversal Test: This is the second of the two important tests of consistency
proposed by Prof Irving Fisher. According to him:
“Just as our formula should permit the interchange of two times without
giving inconsistent results, so it ought to permit interchanging the price
and quantities without giving inconsistent results – i.e., the two results
multiplied together should give the true value ratio, except for a constant
of proportionality.”
212
Caritas Business College BUSINESS STATISTICS – OAD 221
This implies that if the price and quantity indices are obtained for the same data,
same base and current periods and using the same formula, then their product
(without the
213
Caritas Business College BUSINESS STATISTICS – OAD 221
factor 100) should give the true value ratio. Symbolically, we should have (without
factor 100).
p1 q1
P xQ V …………(6-30)
01 01
p0 q0 0
Fisher’s formula satisfies the factor reversal test. In fact fisher’s index is the only
index satisfying this test as none of the formulae discussed in the lesson satisfies this
test.
Remark: Since Fisher’s index is the only index that satisfies both the time reversal
and factor reversal tests, it is termed as Fisher’s Ideal Index.
4. Circular Test: Circular test, first suggested by Westergaard, is an extension of time
reversal test for more than two periods and is based on the shift ability of the base
period. This requires the index to work in a circular manner and this property enables
us to find the index numbers from period to period without referring back to the
original base each time. For three periods a,b,c, the test requires :
Pab xPbc ab …………(6-31)
xPca 1 c
In the usual notations Eq. (6-31) can be stated
as:
…………(6-31a)
P01 xP12 xP20
1
For Instance
p2 q1 p0 q2
La La La x x 1
P xP xP
p1q0
01 12 0 0 1 1 2 2
21 p q
pq p q
Hence Laspeyre’s index does not satisfy the circular test. In fact, circular test is not
satisfied by any of the weighted aggregative formulae with changing weights. This
test is satisfied only by the index number formulae based on:
(i) Simple geometric mean of the price relatives, and
214
Caritas Business College BUSINESS STATISTICS – OAD 221
(ii) Kelly’s fixed base method
215
Caritas Business College BUSINESS STATISTICS – OAD 221
6.8 SPECIAL ISSUSES IN THE CONSTRUCTION OF INDEX NUMBERS
6.8.1 BASE SHIFTING
The need for shifting the base may arise either
(i) when the base period of a given index number series is to be made
more recent, or
(ii) when two index number series with different base periods are to be
compared, or
(iii) when there is need for splicing two overlapping index number series.
Whatever be the reason, the technique of shifting the base is simple:
New Base Index Number Old Index Number of Current Year
x100
Old Index Number of New Base
Year
Example 6-6
Reconstruct the following indices using 1997 as base:
Year : 1991 1992 1993 1994 1995 1996 1997 1998
Index : 100 110 130 150 175 180 200 220
Solution:
Shifting the Base Period
Index Number Index Number
Year
(1991 = 100) (1997 = 100)
1991 100 (100/200) x100 = 50.00
1992 110 (110/200) x100 = 55.00
1993 130 (130/200) x100 = 65.00
1994 150 (150/200) x100 = 75.00
1995 175 (175/200) x100 = 87.50
1996 180 (180/200) x100 = 90.00
1997 200 (200/200) x100 = 100.00
1998 220 (220/200) x100 = 110.00
216
Caritas Business College BUSINESS STATISTICS – OAD 221
6.8.2 SPLICING TWO OVERLAPPING INDEX NUMBER SERIES
Splicing two index number series means reducing two overlapping index series with different
base periods into a single series either at the base period of the old series (one with an old
base year), or at the base period of the new series (one with a recent base year). This
actually amounts to changing the weights of one series into the weights of the other series.
1. Splicing the New Series to Make it Continuous with the Old Series
Here we reduce the new series into the old series after the base year of the former.
As shown in Table 6.8.2(i), splicing here takes place at the base year (1980) of the
new series. To do this, a ratio of the index for 1980 in the old series (200) to the
index of 1980 in the new series (100) is computed and the index for each of the
following years in the new series is multiplied by this ratio.
Table 6.8.2(i)
Splicing the New Series with the Old Series
Price Index Price Index
Year Spliced Index Number
(1976 = 100) (1980 = 100)
[New Series x (200/100)]
(Old Series) (New Series)
1976 100 -- 100
1977 120 -- 120
1978 146 -- 146
1979 172 -- 172
1980 200 100 200
1981 -- 110 220
1982 -- 116 232
1983 -- 125 250
1984 -- 140 280
2. Splicing the Old Series to Make it Continuous with the New Series
217
Caritas Business College BUSINESS STATISTICS – OAD 221
This means reducing the old series into the new series before the base period of the
letter. As shown in Table 6.8.2(ii), splicing here takes place at the base period of the
new series. To do this, a ratio of the index of 1980 of the new series (100) to the
index of 1980 of the old series (200) is computed and the index for each of the
preceding years of the old series are then multiplied by this ratio.
Table 6.8.2(ii)
Splicing the Old Series with the New Series
Price Index Price Index
Year Spliced Index Number
(1976 = 100) (1980 = 100)
[Old Series x (100/200)]
(Old Series) (New Series)
1976 100 -- 50
1977 120 -- 60
1978 146 -- 73.50
1979 172 -- 86
1980 200 100 100
1981 -- 110 110
1982 -- 116 116
1983 -- 125 125
1984 -- 140 140
6.8.3 CHAIN BASE INDEX NUMBERS
The various indices discussed so far are fixed base indices in the sense that either the base
year quantities/prices (or the given year quantities/prices) are used as weights. In a dynamic
situation where tastes, preferences, and habits are constantly changing, the weights should
be revised on a continuous basis so that new commodities are included and the old ones
deleted from consideration.
This is all the more necessary in a developing society where new substitutes keep replacing
the old ones, and completely new commodities are entering the market. To take care of such
changes, the base year should be the most recent, that is, the year immediately preceding
the
218
Caritas Business College BUSINESS STATISTICS – OAD 221
given year. This means that as we move forward, the base year should move along the
given year in a chain year after year.
Conversion of Fixed-base Index into Chain-base Index
As shown in Table 6.8.3(i), to convert fixed-base index numbers into chain-base index
numbers, the following procedure is adopted:
The first year's index number is taken equal to 100
For subsequent years, the index number is obtained by following formula:
Current Year's FBI
Current Year's CBI
Previous Year's CBI x 100
Table 6.8.3(i)
Conversion of Fixed-base Index into Chain-base Index
Fixed Base Index Chain Base
Year Conversion
Number Index Number
(FBI) (CBI)
1975 376 -- 100
1976 392 (392/376) x100 104.3
1977 408 (408/392) x100 104.1
1978 380 (380/408) x100 93.1
1979 392 (392/380) x100 103.2
1980 400 (400/392) x100 102
Conversion of Chain-base Index into Fixed-base Index
As shown in Table 6.8.3(ii), to convert fixed-base index numbers into chain-base index
numbers, the following procedure is adopted:
The first year's index is taken what the chain base index is; but if it is to form the
base it is taken equal to 100
In subsequent years, the index number is obtained by following formula:
Current Year's CBI x Previous Year's FBI
Current Year's FBI 100
219
Caritas Business College BUSINESS STATISTICS – OAD 221
Table 6.8.3(ii)
Conversion of Chain-base Index into Fixed-base Index
Chain Base Index Fixed Base Index
Year Conversion
Number Number
(CBI) (FBI)
1978 90 -- 90
1979 120 (120 x 90) /100 108
1980 125 (125 x 108) /100 135
1981 110 (110 x 135) /100 148.5
1982 112 (112 x 148.5) /100 166.3
1983 150 (150 x 166.3) /100 249.45
6.9 PROBLEMS OF CONSTRUCTING INDEX NUMBERS
The above discussion enables us to identify some of the important problems, which may be
faced in the construction of index numbers:
1. Choice of the Base Period: Choice of the base period is a critical decision because
of its importance in the construction of index numbers. A base period is the reference
period for describing and comparing the changes in prices or quantities in a given
period. The selection of a base year or period does not pose difficult theoretical
questions. To a large extent, the choice of the base year depends on the objective
of the index. A major consideration should be to ensure that the base year is not an
abnormal year. For example, a base period with very low price/quantity will unduly
inflate, while the one with a very high figure will unduly depress, the entire index
number series. An index number series constructed with any such period as the base
may give very misleading results. It is, therefore, necessary that the base period be
selected carefully.
Another important consideration is that the base year should not be too remote in the
past. A more recent year needs to be selected as the base year. The use of a
particular
220
Caritas Business College BUSINESS STATISTICS – OAD 221
year for a prolonged period would distort the changes that it purports to measure.
That is why we find that the base year of major index numbers, such as consumer
price index or index of industrial production, is shifted from time to time.
2. Selection of Weights to be Used: It should be amply clear from the various indices
discussed in the lesson that the choice of the system of weights, which may be used,
is fairly large. Since any system of weights has its own merits and is capable of
giving results amenable to precise interpretations, the weights used should be
decided keeping in view the purpose for which an index is constructed.
It is also worthwhile to bear in, mind that the use of any system of weights should
represent the relative importance of individual commodities that enter into the
construction of an index. The interpretations that are intended to be made from an
index number are also important in deciding the weights. The use of a system of
weights that involves heavy computational work deserves to be avoided.
3. Type of Average to be Used: What type of average should be used is a problem
specific to simple average indices. Theoretically, one can use any of the several
averages that we have, such as mean, median, mode, harmonic mean, and
geometric mean. Besides being locational averages, median and mode are not the
appropriate averages to use especially where the number of years for which an index
is to be computed, is not large.
While the use of harmonic mean and geometric mean has some definite merits over
mean, particularly when the data to be averaged refer to ratios, mean is
generally more frequently used for convenience in computations.
4. Choice of Index: The problem of selection of an appropriate index arises because of
availability of different types of indices giving different results when applied to the
same data. Out of the various indices discussed, the choice should be in favour of
one
221
Caritas Business College BUSINESS STATISTICS – OAD 221
which is capable of giving more accurate and precise results, and which provides
answer to specific questions for which an index is constructed.
While the Fisher's index may be considered ideal for its ability to satisfy the tests of
adequacy, this too suffers from two important drawbacks. First, it involves too
lengthy computations, and second, it is not amenable to easy interpretations as are
the Laspeyre's and Paasche's indices. The use of the term ideal does not, however,
mean that it is the best to use under all types of situations. Other indices are more
appro- priate under situations where specific answers are needed.
5. Selection of Commodities: Commodities to be included in the construction of
an index should be carefully selected. Only those commodities deserve to be
included in the construction of an index as would make it more representative. This,
in fact, is a problem of sampling, for being related to the selection of commodities to
be included in the sample.
In this context, it is important to note that the selection of commodities must not be
based on random sampling. The reason being that in random sampling every
commodity, including those that are not important and relevant, have equal chance of
being selected, and consequently, the index may not be representative. The choice
of commodities has, therefore, to be deliberate and in keeping with the relevance and
im- portance of each individual commodity to the purpose for which the index is
constructed.
6. Data Collection: Collection of data through a sample is the most important issue
in the construction of index numbers. The data collected are the raw material of
an index. Data quality is the basic factor that determines the usefulness of an index.
The data have to be as accurate, reliable, comparable, representative, and adequate,
as possible.
222
Caritas Business College BUSINESS STATISTICS – OAD 221
The practical utility of an index also depends on how readily it can be constructed.
Therefore, data should be collected from where these can be easily available. While
the purpose of an index number will indicate what type of data are to be collected, it
also determines the source from where the data can be available.
6.10 SELF-ASSESSMENT QUESTIONS
1. “Index Numbers are devices for measuring changes in the magnitude of a group of
related variables”. Discuss this statement and point out the important uses of index
numbers.
2. “Index Numbers are Economic Barometers”. Discuss this statement. What
precautions would you take while constructing index numbers?
3. (a) Explain the uses of index numbers.
(b) What problems are involved in the construction of index numbers?
4. Describe each of the following:
a. Base period
b. Price relatives
c. Fixed-base index numbers
d. Chain-base index numbers
5. Describe briefly the following methods of construction of price index numbers:
a. Simple Aggregate Method
b. Simple Average of Price Relatives Method
c. Weighted Aggregative Method
d. Weighted Average of Price Relatives
6. “Laspeyre’s index has an upward bias and the Paasche’s index downward bias”.
Explain this statement.
7. Discuss the various tests of adequacy of index numbers.
223
Caritas Business College BUSINESS STATISTICS – OAD 221
8. State and explain the Fisher’s ideal formula for price index number. Show how it
satisfies the time-reversal and factor- reversal test? Why is it used little in practice?
9. Briefly explain each of the following:
a. Base-shifting
b. Splicing
c. Deflating
10. From the following data, construct the price index for each year with price of 1995
as base.
Year: 1995 1996 1997 1998 1999 2000
Price of Commodity: 40 50 45 55 65 70
11. From the following data, construct an index number for 2004 taking 2003 as base
year:
Articles: A B C D E
Prices (2003): 100 125 50 40 5
Prices (2004): 140 200 80 60 10
12. Find the index number for 1982 and 1983 taking 1981 as base year by the Simple
Average of Price Relatives Method, using (i) Mean, (ii) Median, and (iii) Geometric
Mean:
Commodities 1981 (Prices) 1982 (Prices) 1983 (Prices)
A 40 55 60
B 50 60 80
C 62 72 93
D 80 88 96
E 20 24 30
13. Construct index number of price and index number of quantity from the following
data using:
224
Caritas Business College BUSINESS STATISTICS – OAD 221
a. Laspeyre’s formula,
b. Paasche’s formula,
c. Dorbish and Bowley’s formula,
d. Marshall and Edgeworth’s formula, and
e. Fisher’s Ideal Index formula
Base Year Current Year
Commodities
Price Quantity Price Quantity
A 2 8 4 6
B 5 10 6 5
C 4 14 5 10
D 2 19 2 13
Which of the formula satisfy
(i) the time reversal test, and
(ii) the factor reversal test?
14. Calculate index number using Kelly’s Method of Standard Weights, from the
following data:
Commodities Quantity Base Year Current Year Price
Price
A 5 30 40
B 8 20 30
C 10 10 20
15. From the following data, construct price index by using Weighted Average of Price
Relatives Method:
Commodities Quantity Base Year Current Year Price
Price
A 6 Qtl 5.00 6.00
B 5 Qtl 5.00 8.00
C 1 Qtl 6.00 9.00
D 4 Kg 8.00 10.00
E 1 Kg 20.00 15.00
225
Caritas Business College BUSINESS STATISTICS – OAD 221
16. From the information given below, calculate the Cost of Living Index number for
1985, with 1984 as base year by
a. Aggregative Expenditure Method, and
b. Family Budget Method.
Items Quantity consumed Unit Prices in 1984 Prices in 1985
Wheat 2 Qtl Qtl 75 125
Rice 20 Kg Kg 12 16
Sugar 10 Kg Kg 12 16
Ghee 5 Kg Kg 10 15
Clothing 25 Meter Meter 4.5 5
Fuel 40 Litre Litre 10 12
Rent One house House 25 40
17. An enquiry into budgets of the middle class families in a city gave the following
information:
Expenses Food Rent Clothing Fuel Miscellaneous
on
40% 10% 20% 10% 20%
→
Prices(2001) 160 50 60 20 50
Prices(2002) 175 60 75 25 75
What changes in the cost of living figure of 2002 have taken place as compared
to 2001?
18. Reconstruct the following indices using 1985 as base:
Year : 1982 1983 1984 1985 1986 1987
Index : 100 120 190 200 212 250
19. Given below are two sets of indices one with 1975 as base and the other with 1979 as
base: First set
Year : 1975 1976 1977 1978 1979
Index Numbers : 100 110 125 180 200
226
Caritas Business College BUSINESS STATISTICS – OAD 221
Second Set
Year : 1979 1980 1981 1982 1983
Index Numbers : 100 104 110 116 124
a. Splice the second set of index numbers from 1975
b. Splice the first set of index numbers from 1979
20. Construct chain index numbers from the following data:
Year : 1991 1992 1993 1994 1995
Price : 25 30 45 60 90
21. Convert into Chain Base Index Number from Fixed Base Index Number
Year : 1980 1981 1982 1983 1984
Fixed Base Index : 100 98 102 140 190
22. From the Chain Base Index numbers given below, construct Fixed
Base Index numbers:
Year : 1993 1994 1995 1996 1997
Chain Base Index : 100 105 95 115 102
23. From the following data, prepare index number for real wages of workers:
Year : 1990 1991 1992 1993 1994 1995
Wages (in Rs) : 300 340 450 460 475 540
Price Index Number : 100 120 220 230 250 300
24. During certain period, the Cost of Living Index number went up from 110 to 200 and
salary of a worker also raised from 325 to 500. State by how much the worker has
gained or lost in real term.
227
Caritas Business College BUSINESS STATISTICS – OAD 221
6.11 SUGGESTED READINGS
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons.,
New Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
228
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 8: ANALYSIS OF TIME SERIES
Topic : Analysis of Time Series
Objective: This lesson would enable you to understand the meaning, importance,
models, and components of time series along with details of
methods of measuring trends.
Structure
7.1. Introduction
7.2. Objectives of time series analysis
7.3. Components of time series
7.4. Time series decomposition models
7.5. Measurement of secular trend
7.6. Seasonal variations
7.7. Measurement of cyclical variations
7.8. Measurement of irregular variations
7.9. Questions
7.10. Suggested readings
7.1. INTRODUCTION
A series of observations, on a variable, recorded after successive intervals of time is called a
time series. The successive intervals are usually equal time intervals, e.g., it can be 10
years, a year, a quarter, a month, a week, a day, and an hour, etc. The data on the
population of India is a time series data where time interval between two successive figures
is 10 years. Similarly figures of national income, agricultural and industrial production, etc.,
are available on yearly basis.
7.2 OBJECTIVES OF TIME SERIES ANALYSIS
The analysis of time series implies its decomposition into various factors that affect the
value of its variable in a given period. It is a quantitative and objective evaluation of the
effects of various factors on the activity under consideration.
There are two main objectives of the analysis of any time series data:
(i) To study the past behaviour of data.
(ii) To make forecasts for future.
The study of past behaviour is essential because it provides us the knowledge of the effects
of various forces. This can facilitate the process of anticipation of future course of events,
and, thus, forecasting the value of the variable as well as planning for future.
7.3 Components of a Time Series
229
Caritas Business College BUSINESS STATISTICS – OAD 221
In the typical time-series there are three main components which seem to be independent
of the and seems to be influencing time-series data.
Trend- It is the broad long-term tendency of either upward or downward movement in the
average (or mean) value of the forecast variable y over time. The rate of trend growth
usually varies over time, as shown in fig 7.1(a) and (b).
Cycles- An upward and downward oscillation of uncertain duration and magnitude about
the trend line due to seasonal effect with fairly regular period or long period with irregular
swings is called a cycle. A business cycle may vary in length, usually greater than one year
but less than 5 to 7 years. The movement is through four phases: from peak (prosperity) to
contradiction (recession) to trough (depression) to expansion (recovery or growth) as
shown in Fig. 7.1 (b) and (c).
Seasonal- It is a special case of a cycle component of time series in which the magnitude
and duration of the cycle do not vary but happen at a regular interval each year. For
example, average sales for a retail store may increase greatly during festival seasons.
Irregular- An irregular or erratic (or residual) movements in a time series is caused by short-
term unanticipated and non-recurring factors. These follow no specific pattern.
7.4 TIME SERIES DECOMPOSITION MODELS
The analysis of time series consists of two major steps:
1. Identifying the various forces (influences) or factors which produce the variations in the
time series, and
2. Isolating, analysing and measuring the effect of these factors separately
and independently, by holding other things constant.
The purpose of decomposition models is to break a time series into its components: Trend
(T), Cyclical (C), Seasonality (S), and Irregularity (I). Decomposition of time series provides
a basis for forecasting. There are many models by which a time series can be analysed;
two models commonly used for decomposition of a time series are discussed below.
7.4.1. Multiplicative Model
This is a most widely used model which assumes that forecast (Y) is the product of the
four components at a particular time period. That is, the effect of four components on the
time series is interdependent.
Y=T x C x S × I Multiplicative model
The multiplicative model is appropriate in situations where the effect of S, C, and I is
measured in relative sense and is not in absolute sense. The geometric mean of S, C, and
I is assumed to be less than one. For example, let the actual sales for period 20 be Y20 =
423.36. Further let, this value be broken down into its components as: let trend component
(mean sales) be 400; effect of current cycle (0.90) is to depress sales by 10 per cent;
seasonality of the series (1.20) boosts sales by 20 per cent. Thus besides the random
fluctuation, the expected value of sales for the period is 400 × 0.90 × 1.20 = 432. If the
random factor depresses sales by 2 per cent in this period, then the actual sales value will
be 432 × 0.98 = 423.36.
7.4.2. Additive Model
In this model, it is assumed that the effect of various components can be estimated by
adding the various components of a time-series. It is stated as:
Y=T + C + S + I Additive model
Here S, C, and I are absolute quantities and can have positive or negative values. It is
assumed that these four components are independent of each other. However, in real-life
time series data this assumption does not hold good.
230
Caritas Business College BUSINESS STATISTICS – OAD 221
7.5. MEASUREMENT OF SECULAR TREND
The principal methods of measuring trend fall into following categories:
1. Free Hand Curve methods
2. Method of Averages
3. Method of least squares
The time series methods are concerned with taking some observed historical pattern for
some variable and projecting this pattern into the future using a mathematical formula.
These methods do not attempt to suggest why the variable under study will take some
future value. This limitation of the time series approach is taken care by the application of a
causal method. The causal method tries to identify factors which influence the variable is
some way or cause it to vary in some predictable manner. The two causal methods,
regression analysis and correlation analysis, have already been discussed previously.
A few time series methods such as freehand curves and moving averages simply describe
the given data values, while other methods such as semi-average and least squares help to
identify a trend equation to describe the given data values.
7.5.1. Freehand Method
A freehand curve drawn smoothly through the data values is often an easy and,
perhaps, adequate representation of the data. The forecast can be obtained simply by
extending the trend line. A trend line fitted by the freehand method should conform to
the following conditions:
(i) The trend line should be smooth- a straight line or mix of long gradual curves.
(ii) The sum of the vertical deviations of the observations above the trend line should
equal the sum of the vertical deviations of the observations below the trend line.
(iii) The sum of squares of the vertical deviations of the observations from the trend
line should be as small as possible.
(iv) The trend line should bisect the cycles so that area above the trend line should
be equal to the area below the trend line, not only for the entire series but as
much as possible for each full cycle.
Example 7.1: Fit a trend line to the following data by using the freehand method.
Year 1991 1992 1993 1994 1995 1996 1997 1998
Sales turnover : 80 90 92 83 94 99 92 104
(Rs. in lakh)
Solution: Figure 7.2
presents the 110 freehand
graph of 105
sales
turnover 100
(Rs. in lakh)
from 1991 to 95
90 1998.
Forecast be
can simply 85
Sal
80 obtained
by the trend extending
1991 1992 1993 1994 1995 1996 1997 1998
Years line.
2
Caritas Business College BUSINESS STATISTICS – OAD 221
Fig. 7.2: Graph of Sales Turnover
Limitations of freehand method
(i) This method is highly subjective because the trend line depends on personal
judgement and therefore what happens to be a good-fit for one individual may not
be so for another.
(ii) The trend line drawn cannot have much value if it is used as a basis for
predictions.
(iii) It is very time-consuming to construct a freehand trend if a careful and
conscientious job is to be done.
7.5.2. Method of Averages
The objective of smoothing methods into smoothen out the random variations due to
irregular components of the time series and thereby provide us with an overall impression of
the pattern of movement in the data over time. In this section, we shall discuss three
smoothing methods.
(i) Moving averages
(ii) Weighted moving averages
(iii) Semi-averages
The data requirements for the techniques to be discussed in this section are minimal and
these techniques are easy to use and understand.
Moving Averages
If we are observing the movement of some variable values over a period of time and trying
to project this movement into the future, then it is essential to smooth out first the irregular
pattern in the historical values of the variable, and later use this as the basis for a future
projection. This can be done by calculating a series of moving averages.
This method is a subjective method and depends on the length of the period chosen for
calculating moving averages. To remove the effect of cyclical variations, the period chosen
should be an integer value that corresponds to or is a multiple of the estimated average
length of a cycle in the series.
The moving averages which serve as an estimate of the next period’s value of a
variable given a period of length n is expressed as:
{Dt Dt1 Dt2. .. Dtn1 }
Moving average, Mat+1 =
n
221
Caritas Business College BUSINESS STATISTICS – OAD 221
where t = current time period
D = actual data which is exchanged each period
n = length of time period
In this method, the term ‘moving’ is used because it is obtained by summing and
averaging the values from a given number of periods, each time deleting the oldest value
and adding a new value.
The limitation of this method is that it is highly subjective and dependent on the length of
period chosen for constructing the averages. Moving averages have the following three
limitations:
(i) As the size of n (the number of periods averaged) increases, it smoothens the
variations better, but it also makes the method less sensitive to real changes in
the data.
(ii) Moving averages cannot pick-up trends very well. Since these are averages, it
will always stay within past levels and will not predict a change to either a higher
or lower level.
(iii) Moving average requires extensive records of past data.
Example 7.2: Using three-yearly moving averages, determine the trend and short-term-error.
Year Production Year Production
(in ‘000 tonnes) (in ‘000 tonnes)
1987 21 1992 22
1988 22 1993 25
1989 23 1994 26
1990 25 1995 27
1991 24 1996 26
Solution: The moving average calculation for the first 3 years is:
21 + 22 + 23
Moving average (year 1-3) = = 22
3
Similarly, the moving average calculation for the next 3 years is:
22 + 23 + 25
Moving average (year 2-4) = = 22.33
3
A complete summary of 3-year moving average calculations is given in Table 7.1
Table 7.1: Calculation of Trend and Short-term Fluctuations
Year Production 3-Year 3-yearly Moving Forecast
Y Moving Average (Trend Error (y- yˆ )
Total values yˆ
1987 21 - - -
1988 22 66 22.00 0
1989 23 70 23.33 -0.33
1990 25 72 24.00 1.00
1991 24 71 23.67 0.33
1992 22 71 23.67 -1.67
1993 25 73 24.33 0.67
222
Caritas Business College BUSINESS STATISTICS – OAD 221
1994 26 78 26.00 0
1995 27 79 26.33 0.67
1996 26 - - -
Odd and Even Number of Years
When the chosen period of length n is an odd number, the moving average at
year i is centred on i, the middle year in the consecutive sequence of n yearly
values used to compute i. For instance with n =5, MA3(5) is centred on the third
year, MA4(5) is centred on the fourth year…, and MA 9(5) is centred on the ninth
year.
No moving average can be obtained for the first (n-1)/2 years or the last (n- 1/2)
year of the series. Thus for a 5-year moving average, we cannot make
computations for the just two years or the last two years of the series.
When the chosen period of length n is an even numbers, equal parts can easily
be formed and an average of each part is obtained. For example, if n = 4, then
the first moving average M 3 (placed at period 3) is an average of the first four
data values, and the second moving average M 4 (placed at period 4) is the
average of data values 2 through 5). The average of M3 and M4 is placed
at period 3 because it is an average of data values for period 1 through 5.
Example 7.3: Assume a four-yearly cycle and calculate the trend by the method of
moving average from the following data relating to the production of tea in India.
Year Production (million Year Production (million
lbs) lbs)
1987 464 1992 540
1988 515 1993 557
1989 518 1994 571
1990 467 1995 586
1991 502 1996 612
Solution: The first 4-year moving average is:
464 + 515 + 518+ 467 1964
MA3(4) = = = 491.00
4 4
This moving average is centred on the middle value, that is, the third year of the
series. Similarly,
515 + 518 + 467+ 502 2002
223
Caritas Business College BUSINESS STATISTICS – OAD 221
MA4(4) = = = 500.50
4 4
This moving average is centred on the fourth year of the series.
Table 7.2. presents the data along with the computations of 4-year moving averages.
Table 7.2: Calculation of Trend and Short-term Fluctuations
Year Production 4-yearly 4-Yearly 4-Yearly Moving
(mm lbs) Moving Totals Moving Average Centred
Average
1987 464 - - -
1988 515 - - -
1964 491.00
1989 518 495.75
2002 500.50
1990 467 503.62
2027 506.75
1991 502 511.62
2066 516.50
1992 540 529.50
2170 542.50
1993 557 553.00
2254 563.50
1994 571 572.00
2326 581.50 -
1995 586 - - -
1996 612 - - -
Weighted Moving Averages
In moving averages, each observation is given equal importance (weight). However,
different values may be assigned to calculate a weighted average of the most recent n
values. Choice of weights is somewhat arbitrary because there is no set formula to
determine them. In most cases, the most recent observation receives the most weightage,
and the weight decreases for older data values.
A weighted moving average may be expressed mathematically as
(Weight for period n) (Data value in period n)
Weighted moving average =
Weights
Example 7.4: Vaccum cleaner sales for 12 months is given below. The owner of
the supermarket decides to forecast sales by weighting the past three months as
follows:
Weight Applied Month
3 Last month
2 Two months ago
1 Three months ago
6
Month : 1 2 3 4 5 6 7 8 9 10 11 12
Actual sales : 10 12 13 16 19 23 26 30 28 18 16 14
(in units)
Solution: The results of 3-month weighted average are shown in Table 7.3.
224
Caritas Business College BUSINESS STATISTICS – OAD 221
3 × Sales last month + 2 × Sales two months ago + 1
× Sales three months ago
Forecast for the =
Current month 6
Table 7.3: Weighted Moving Average
Month Actual Sales Three-month Weighted
Moving Average
1 10 -
2 12 -
3 13 -
4 16 1 [313) (2 12) 110]
121
6 6
5 19 1 [316) (2 13) 112]
141
6 3
6 23 1 [319)
17
(2 16) 113]
6
7 26 1 [3
23) (2 19) 116] 20
1
6 2
8 30 1 [3
26) (2 23) 119] 235
6 6
9 28 1 [3 30) (2 26) 1
23] 271
6 2
10 18 1 [3 28) (2 30) 1
26] 289
6 3
11 16 1 [318)
(2 28) 1 30] 23
1
6 3
12 14 1 [316) (2 18) 1 28]
182
6 3
Example 7.5: A food processor uses a moving average to forecast next month’s demand.
Past actual demand( in units) is shown below:
Month : 43 44 45 46 47 48 49 50 51
Actual demand : 105 106 110 110 114 121 130 128 137
(in units)
(a) Compute a simple five-month moving average to forecast demand for month 52.
(b) Compute a weighted three-month moving average where the weights are highest for
the latest months and descend in order of 3, 2, 1.
Solution: Calculation for five-month moving average are shown in Table 7.4.
Month Actual Demand 5-month Moving 5-month Moving
Total Average
43 105 - -
44 106 - -
45 110 545 109.50
46 110 561 112.2
47 114 585 117.0
48 121 603 120.6
225
Caritas Business College BUSINESS STATISTICS – OAD 221
49 130 630 126.0
50 128 - -
51 137 - -
(a) Five-month average demand for month 52 is
x 114 + 121 + 130 + 128 + 137
= = 126 units
Number of periods 5
(b) Weighted three-month average as per weights is as follows:
Weight × Data value
MA Wt =
weight
Where Month Weight × Value = Total
51 3 × 137 = 141
50 2 × 128 = 256
49 1 × 130 = 130
6 797
797
6
MAWT = = 133 units
Semi-Average Method
The semi-average method permits us to estimate the slope and intercept of the trend the
quite easily if a linear function will adequately described the data. The procedure is simply
to divide the data into two parts and compute their respective arithmetic means. These two
points are plotted corresponding to their midpoint of the class interval covered by the
respective part and then these points are joined by a straight line, which is the required
trend line. The arithmetic mean of the first part is the intercept value, and the slope is
determined by the ratio of the difference in the arithmetic mean of the number of years
between them,
that is, the change per unit time. The resultant is a time series of the form yˆ a bx . The
:
yˆ is the calculated trend value and a and b are the intercept and slope values respectively.
The equation should always be stated completely with reference to the year where x =0 and
a description of the units of x and y.
The semi-average method of developing a trend equation is relatively easy to commute and
may be satisfactory if the trend is linear. If the data deviate much from linearity, the forecast
will be biased and less reliable.
Example 7.6: Fit a trend line to the following data by the method of semi-average
and forecast the sales for the year 2002.
Year Sales of Firm Year Sales of Firm
(thousand (thousand
units) units)
1993 102 1997 108
1994 105 1998 116
1995 114 1999 112
1996 110
226
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution: Since number of years are odd in number, therefore divide the data into equal
parts (A and B) of 3 years ignoring the middle year (1996). The average of part A and B is
102 + 105 + 114 321
yA = = = 107 units
3 3
108 + 116 + 112 336
y B = = = 112 units
3 2
Part A is centred upon 1994 and part B on 1998. Plot points 107 and 112 against their middle
years, 1994 and 1998. By joining these points, we obtain the required trend line as shown Fig.
7.3. The line can be extended and be used for prediction.
120
115
110
Sal
105
100
199 1994 1995 1996 1997 1998 1999
3
Years
Fig. 7.3: Trend Line by the Method of Semi-Average
To calculate the time-series yˆ = a + bx, we need
y Change in sales
Slope b = =
x Change in year
112 – 107 5
= = = 1.25
1998 – 1994 4
Intercept = a = 107 units at 1994
Thus, the trend line is : yˆ = 107 +
1.25x
Since 2002 is 8 year distant from the origin (1994), therefore we have
yˆ = 107 + 1.25(8) = 117
Exponential Smoothing Methods
Exponential smoothing is a type of moving-average forecasting technique which weighs
past data in an exponential manner so that the most recent data carries more weight in the
moving average. Simple exponential smoothing makes no explicit adjustment for trend
effects whereas adjusted exponential smoothing does take trend effect into account (see
next section for details).
Simple Exponential Smoothing
227
Caritas Business College BUSINESS STATISTICS – OAD 221
With simple exponential smoothing, the forecast is made up of the last period forecast plus
a portion of the difference between the last period’s actual demand and the last period’s
forecast.
Ft = Ft-1 + (Dt-1 – Ft-1) = (1-)Ft-1+ Dt-1 …(7.1)
Where Ft = current period forecast
Ft-1 = last period forecast
= a weight called smoothing constant (0
1) Dt-1 = last period actual demand
From Eqn. (7.1), we may notice that each forecast is simply the previous forecast plus
some correction for demand in the last period. If demand was above the last period
forecast the correction will be positive, and if below it will negative.
When smoothing constant is low, more weight is given to past data, and when it is high,
more weight is given to recent data. When is equal to 0.9, then 99.99 per cent of the
forecast value is determined by the four most recent demands. When is as low as 0.1,
only
34.39 per cent of the average is due to these last 4 periods and the smoothing effect
is equivalent to a 19-period arithmetic moving average.
If were assigned a value as high as 1, each forecast would reflect total adjustment to the
recent demand and the forecast would simply be last period’s actual demand, that is, Ft =
1.0Dt-1. Since demand fluctuations are typically random, the value of is generally kept in
the range of 0.005 to 0.30 in order to ‘smooth’ the forecast. The exact value depends upon
the response to demand that is best for the individual firm.
The following table helps illustrate this concept. For example, when = 0.5, we can see
that the new forecast is based on demand in the last three or four periods. When = 0.1,
the forecast places little weight on recent demand and takes a 19-period arithmetic moving
average.
Weight Assigned
to
Smoothin Most nd
2 Most 3rd Most 4th Most 5th Most
g Recent Recent Recent Recent Recent
Constant Period Period Period Period Period
() (1-) (1-)2 (1-)3 (1-)4
= 0.1 0.1 0.09 0.081 0.073 0.066
0.5 0.25 0.125 0.063 0.031
Selecting the smoothing constant
The exponential smoothing approach is easy to use and it has been successfully applied by
banks, manufacturing companies, wholesalers, and other organizations. The appropriate
value of the smoothing constant, , however, can make the difference between an accurate
and an inaccurate forecast. In picking a value for the smoothing constant, the objective is to
obtain the most accurate forecast.
The correct -value facilitates scheduling by providing a reasonable reaction to
demand without incorporating too much random variation. An approximate value of
which is equivalent to an arithmetic moving average, in terms of degree of smoothing,
can be estimated as: = 2 (n +1). The accuracy of a forecasting model can be
determined by comparing the forecasting values with the actual or observed values.
The forecast error is defined as:
Forecast error = Actual values – Forecasted values
One measure of the overall forecast error for a model is the mean absolute deviation
(MAD). This is computed by taking the sum of the absolute values of the individual forecast
errors and dividing by the number of periods n of data.
228
Caritas Business College BUSINESS STATISTICS – OAD 221
Forecast errors
MAD =
n
where Standard deviation = 1.25 MAD
The exponential smoothing method also facilities continuous updating of the estimate
of MAD. The current MADt is given by
MADt = Actual values- Forecasted values+ (1-) MADt-1
Higher values of smoothing constant make the current MAD more responsive to
current forecast errors.
Example 7.7: A firm uses simple exponential smoothing with =0.1 to forecast demand.
The forecast for the week of February 1 was 500 units whereas actual demand turned out
to be 450 units.
(a) Forecast the demand for the week of February 8.
(b) Assume the actual demand during the week of February 8 turned out to be 505 units.
Forecast the demand for the week of February 15. Continue forecasting through March
15, assuming that subsequent demands were actually 516, 488, 467, 554 and 510 units.
Solution: Given Ft-1 = 500, D t-1 = 450, and = 0.1
(a) Ft = F t-1 – (Dt-1 - Ft-1) = 500 + 0.1(450-500) = 495 units
(b) Forecast of demand for the week of February 15 is shown in Table 7.5
Table 7.5: Forecast of Demand
Week Demand Old Forecast Correction New Forecast (Ft)
Dt-1 Forecas Error (Dt-1 -Ft- Ft-1 +(Dt-1-Ft-
t 1) 1)
Ft-1 (Dt-1 –Ft-
1)
Feb. 1 450 500 -50 -5 495
Feb. 8 505 495 10 1 496
Feb. 15 516 496 20 2 498
Feb. 22 488 498 -10 -1 497
Mar. 1 467 497 -30 -3 494
Mar. 8 554 494 60 6 500
Mar. 15 510 500 10 1 501
If no previous forecast value is known, the old forecast starting point may be estimated
or taken to be an average of some preceding periods.
Example 7.8: A hospital has used a 9 month moving average forecasting method to
predict drug and surgical inventory requirements. The actual demand for one item is
shown in the table below. Using the previous moving average data, convert to an
exponential smoothing forecast for month 33.
Month : 24 25 26 27 28 29 30 31 32
Demand : 78 65 90 71 80 101 84 60 73
(in units)
Solution: The moving average of a 9-month period is given by
Demand (x) 78 + 65 … + 73
MA = = = 78
Number of periods 9
Assume Ft-1 = 78. Therefore, 2 2
0.2
estimated 9
1
n1
Thus, Ft = Ft-1 + (Dt-1-Ft-1) = 78 + 0.2 (73 - 78) = 77 units
229
Caritas Business College BUSINESS STATISTICS – OAD 221
Methods of least square
The trend project method fits a trend line to a series of historical data points and then
projects the line into the future for medium-to-long range forecasts. Several mathematical
trend equations can be developed (such as exponential and quadratic), depending upon
movement of time-series data.
Reasons to study trend: A few reasons to study trends are as follows:
1. The study of trend allows us to describe a historical pattern so that we may evaluate
the success of previous policy.
2. The study allows us to use trends as an aid in making intermediate and long-
range forecasting projections in the future.
3. The study of trends helps us to isolate and then eliminate its influencing effects on the
time-series model as a guide to short-run (one year or less) forecasting of general
business cycle conditions.
Linear Trend Model
If we decide to develop a linear trend line by a precise statistical method, we can apply the
least squares method. A least squares line is described in terms of its y-intercept (the height
at which it intercepts the y-axis) and its slope (the angle of the line). If we can compute the y-
intercept and slope, we can express the line with the following equation
yˆ a bx
where yˆ = predicted value of the dependent
variable a = y-axis intercept
b = slope of the regression line (or the rate of change in y for a given change in
x)
x = independent variable (which is time in this case)
Least squares is one of the most widely used methods of fitting trends to data because it
yields what is mathematically described as a ‘line of best fit’. This trend line has the
properties that (i) the summation of all vertical deviations about it is zero, that is, (y- yˆ ) =
0,
(ii) the summation f all vertical deviations squared is a minimum, that is, (y- yˆ ) is least, and
(iii) the line goes through the mean values of variables x and y. For linear equations, it
is found by the simultaneous solution for a and b of the two normal equations:
y = na + bx and xy = ax + bx2
Where the data can be coded so that x = 0, two terms in three equations drop
out and we have y = na and xy = bx2
Coding is easily done with time-series data. For coding the data, we choose
the centre of the time period as x = 0 and have an equal number of plus and minus
periods on each side of the trend line which sum to zero.
Alternately, we can also find the values of constants a and b for any regression
line as:
xy
b=
nxy and a = y bx
x 2 n(x) 2
230
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 7.9: Below are given the figures of production (in thousand quintals) of
a sugar factory:
Year : 1992 1993 1994 1995 1996 1997 1998
Production : 80 90 92 83 94 99 92
(a) Fit a straight line trend to these figures.
(b) Plot these figures on a graph and show the trend line.
(c) Estimate the production in 2001.
Solution: (a) Using normal equations and the sugar production data we can compute
constants a and b as shown in Table 7.6:
Table 7.6: Calculations for Least Squares Equation
Year Time Production x2 xy Trend
Period (x) (x) Values y
1992 1 80 1 80 84
1993 2 90 4 180 86
1994 3 92 9 276 88
1995 4 83 16 332 90
1996 5 94 25 470 92
1997 6 99 36 594 94
1998 7 92 49 644 96
Total 28 630 140 2576
x 28 y 630
x 4, y 90
n 7 n 7
xy nxy 2576 7(4)(90) 56
b = 2
x2 n(x) 2 140 7(4)2 28
a = y bx 90 2(4) 82
Therefore, linear trend component for the production of sugar is:
yˆ a bx 82 2x
The slope b = 2 indicates that over the past 7 years, the production of sugar had
an average growth of about 2 thousand quintals per year.
231
Caritas Business College BUSINESS STATISTICS – OAD 221
100
95
Producti
90
85
80
75
1992 1993 1994 1995 1996 1997 1998
Years
Fig.7.4: Linear Trend for Production of Sugar
(b) Plotting points on the graph paper, we get an actual graph representing
production of sugar over the past 7 years. Join the point a = 82 and b = 2
(corresponds to 1993) on the graph we get a trend line as shown in Fig. 7.4.
(c) The production of sugar for year 2001 will be
yˆ = 82 + 2 (10) = 102 thousand quintals
Parabolic Trend Model
The curvilinear relationship for estimating the value of a dependent variable
y from an independent variable x might take the form
yˆ = a + bx + cx2
This trend line is called the parabola.
For a non-linear equation yˆ = a + bx - cx2, the values of constants a, b, and
c can be determined by solving three normal equations.
y = na + bx + cx2
xy = ax + bx2 + cx3
x2y = ax2 + bx3 + cx4
232
Caritas Business College BUSINESS STATISTICS – OAD 221
When the data can be coded so that x = 0 and x3 = 0, two term in the above
expressions drop out and we have
y = na + cx2
xy = bx2
x2y = ax2 + cx4
To find the exact estimated value of the variable y, the values of constants a, b,
and c need to be calculated. The values of these constants can be calculated
by using the following shortest method:
yc xy
2
x ;b n x2 y x2
a= c
n and y n x4 ( x2
2
x2 )
Example 7.10: The prices of a commodity during 1999-2004 are given below.
Fit a parabola to these data. Estimate the price of the commodity for the year
2005.
Year Price Year Price
1999 100 2002 140
2000 107 2003 181
2001 128 2004 192
Also plot the actual and trend values on a graph.
Solution: To fit a parabola yˆ = a + bx + cx2, the calculations to determine the
values of constants a, b, and c are shown in Table 7.7.
Table 7.7: Calculations for Parabola Trend Line
Year Time Price x2 x3 x4 xy x2y Trend
Scale (y) Values
(x) ( yˆ )
1999 -2 100 4 -8 16 -200 400 97.72
2000 -1 107 1 -1 1 -107 107 110.34
2001 0 128 0 0 0 0 0 126.68
2002 1 140 1 1 1 140 140 146.50
2003 2 181 4 8 16 362 724 169.88
233
Caritas Business College BUSINESS STATISTICS – OAD 221
2004 3 192 9 27 81 576 1728 196.82
3 848 19 27 115 771 3099 847.94
(i) y = na- bx + cx2
848 = 6a + 3b + 19c
(ii) xy = ax + bx2 + cx3
771 = 3a + 19b + 27c
(iii) x2y = ax2 + bx2 + cx4
3099 = 19a + 27b + 115c
Eliminating a from eqns. (i) and (ii), we get (iv)
694 = 35b + 35c
Eliminating a from eqns. (ii) and (iii), we get (v)
5352 = 280b + 168c
Solving eqns. (iv) and (v) for b and c we get b =18.04 and c = 1.78.
Substituting values of b and c in eqn. (i), we get a = 126.68.
Hence, the required non-linear trend line becomes
y = 126.68 +18.04x + 1.78x2
Several trend values as shown in Table 7.7 can be obtained by putting x = - 2,
-1, 0, 1, 2 and 3 in the trend line. The trend values are plotted on a graph paper.
The graph is shown in Fig. 7.5.
234
Caritas Business College BUSINESS STATISTICS – OAD 221
600
500
Yea 400
300
200
100
0
1994 1995 1996 1997 1998 1999
Price (Rs.)
Fig. 7.5
Exponential Trend Model
When the given values of dependent variable y from approximately a geometric
progression while the corresponding independent variable x values form an
arithmetic progression, the relationship between variables x and y is given by
an exponential function, and the best fitting curve is said to describe the
exponential trend. Data from the fields of biology, banking, and economics
frequently exhibit such a trend. For example, growth of bacteria, money
accumulating at compound interest, sales or earnings over a short period, and so
on, follow exponential growth.
The characteristics property of this law is that the rate of growth, that is, the rate
of change of y with respect to x is proportional to the values of the function. The
following function has this property.
y = abcx, a > 0
The letter b is a fixed constant, usually either 10 or e, where a is a constant
to be determined from the data.
To assume that the law of growth will continue is usually unwarranted, so only
short range predictions can be made with any considerable degree or reliability.
235
Caritas Business College BUSINESS STATISTICS – OAD 221
If we take logarithms (with base 10) of both sides of the above equation, we obtain
Log y = log a + (c log b) x (7.2)
For b =10, log b =1, but for b=e, log b =0.4343 (approx.). In either case, this
equation is of the form y = c + dx
Where y = log y, c = log a, and d = c log b.
Equation (7.2) represents a straight line. A method of fitting an exponential trend line
to a set of observed values of y is to fit a straight trend line to the logarithms of the
y-values.
In order to find out the values of constants a and b in the exponential function,
the two normal equations to be solved are
log y = n log a + log bx
x log y = log ax + log bx2
When the data is coded so that x = 0, the two normal equations become
1
log y = n log a or log a = log y
n
x log y
and x log y = log b x2 or log b
=
x2
Coding is easily done with time-series data by simply designating the center of
the time period as x =0, and have equal number of plus and minus period on
each side which sum to zero.
Example 7.11: The sales (Rs. In million) of a company for the years 1995 to 1999
are:
Year : 1995 1996 1997 1998 1999
Sales : 1.6 4.5 13.8 40.2 125.0
Find the exponential trend for the given data and estimate the sales for
2002.
236
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution: The computational time can be reduced by coding the data. For this
consider u = x-3. The necessary computations are shown in Table 7.8.
Table 7.8: Fitting the Exponential Trend Line
Year Time u=x-3 u2 Sales y Log y u log y
Period x
1995 1 -2 4 1.60 0.2041 -0.4082
1996 2 -1 1 4.50 0.6532 -0.6532
1997 3 0 0 13.80 1.1390 0
1998 4 1 1 40.20 1.6042 1.6042
1999 5 2 4 125.00 2.0969 4.1938
10 5.6983 4.7366
1 1
log a = log y (5.6983) = 1.1397
= 5
n
u log 4.7366
log b = y = = 0.4737
u2
1
Therefore log y = log a + (x+3) log b = 1.1397 + 0.4737x
For sales during 2002, x =3, and we obtain
log y = 1.1397 + 0.4737 (3) = 2.5608
y = antilog (2.5608) = 363.80
Changing the Origin and Scale of Equations
When a moving average or trend value is calculated it is assumed to be centred
in the middle of the month (fifteenth day) or the year (July 1). Similarly, the
forecast value is assumed to be centred in the middle of the future period.
However, the reference point (origin) can be shifted, or the units of variables x
and y are changed to monthly or quarterly values it desired. The procedure is as
follows:
(i) Shift the origin, simply by adding or subtracting the desired number of
periods from independent variable x in the original forecasting equation.
237
Caritas Business College BUSINESS STATISTICS – OAD 221
(ii) Change the time units from annual values to monthly values by
dividing independent variable x by 12.
(iii) Change the y units from annual to monthly values, the entire right-
hand side of the equation must be divided by 12.
Example 7.12: The following forecasting equation has been derived by a
least-squares method:
yˆ =10.27 + 1.65x (Base year:1992; x = years; y = tonnes/year)
Rewrite the equation by
(a) shifting the origin to 1997.
(b) expressing x units in months, retaining y n tonnes/year.
(c) expressing x units in months and y in tonnes/month.
Solution: (a) Shifting of origin can be done by adding the desired number of
period 5(=1997-1992) to x in the given equation. That is
yˆ =10.27 + 1.65 (x + 5) = 18.52 + 1.65x
where 1997 = 0, x = years, y = tonnes/year
(b) Expressing x units in months
1.65x
yˆ =10.27 + = 10.27 + 0.14x
12
where July 1, 1992 = 0, x = months, y = tonnes/year
(c) Expressing y in tonnes/month, retaining x months.
1
yˆ = = (10.27 + 0.14x) = 0.86+0.01x
1
where July 1, 1992 = 0, x = months, y = tones/month
Remarks
1. If both x and y are to be expressed in months together, then divide constant
‘a’ by 12 and constant ‘b’ by 24. It is because data are sums of 12 months. Thus
monthly trend equation becomes.
Linear trend : a b
yˆ x
12 24
238
Caritas Business College BUSINESS STATISTICS – OAD 221
Parabolic trend : yˆ x x2
a 1
1 7
2 2
c 8
144
But if data are given as monthly averages per year, then value of ‘a’ remains
unchanged ‘b’ is divided by 12 and ‘c’ by 144.
2. The annual trend equation can be reduced to quarterly trend equation as
:
a
yˆ b a b
x x
4
4 48
4
12
7.6. SEASONAL VARIATIONS
If the time series data are in terms of annual figures, the seasonal variations
are absent. These variations are likely to be present in data recorded on quarterly
or monthly or weekly or daily or hourly basis. As discussed earlier, the seasonal
variations are of periodic in nature with period less than or equal to one year.
These variations reflect the annual repetitive pattern of the economic or
business activity of any society. The main objectives of measuring seasonal
variations are:
(i) To understand their pattern.
(ii) To use them for short-term forecasting or planning.
(iii) To compare the pattern of seasonal variations of two or more time
series in a given period or of the same series in different periods.
(iv) To eliminate the seasonal variations from the data. This process is
known as deseasonalisation of data.
Methods of Measuring Seasonal Variations
The measurement of seasonal variation is done by isolating them from other
components of a time series. There are four methods commonly used for the
measurement of seasonal variations. These method are:
1. Method of Simple Averages
2. Ratio to Trend Method
3. Ratio to Moving Average Method
239
Caritas Business College BUSINESS STATISTICS – OAD 221
4. Method of Line Relatives
Note: In the discussion of the above methods, we shall often assume a
multiplicative model. However, with suitable modifications, these methods are
also applicable to the problems based on additive model.
Method of Simple Averages
This method is used when the time series variable consists of only the seasonal
and random components. The effect of taking average of data corresponding to the
same period (say 1st quarter of each year) is to eliminate the effect of random
component and thus, the resulting averages consist of only seasonal component.
These averages are then converted into seasonal indices, as explained in the
following examples.
Example 7.13.
Assuming that trend and cyclical variations are absent compute the seasonal
index for each month of the following data of sales (in Rs. ‘000) of a company.
Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1987 46 45 44 46 45 47 46 43 40 40 41 45
1988 45 44 43 46 46 45 47 42 43 42 43 44
1989 42 41 40 44 45 45 46 43 41 40 42 45
Solution
Calculation Table
Yea Jan Feb Mar Apr May Jun Jul Aug Se Oct Nov Dec
r p
198 46 45 44 46 45 47 46 43 40 40 41 45
7
198 45 44 43 46 46 45 47 42 43 42 43 44
8
198 42 41 40 44 45 45 46 43 41 40 42 45
9
Tot 133 130 127 136 136 137 139 128 124 122 126 134
al
At 44.3 43.3 42. 45.3 45.3 45.7 46.3 42. 41. 40. 42. 44.7
3 7 3 7 0
S.l. 101. 99.1 96. 103. 103. 104. 105. 97. 94. 93. 96. 102.
4 8 7 7 6 9 7 5 1 1 3
In the above table, A denotes the average and S.I the seasonal index for a
particular month of various years. To calculate the seasonal index, we
240
Caritas Business College BUSINESS STATISTICS – OAD 221
compute grand average G, given by 523
43.7 . Then the seasonal
Gi
A 12
12
At
index for a particular month is given by S.I. = 100 .
G
Further, S.I.=11998.91200. Thus, we have to adjust these values such that
their total is 1200. This can be done by multiplying each figure by
1200
. The resulting figures are the adjusted seasonal indices, as given
1198.9
below:
Jan Feb Mar Apr May Jun Jul
Au Se Oct Nov Dec
g p
101.5 99. 96. 103.8 103.8 104.7 106.0 97. 94. 93. 96. 102.3
2 9 8 6 2 2
Remarks: The total equal to 1200, in case of monthly indices and 400, in
case of quarterly indices, indicate that the ups and downs in the time series, due
to seasons, neutralise themselves within that year. It is because of this that the
annual data are free from seasonal component.
Example 7.14
Compute the seasonal index from the following data by the method of simple
averages.
Year Quarter Y Year Quarter Y Year Quarter Y
1980 I 106 1982 I 90 1984 I 80
II 124 II 112 II 104
III 104 III 101 III 95
IV 90 IV 85 IV 83
1981 I 84 1983 I 76 1985 I 104
II 114 II 94 II 112
III 107 III 91 III 102
IV 88 IV 76 IV 84
Solution
Calculation of Seasonal Indices
Years Ist Qr 2nd 3rd Qr 4th Qr
Qr
1980 106 124 104 90
1981 84 114 107 88
1982 90 112 101 85
1983 76 94 91 76
1984 80 104 95 83
1985 104 112 102 84
241
Caritas Business College BUSINESS STATISTICS – OAD 221
Total 104 660 600 506
Ai 90 110 100 84.33
Ai
93.67 114.49 104.07 87.77
100
G
384.33
We have G 96.08 . Further, since the sum of terms in the last
At 4
4
row of the table is 400, no adjustment is needed. These terms are the seasonal
indices of respective quarters.
Merits and Demerits
This is a simple method of measuring seasonal variations which is based on the
unrealistic assumption that the trend and cyclical variations are absent from the
data. However, we shall see later that this method, being a part of the other
methods of measuring seasonal variations, is very useful.
Ratio to Trend Method
This method is used when cyclical variations are absent from the data, i.e.
the time series variable Y consists of trend, seasonal and random
components.
Using symbols, we can write Y = T.S.R
Various steps in the computation of seasonal indices are:
(i) Obtain the trend values for each month or quarter, etc. by the method
of least squares.
(ii) Divide the original values by the corresponding trend values. This would
eliminate trend values from the data. To get figures in percentages, the
quotients are multiplied by 100.
Thus, we have Y T.S.R
100 100 S.R.100
T T
(iii) Finally, the random component is eliminated by the method of simple
averages.
242
Caritas Business College BUSINESS STATISTICS – OAD 221
Example 7.15
Assuming that the trend is linear, calculate seasonal indices by the ratio to moving
average method from the following data:
Quarterly output of coal in 4 years (in thousand tonnes)
Year I II III IV
1982 65 58 56 61
1983 68 63 63 67
1984 70 59 56 52
1985 60 55 51 58
Solution
By adding the values of all the quarters of a year, we can obtain annual
output for each of the four years. Fit a linear trend to the data and obtain
trend values for each quarter.
Year Output X=2(t-1983.5) XY X2
1982 240 -3 -720 9
1983 261 -1 -261 1
1984 237 1 237 1
1985 224 3 672 9
Total 962 0 -72 20
962 72
From the above table, we get a and b 3.6
20
240.5
4
Thus, the trend line is Y=240.5 – 3.6X, Origin: Ist January 1984, unit of X:6 months.
The quarterly trend equation is given by
240.5 3.6
Y X or Y = 60.13-0.45X, Origin: Ist January 1984, unit of X:1
4 8
quarter (i.e., 3 months).
Shifting origin to 15th Feb. 1984, we get
243
Caritas Business College BUSINESS STATISTICS – OAD 221
1
Y=60.13-0.45(X+
) = 59.9-0.45X, origin I-quarter, unit of X=1 quarter.
2
The table of quarterly values is given by
Year I II III IV
1982 63.50 63.05 62.50 62.15
1983 61.70 61.25 60.80 60.35
1984 59.90 59.45 59.00 58.55
1985 58.10 57.65 57.20 56.75
Y
The table of Ratio to Trend Values, i.e. 100
T
Year I II III IV
1982 102.36 91.99 89.46 98.15
1983 110.21 102.86 103.62 111.02
1984 116.86 99.24 94.92 88.81
1985 103.27 95.40 89.16 102.20
Total 432.70 389.49 377.16 400.18
Average 108.18 97.37 94.29 100.05
S.I. 108.20 97.40 94.32 100.08
Note : Grand Average, 399.89
G 99.97
4
Example 7.16.
Find seasonal variations by the ratio to trend method, from the following
data:
Year I-Qr II-Qr III-Qr IV-Qr
1995 30 40 36 34
1996 34 52 50 44
1997 40 58 54 48
1998 54 76 68 62
1999 80 92 86 82
244
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution
245
Caritas Business College BUSINESS STATISTICS – OAD 221
First we fit a linear trend to the annual totals.
Year Annual Totals (Y) X XY X2
1995 140 -2 -280 4
1996 180 -1 -180 1
1997 200 0 0 0
1998 260 1 260 1
1999 340 2 680 4
Total 1120 0 480 10
1120 480
Now a = 224 and b = 48
5 10
Trend equation is Y = 224+48X, origin: Ist July 1997, unit of X = 1 year
224
The quarterly trend equation is Y= 48 X=56+3X, origin: Ist July 1997,
4 16
unit of X = 1 quarter.
Shifting the origin to III quarter of 1997, we get
Y = 56 + 3 (X+
1 ) = 57.5 + 3X
Table of Quarterly Trend Values
Year I II III IV
1995 27.5 30.5 33.5 36.5
1996 39.5 42.5 45.5 48.5
1997 51.5 54.5 57.5 60.5
1998 63.5 66.5 69.5 72.5
1999 75.5 78.5 81.5 84.5
Ratio to Trend Values
Year I II III IV
1995 109.1 131.1 107.5 93.2
1996 86.1 122.4 109.9 90.7
246
Caritas Business College BUSINESS STATISTICS – OAD 221
1997 77.7 106.4 93.9 79.3
247
Caritas Business College BUSINESS STATISTICS – OAD 221
1998 85.0 114.3 97.8 85.5
1999 106.0 117.2 105.5 97.0
Total 463.9 591.4 514.6 445.7
At 92.78 118.28 102.92 89.14
S.I. 92.10 117.35 102.11 88.44
403.12
Note that the Grand Average G= 100.78. Also check that the sum of
4
indices is 400.
Remarks: If instead of multiplicative model we have an additive model, then Y =
T + S + R or S + R = Y-T. Thus, the trend values are to be subtracted from
the Y values. Random component is then eliminated by the method of simple
averages.
Merits and Demerits
It is an objective method of measuring seasonal variations. However, it is very
complicated and doesn’t work if cyclical variations are present.
Ratio to Moving Average Method
The ratio to moving average is the most commonly used method of measuring
seasonal variations. This method assumes the presence of all the four
components of a time series. Various steps in the computation of seasonal
indices are as follows:
(i) Compute the moving averages with period equal to the period of
seasonal variations. This would eliminate the seasonal component and
minimise the effect of random component. The resulting moving
averages would consist of trend, cyclical and random components.
(ii) The original values, for each quarter (or month) are divided by the
respective moving average figures and the ratio is expressed as a
248
Caritas Business College BUSINESS STATISTICS – OAD 221
percentage, Y TCSR SR' ' , where R´ and R´´ denote the
i.e.
M .A TCR'
.
changed random components.
(iii) Finally, the random component R´´ is eliminated by the method of
simple averages.
Example 7.17
Given the following quarterly sale figures, in thousand of rupees, for the year
1996-1999, find the specific seasonal indices by the method of moving
averages.
Year I II III IV
1996 34 33 34 37
1997 37 35 37 39
1998 39 37 38 40
1999 42 41 42 44
Solution
Calculation of Ratio of Moving Averages
Year/Quarter Sale 4-Period Centred 4 Period Y
100
s Moving Total M
M
Total
1996 I 34 … … …
II 33 138 … … …
III 34 141 279 34.9 97.4
IV 37 143 284 35.5 104.2
1997 I 37 146 289 36.1 102.5
II 35 148 294 36.8 95.1
III 37 150 298 37.3 99.2
152
IV 39 302 37.8 103.2
153
1998 I 39 305 38.1 102.4
157
II 37 307 38.4 96.4
161
III 38 311 38.9 97.7
165
IV 40 318 39.8 100.5
169
1999 I 42 326 40.8 102.9
II 41 334 41.8 98.1
III 42 … … …
IV 44 … … …
Calculation of Seasonal Indices
249
Caritas Business College BUSINESS STATISTICS – OAD 221
Year I II III IV
1996 - - 97.4 104.2
1997 102.5 95.1 99.2 103.2
1998 102.4 96.4 97.7 100.5
1999 102.9 98.1 - -
Total 307.8 289.6 294.3 307.9
At 102.6 96.5 98.1 102.6
S.I. 102.7 96.5 98.1 102.7
399.8
Note that the Grand Average G= =99.95. Also check that the sum of
4
indices is 400.
Merits and Demerits
This method assumes that all the four components of a time series are present
and, therefore, widely used for measuring seasonal variations. However, the
seasonal variations are not completely eliminated if the cycles of these
variations are not of regular nature. Further, some information is always lost at the
ends of the time series.
Line Relatives Method
This method is based on the assumption that the trend is linear and cyclical
variations are of uniform pattern. As discussed in earlier chapter, the link relatives
are percentages of the current period (quarter or month) as compared with
previous period. With the computation of link relatives and their average, the effect
of cyclical and random component is minimised. Further, the trend gets eliminated
in the process of adjustment of chained relatives. The following steps are involved in
the computation of seasonal indices by this method:
(i) Compute the link relative (L.R.) of each period by dividing the figure of that
period with the figure of previous period. For example, link relative of
figure of 3rd quarter
3rd quarter = 100
figure of 2nd quarter
250
Caritas Business College BUSINESS STATISTICS – OAD 221
(ii) Obtain the average of link relatives of a given quarter (or month) of various
years. A.M. or Md can be used for this purpose. Theoretically, the later is
preferable because the former gives undue importance to extreme items.
(iii) These averages are converted into chained relatives by assuming the chained
relative of the first quarter (or month) equal to 100. The chained relative (C.R.) for
the current period (quarter or month)
C.R. of the previous period × L.R. of the current period
=
100
(iv) Compute the C.R. of first quarter (or month) on the basis of the last
quarter (or month). This is given by
C.R. of the last quarter (or month) × L.R. of 1st quarter (or month)
=
100
This value, in general, be different from 100 due to long term trend in the data.
The chained relatives, obtained above, are to be adjusted for the effect of this
trend. The adjustment factor is
d=
1 [New C.R. for Ist quarter-100] for quarterly data
and d = 1
[New C.R. for Ist month –100] for monthly data.
12
On the assumption that the trend is linear, d, 2d, 3d, etc. is respectively
subtracted from the 2nd, 3rd, 4th, etc., quarter (or month).
(v) Express the adjusted chained relatives as a percentage of their average to
obtain seasonal indices.
(vi) Make sure that the sum of these indices is 400 for quarterly data and 1200
for monthly data.
Example 7.18
251
Caritas Business College BUSINESS STATISTICS – OAD 221
Determine the seasonal indices from the following data by the method of link relatives:
Year Ist 2nd Qr 3rd Qr 4th Qr
2000 26 19 15 10
2001 36 29 23 22
2002 40 25 20 15
2003 46 26 20 18
2004 42 28 24 21
Solution
Calculation Table
Year I II III IV
2000 - 73.1 78.9 66.7
2001 360.0 80.5 79.3 95.7
2002 181.8 62.5 80.0 75.0
2003 306.7 56.5 76.9 90.0
2004 233.3 66.7 85.7 87.5
The chained
Total relative 1081.8
(C.R.) of the Ist quarter on the400.8
339.3 of the 4th quarter
basis of C. R.414.0
270 Mean
45.2
= 122.3 270.5 67.9 80.2 83.0
100
C.R. 100.0 67.9 54.5 45.2
1
The
C.R.trend adjustment factor
(adjusted) 100.0d = (122.3 100) 5.6 43.3
62.3 28.4
4
S.I. 170.9 106.5 74.0 48.6
Thus, the adjusted C.R. of 1st quarter = 100
and for 2nd = 67.9 – 5.6 = 62.3
for 3rd = 54.5-2 × 5.6 = 43.3
for 4th = 45.2 – 3 × 5.6 = 28.4
100 62.3 43.3 28.4
The grand average of adjusted C.R., G = 58.5
4
252
Caritas Business College BUSINESS STATISTICS – OAD 221
Adjusted C.R. ×
The seasonal index of a quarter =
100 G
Merits and Demerits
This method is less complicated than the ratio to moving average and the ratio
to trend methods. However, this method is based upon the assumption of a
linear trend, which may not always hold true.
Deseasonalisation of Data
The deseasonalization of data implies the removal of the effect of seasonal
variations from the time series variable. If Y consists of the sum of various
components, then for its deaseasonalization, we subtract seasonal variations
from it. Similarly, in case of multiplicative model, the deseasonalisation is done
by taking the ratio of Y value to the corresponding seasonal index. A clue to this
is provided by the fact that the sum of seasonal indices is equal to zero for an
additive model while their sum is 400 or 1200 for a multiplicative model.
It may be pointed out here that the deseasonalization of a data is done under
the assumption that the pattern of seasonal variations, computed on the basis of
past data, is similar to the pattern of seasonal variations in the year of
deseasonalization.
Example 7.19
Deseasonalise the following data on the sales of a company during various
months of 1990 by using their respective seasonal indices. Also interpret the
deseasonalised values.
Month Sales S.I. Month Sales S.I.
(Rs. ‘000) (Rs. ‘000)
Jan 16.5 109 Jul 36.5 85
Feb 21.3 105 Aug 44.4 88
Mar 27.1 108 Sep 54.9 98
Apr 31.0 102 Oct 62.0 102
May 35.5 100 Nov 67.6 104
253
Caritas Business College BUSINESS STATISTICS – OAD 221
Jun 36.3 89 Dec 78.7 110
Solution
Let Y denote monthly sales and DS denote the deseasonalised sales. Then,
we can write
DS Y
100
S.I
Computation of Deseasonalised Values
Month Sales S.I. DS Month Sales (Y) S.I. DS
(Y)
Jan 16.5 109 15.14 Jul 36.5 85 42.94
Feb 21.3 105 20.29 Aug 44.5 88 50.45
Mar 27.1 108 25.09 Sep 54.9 98 56.02
Apr 31.0 102 30.39 Oct 62.0 102 60.78
May 35.5 100 35.50 Nov 67.6 104 65.00
Jun 36.3 89 40.79 Dec 78.7 110 71.55
The deseasonalised figures of sales for each month represent the monthly sales
that would have been in the absence of seasonal variations.
7.7. MEASUREMENT OF CYCLICAL VARIATIONS-RESIDUAL METHOD
As mentioned earlier that a typical time-series has four components: secular trend
(T), seasonal variation (S), cyclical variation (C), and irregular variation (I). In a
multiplicative time-series model, these components are written as:
Y=T×C×S×I
The deseasonalization data can be adjusted for trend analysis these by the
corresponding trend and seasonal variation values. Thus we are left with only
cyclical (C) and irregular (I) variations in the data set as shown below:
Y T×C×S×I
= =C×I
T×S T×S
254
Caritas Business College BUSINESS STATISTICS – OAD 221
The moving averages of an appropriate period may be used to eliminate or reduce
the effect of irregular variations and thus left behind only the cyclical variations.
The procedure of identifying cyclical variation is known as the residual method.
Recall that cyclical variations in time-series tend to oscillate above and below the
secular trend line for periods longer than one year. The steps of residual method
are summarized as follows:
(i) Obtain seasonal indexes and deseasonalized data.
(ii) Obtain trend values and expressed seasonalized data as percentages
of the trend values.
(iii) Divided the original data (Y) by the corresponding trend values (T) in
the time-series to get S × C × I. Further divide S × C × I by S to get C
× I.
(iv) Smooth out irregular variations by using moving averages of an
appropriate period but of short duration, leaving only the cyclical variation.
7.8. MEASUREMENT OF IRREGULAR VARIATIONS
Since irregular variations are random in nature, no particular procedure can be
followed to isolate and identify these variations. However, the residual method
can be extended one step further by dividing C × I by the cyclical component (C)
to identify the irregular component (I).
Alternately, trend (T), seasonal (S), and cyclical (C) components of the given time-
series are estimated and then the residual is taken as the irregular variation. Thus,
in the case of multiplicative time-series model, we have
Y T×C×S×I
= = I
T×C×S T×C×S
where S and C are in fractional form and not in percentages.
7.9. QUESTIONS
1. What effect does seasonal variability have on a time-series? What is the
basis for this variability for an economic time-series?
255
Caritas Business College BUSINESS STATISTICS – OAD 221
2. What is measured by a moving average? Why are 4-quarter and 12- month
moving averages used to develop a seasonal index?
3. Briefly describe the moving average and least squares methods of
measuring trend in time-series.
4. Distinguish between ratio-to-trend and ratio-to-moving average as
methods of measuring seasonal variations, which is better and why?
5. Why do we deseasonalize data? Explain the ratio-to-moving average method
to compute the seasonal index.
6. Apply the method of link relatives to the following data and calculate seasonal
indexes.
Quarter 1995 1996 1997 1998 1999
I 6.0 5.4 6.8 7.2 6.6
II 6.5 7.9 6.5 5.8 7.3
III 7.8 8.4 9.3 7.5 8.0
IV 8.7 7.3 6.4 8.5 7.1
7. Calculate seasonal index numbers from the following data:
Year Ist Quarter 2nd Quarter 3rd Quarter 4th Quarter
1991 108 130 107 93
1992 86 120 110 91
1993 92 118 104 88
1994 78 100 94 78
1995 82 110 98 86
1996 106 118 105 98
8. For what purpose do we apply time series analysis to data collected over a period
of time?
9. What is the difference between a causal model and a time series model?
10. Explain clearly the different components into which a time series may be
analysed. Explain any method for isolating trend values in a time series.
11. Explain what you understand by time series. Why is time-series considered to be
an effective tool of forecasting?
256
Caritas Business College BUSINESS STATISTICS – OAD 221
12. Explain briefly the additive and multiplicative models of time series. Which of these
models is more popular in practice and why?
13. A company that manufactures steel observed the production of steel (in
metric tonnes) represented by the time-series:
Year : 1990 1991 1992 1993 1994 1995 1996
Production in steel : 60 72 75 65 80 85 95
(a) Find the linear equation that describes the trend in the production of steel by
the company.
(b) Estimate the production of steel in 1997.
14. The sales (Rs. In lakh) of a company for the years 1990 to 1996 are
given below:
Year : 1990 1991 1992 1993 1994 1995 1996
Sales : 32 47 65 88 132 190 275
Find trend values by using the equation Yc = abx and estimate the value for
1997.
15. A company that specializes in the production of petrol filters has
recorded the following production (in 1000 units) over the last 7 years.
Year : 1994 1995 1996 1997 1998 1999 2000
Production : 42 49 62 75 92 122 158
(a) Develop a second degree estimating equation that best describes these data.
(b) Estimate the production in 2004.
7.10. SUGGESTED READINGS
1. Spiegel, Murray R.: Theory and Practical of Statistics., London
McGraw Hill Book Company.
2. Yamane, T.: Statistics: An Introductory Analysis, New York, Harpered
Row Publication
3. R.P. Hooda: Statistic for Business and Economic, McMillan India Ltd.
4. G.C. Beri: Statistics for Mgt., TMH.
5. J.K. Sharma: Business Statistics, Pearson Education.
6. S.P. Gupta : Statistical Methods, Sultan Chand and Sons.
257
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 8: PROBABILITY THEORY
Topic : Probability
Objectives : The present lesson is an attempt to overview the concept of
probability, thereby enabling the students to appreciate the
relevance of probability theory in decision-making under
conditions of uncertainty. After successful completion of the
lesson the students will be able to understand and use the
different approaches to probability as well as different probability
rules for calculating probabilities in different situations.
Structure
8.1 Introduction
8.2 Some Basic Concepts
8.3 Approaches to Probability Theory
8.4 Probability Rules
8.5 Bayes’ Theorem
8.6 Some Counting Concepts
8.7 Self-Assessment Questions
8.8 Suggested Readings
8.1 INTRODUCTION
Life is full of uncertainties. ‘Probably’, ‘likely’, ‘possibly’, ‘chance’ etc. is some of the most
commonly used terms in our day-to-day conversation. All these terms more or less
convey the same sense - “the situation under consideration is uncertain and
commenting on the
258
Caritas Business College BUSINESS STATISTICS – OAD 221
future with certainty is impossible”. Decision-making in such areas is facilitated through
formal and precise expressions for the uncertainties involved. For example, product
demand is uncertain but study of demand spelled out in a form amenable for analysis may
go a long to help analyze, and facilitate decisions on sales planning and inventory
management. Intuitively, we see that if there is a high chance of a high demand in the
coming year, we may decide to stock more. We may also take some decisions regarding the
price increase, reducing sales expenses etc. to manage the demand. However, in order to
make such decisions, we need to quantify the chances of different quantities of demand in
the coming year. Probability theory provides us with the ways and means to quantify the
uncertainties involved in such situations.
A probability is a quantitative measure of uncertainty - a number that
conveys the strength of our belief in the occurrence of an uncertain
event.
Since uncertainty is an integral part of human life, people have always been interested -
consciously or unconsciously - in evaluating probabilities.
Having its origin associated with gamblers, the theory of probability today is an
indispensable tool in the analysis of situations involving uncertainty. It forms the basis for
inferential statistics as well as for other fields that require quantitative assessments of
chance occurrences, such as quality control, management decision analysis, and almost all
areas in physics, biology, engineering and economics or social life.
8.2 SOME BASIC CONCEPTS
Probability, in common parlance, refers to the chance of occurrence of an event or
happening. In order that we are able to compute it, a proper understanding of certain basic
concepts in probability theory is required. These concepts are an experiment, a sample
space, and an event.
259
Caritas Business College BUSINESS STATISTICS – OAD 221
8.2.1 EXPERIMENT
An experiment is a process that leads to one of several possible outcomes. An
outcome of an experiment is some observation or measurement.
The term experiment is used in probability theory in a much broader sense than in physics or
chemistry. Any action, whether it is the drawing a card out of a deck of 52 cards, or reading
the temperature, or measurement of a product's dimension to ascertain quality, or the
launching of a new product in the market, constitute an experiment in the probability theory
terminology.
The experiments in probability theory have three things in common:
there are two or more outcomes of each experiment
it is possible to specify the outcomes in advance
there is uncertainty about the outcomes
For example, the product we are measuring may turn out to be undersize or right size or
oversize, and we are not certain which way it will be when we measure it. Similarly,
launching a new product involves uncertain outcome of meeting with a success or failure in
the market.
A single outcome of an experiment is called a basic outcome or an elementary event. Any
particular card drawn from a deck is a basic outcome.
8.2.2 SAMPLE SPACE
The sample space is the universal set S pertinent to a given experiment. It
is the set of all possible outcomes of an experiment.
So each outcome is visualized as a sample point in the sample space. The sample spaces
for the above experiments are:
Experiment Sample Space
Drawing a Card {all 52 cards in the deck}
Reading the Temperature {all numbers in the range of temperatures}
Measurement of a Product's Dimension {undersize, outsize, right size}
260
Caritas Business College BUSINESS STATISTICS – OAD 221
Launching of a New Product {success, failure}
8.2.3 EVENT
An event, in probability theory, constitutes one or more possible outcomes of an experiment.
An event is a subset of a sample space. It is a set of basic outcomes. We say
that the event occurs if the experiment gives rise to a basic outcome
belonging to the event.
For the experiment of drawing a card, we may obtain different events A, B, and C like:
A : The event that card drawn is king of club
B : The event that card drawn is
red C : The event that card drawn is
ace
In the first case, out of the 52 sample points that constitute the sample space, only one
sample point or outcome defines the event, whereas the number of outcomes used in the
second and third case is 13 and 4 respectively.
8.3 APPROACHES TO PROBABILITY THEORY
Three different approaches to the definition and interpretation of probability have evolved,
mainly to cater to the three different types of situations under which probability measures are
normally required. We will study these approaches with the help of examples of distinct types
of experiments.
Consider the following situations marked by three distinct types of experiments. The events
that we are interested in, within these experiments, are also given.
Situation I
Experiment : Drawing a Card Out of a Deck of 52
Cards Event A: On any draw, a king is there
Situation II
Experiment : Administering a Taste Test for a New
Soup Event B : A consumer likes the taste
Situation III
261
Caritas Business College BUSINESS STATISTICS – OAD 221
Experiment : Commissioning a Solar Power Plant
Event C : The plant turns out to be a successful venture
Situation I : THE CLASSICAL APPROACH
The first situation is characterized by the fact that for a given experiment we have a sample
space with equally likely basic outcomes. When a card is drawn out of a well-shuffled deck,
every one of the cards (the basic outcomes) is as likely to occur as any other. This type of
situations, marked by the presence of "equally likely" outcomes, gave rise to the Classical
Approach to the probability theory. In the Classical Approach, probability of an event is
defined as the relative size of the event with respect to the size of the sample space. Since
there are 4 kings and there are 52 cards, the size of A is 4 and the size of the sample
space is
52. Therefore, the probability of A is equal to 4/52.
The rule we use in computing probabilities, assuming equal likelihood of all basic outcomes,
is as follows:
Probability of the event A:
P(A) = n ( A ) …………(8-1)
N(S
)
where n(A) = the number of outcomes favorable to the event A
n(S) = total number of outcomes
Situation II : THE RELATIVE FREQUENCY APPROACH
If we try to apply the classical definition of probability in the second experiment, we find that
we cannot say that consumers will equally like the taste of the soup. Moreover, we do not
know as to how many persons have been tested. This implies that we should have the
past data on people who were administered the soup and the number that liked the taste. In
the absence of past data, we have to undertake an experiment, where we administer the
taste test on a group of people to check its effect.
262
Caritas Business College BUSINESS STATISTICS – OAD 221
The Relative Frequency Approach is used to compute probability in such cases. As per
this approach, the probability of occurrence of an event is given by the observed relative
frequency of an event in a very large number of trials. In other words, the probability of
occurrence of an event is the ratio of the number of times the event occurs to the total
number of trials. The probability of the event B:
P(B) = n........................................................ (8-2)
N
Where n = the number of times the event occurs
N = total number of trials
It is appreciated in this approach that, in order to take such a measure, we should have the
soup tested for a large number of people. In other words, the total number of trials in the
experiment should be very large.
Situation III : THE SUBJECTIVE APPROACH
The third situation seems apparently similar to the second one. We may be tempted here to
apply the Relative Frequency Approach. We may calculate the probability of the event that
the venture is a success as the ratio of number of successful ventures to the total number of
such ventures undertaken i.e. the relative frequency of successes will be a measure of the
probability.
However, the calculation here presupposes that either
(a) it is possible to do an experiment with such ventures, or
(b) that past data on such ventures will be available
In practice, a solar power plant being a relatively new development involving the latest
technology, past experiences are not available. Experimentation is also ruled out because of
high cost and time involved, unlike the taste testing situation. In such cases, the only way out
is the Subjective Approach to probability. In this approach, we try to assess the probability
from our own experiences. We may bring in any information to assess this. In the situation
263
Caritas Business College BUSINESS STATISTICS – OAD 221
cited, we may, perhaps, look into the performance of the commissioning authority in other
new and related technologies.
Therefore the Subjective Approach involves personal judgment, information, intuition, and
other subjective evaluation criteria. A physician assessing the probability of a patient's
recovery and an expert assessing the probability of success of a merger offer are both
making a personal judgment based upon what they know and feel about the situation. The
area of subjective probability - which is relatively new, having been first developed in the
1930s - is somewhat controversial. One person's subjective probability may very well be
different from another person's subjective probability of the same event. We may note here
that since the assessment is a purely subjective one, it will vary from person to person and,
therefore, subjective probability is also called Personal Probability.
8.3.1 Three Approaches – A Comparative View
As already noted, the different approaches have evolved to cater to different kinds of
situations. So these approaches are not contradictory to one another. In fact, these
complement each other in the sense that where one fails, the other becomes applicable.
These are identical inasmuch as probability is defined as a ratio or a weight assigned to the
occurrence of an event. However, in contrast to the Subjective measure of the third
approach, the first two approaches - Classical and Relative Frequency - provide an objective
measure of probability in the sense that no personal judgment is involved.
We can bring out the commonality between the Classical Approach and the Relative
Frequency Approach with the help of an example. Let us assume that we are interested in
finding out the chances of getting a head in the toss of a coin. By now, you would have come
up with the answer by the Classical Approach, using the argument, that there are two
outcomes, heads and tails, which are equally likely. Hence, given that a head can occur only
once, the probability is ½ : Consider the following alternative line of argument, where the
264
Caritas Business College BUSINESS STATISTICS – OAD 221
probability can be estimated using the Relative Frequency Approach. If we toss the coin for
a sufficiently large number of times and note down the number of times the head occurs, the
proportion of times that a head occurs will give us the required probability.
Figure 8-1 P(H) = n/N→1/2 as N→α
Thus, given our definition of the approaches, we find both the arguments to be valid. This
brings out, in a way, the commonality between the Relative Frequency and the Classical
Approach. The difference, however, is that the probability computed by using the Relative
Frequency Approach will be tending to be ½ with a large number of trials; moreover an
experiment is necessary in this case. In comparison, in the Classical Approach, we know
apriori that the chances are ½ , based on our assumption of "equally likely" outcomes.
Example 8-1
A fair coin is tossed twice. Find the probabilities of the following events:
(a) A, getting two heads
(b) B, getting one head and one tail
(c) C, getting at least one head or one tail
(d) D, getting four heads
Solution: Being a Two-Trial Coin Tossing Experiment, it gives rise to the following On = 2n
= 4, possible equally likely outcomes:
HH HT TH TT
265
Caritas Business College BUSINESS STATISTICS – OAD 221
Thus, for the sample space N(S) = 4
We can use the Classical Approach to find out the required probabilities.
(a) For the event A, the number of favourable cases are:
n(A) = 1 { HH }
So the required probability
P(A) = n ( A )
N(S)
=
1
(b) For the event B, the number of favourable cases are:
n(B) = 2 { HT, TH }
So the required probability
P(B) = n ( B )
N(S)
=
2
1
=2
(c) For the event C, the number of favourable cases are:
n(C) = 4 { HH, HT, TH, TT }
So the required probability
P(C) = n ( A )
N(S)
=
4
=1
(d) For the event D, the number of favourable cases are:
266
Caritas Business College BUSINESS STATISTICS – OAD 221
n(D) = 0
267
Caritas Business College BUSINESS STATISTICS – OAD 221
So the required probability
P(D) = n ( D )
N(S)
=
0
=0
It may be noted that the occurrence of C is certainty, whereas D is an impossible event.
Example 8-2
A newspaper boy wants to find out the chances that on any day he will be able to sell more
than 90 copies of The Times of India. From his dairy where he recorded the daily sales of
the last year, he finds out that out of 365 days, on 75 days he had sold 80 copies, on 144
days he had sold 85 copies, on 62 days he had sold 95 copies and on 84 days he had sold
100 copies of The Times of India. Find out the required probability for the newspaper boy.
Solution: Taking the Relative Frequency Approach, we find:
Sales(Event) No. of Days (Frequency) Relative Frequency
80 75 75/365
85 144 144/365
95 62 62/365
100 84 84/365
Thus, the number of days when his sales were more than 90 = (62 + 84) days = 146 days
So the required probability
P(Sales > 90) = n
N
= 146
365
= 0.4
268
Caritas Business College BUSINESS STATISTICS – OAD 221
8.3.2 Probability Axioms
All the three approaches to probability theory share the same basic axioms. These axioms are
fundamental to probability theory and provide us with unified approach to probability.
The axioms are:
(a) The probability of an event A, written as P(A), must be a number between
zero and one, both values inclusive. Thus
0 ≤ P(A) ≤ 1.....................................................(8-3)
(b) The probability of occurrence of one or the other of all possible events is
equal to one. As S denotes the sample space or the set of all possible
events, we write
P(S) = 1...........................................................(8-4)
Thus in tossing a coin once; P(a head or a tail) = 1.
(c) If two events are such that occurrence of one implies that the other cannot
occur, then the probability that either one or the other will occur is equal to the
sum of their individual probabilities. Thus, in a coin-tossing situation, the
occurrence of a head rules out the possibility of occurrence of tail. These
events are called mutually exclusive events. In such cases then, if A and B
are the two events respectively, then
P (A or B) = P (A) + P (B)
i.e. P(Head or Tail) = P (Head) + P (Tail)
It follows from the last two axioms that if two mutually exclusive events form the sample
space of the experiment, then
P(A or B) = P(A) + P(B) = 1; thus P (Head) + P (Tail) = 1
If two or more events together define the total sample space, the events are said to be
collectively exhaustive.
Given the above axioms, we may now define probability as a function, which assigns
probability value P to each sample point of an experiment abiding by the above
axioms. Thus, the axioms themselves define probability.
269
Caritas Business College BUSINESS STATISTICS – OAD 221
8.3.3 Interpretation of a Probability
From our discussion so far, we can give a general definition of probability:
Probability is a measure of uncertainty. The probability of event A
is a quantitative measure of the likelihood of the event's occurring.
We have also seen that 0 and 1, both values inclusive, sets the range of values that the
proba- bility measure may take. In other words 0 ≤ P(A) ≤ 1
When an event cannot occur (impossible event), its probability is zero. The probability of the
empty set is zero: P(Φ) = 0. In a deck where half the cards are red and half are black, the
probability of drawing a green card is zero because the set corresponding to that event is the
empty set: there are no green cards.
Events that are certain to occur have probability 1.00. The probability of the entire sample
space S is equal to 1.00: P(S) = 1.00. If we draw a card out of a deck, 1 of the 52 cards in
the deck will certainly be drawn, and so the probability of the sample space, the set of all 52
cards, is equal to 1.00.
Within the range of values 0 to 1, the greater the probability, the more confidence we have in
the occurrence of the event in question. A probability of 0.95 implies a very high confidence
in the occurrence of the event. A probability of 0.80 implies a high confidence. When the
probability is 0.5, the event is as likely to occur as it is not to occur. When the probability is
0.2, the event is not very likely to occur. When we assign a probability of 0.05, we believe
the event is unlikely to occur, and so on. Figure 8-2 is an informal aid in interpreting
probability.
267
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 8-2 Interpretation of a Probability
Note that probability is a measure that goes from 0 to 1. In everyday conversation we often
describe probability in less formal terms. For example, people sometimes talk about odds. If
the odds are 1 to 1, the probability 1 i .e . ; if the odds are 1 to 2, the probability is
is 1 1
1 2
1
i .e . ; and so on. Also, people sometimes say, "The probability is 80 percent."
1 1
2 3
Mathematically, this probability is 0.80.
8.4 PROBABILITY RULES
We have seen how to compute probabilities in certain situations. The nature of the events
were relatively simple, so that direct application of the definition of probability could be used
for computation. Quite often, we are interested in the probability of occurrence of more
complex events. Consider for example, that you want to find the probability that a king or a
club will occur in a draw from a deck of 52 cards. Similarly, on examining couples with two
children, if one child is known as a boy, you may be interested in the probability of the event
of both the children being boys. These two situations, we find, are not as simple as those
discussed in the earlier section. As a sequel to the theoretical development in the field of
probability, certain results are available which help us in computing probabilities in such
situations. Now we will explore these results through examples.
8.4.1 THE UNION RULE
A very important rule in probability theory, the Rule of Unions (also called Addition
Theorem) allows us to write the probability of the union of two events in terms of the
probabilities of the two events and the probability of their intersection.
Consider two events A and B defined over the sample space S, as shone in Figure 8-3
2
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 8-3 Two Overlapping Events A and B
We may
define
P(A B) n(A B)
=
N(S)
n(A) n(B) n(A B
= N(S)
n(A) n(A B
= n(B
)
N(S) N(S N(S)
)
= P(A) P(B) P(A B
Thus, the rule of unions is:
P(A B)=P(A) P(B) P(A B …………(8-5)
The probability of the intersection of two events P ( A B is called their joint probability.
The meaning of this rule is very simple and intuitive: When we add the probabilities of A and
B, we are measuring, or counting, the probability of their intersection twice—once when
measuring the relative size of A within the sample space and once when doing this with B.
Since the relative size, or probability, of the intersection of the two sets is counted twice, we
subtract it once so that we are left with the true probability of the union of the two events.
The rule of unions is especially useful when we do not have the sample space for the union
of events but do have the separate probabilities.
Example 8-3
269
Caritas Business College BUSINESS STATISTICS – OAD 221
A card is drawn from a well-shuffled pack of playing cards. Find the probability that the card
drawn is either a club or a king.
270
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution: Let A be the event that a club is drawn and B the event that a king is drawn. Then,
P(A B)=P(A) P(B) P(A B
= 13/52 + 4/52 – 1/52
= 16/52
= 4/13
Example 8-4
Suppose your chance of being offered a certain job is 0.45, your probability of getting
another job is 0.55, and your probability of being offered both jobs is 0.30. What is the
probability that you will be offered at least one of the two jobs?
Solution: Let A be the event that the first job is offered and B the event that the second job is
offered. Then,
P ( A ) 0 .45 P(B) 0 .5 and P ( A B 0 .30
5
So, the required probability is given as:
P(A B)=P(A) P(B) P(A B
= 0.45 + 0.55 – 0.30
= 0.70
Mutually Exclusive Events
When the sets corresponding to two events are disjoint (i.e., have no intersection), the two
events are called mutually exclusive (see Figure 8-4).
Figure 8-4 Two Mutually Exclusive Events A and B
271
Caritas Business College BUSINESS STATISTICS – OAD 221
For mutually exclusive events, the probability of the intersection of the events is zero. This is
so because the intersection of the events is the empty set, and we know that the probability
of the empty set is zero.
For mutually exclusive events A and B:
P(A B
…………(8.6)
0
This fact gives us a special rule for unions of mutually exclusive events. Since the probability
of the intersection of the two events is zero, there is no need to P(A B when
subtract
the probability of the union of the two events is computed. Therefore,
For mutually exclusive events A and B:
P(A B)=P(A) P(B …………(8.7)
)
This is not really a new rule since we can always use the rule of unions for the union of two
events: If the events happen to be mutually exclusive, we subtract zero as the probability of
the intersection.
Example 8-5
A card is drawn from a well-shuffled pack of playing cards. Find the probability that the card
drawn is either a king or a queen.
Solution: Let A be the event that a king is drawn and B the event that a queen is drawn.
Since A and B are two mutually exclusive events, we have,
P(A B)=P(A) P(B)
= 4/52 + 4/52
= 8/52
= 2/13
We can extend the Rule of Unions to three (or more) events. Let A, B, and C be the
three events defined over the sample space S, as shown in Figure 8-5
Then, the Rule of Unions is
272
Caritas Business College BUSINESS STATISTICS – OAD 221
P(A B C)=
P(A) P(B) P (C ) P(A B) P(B C) P(A C) P(A B C)
…………(8.8)
Figure 8-5 Three Overlapping Events A, B and C
When the three events are mutually exclusive (see Figure 8-6), the Rule of Unions is
P(A B C)=P(A) P(B) P (C ) …………(8.9)
Figure 8-6 Three Mutually Exclusive Events A, B and C
Example 8-6
A card is drawn from a well-shuffled pack of playing cards. Find the probability that the card
drawn is
(a) either a heart or an honour or king
(b) either an ace or a king or a queen
Solution: (a) Let A be the event that a heart is drawn, B the event that an honour is drawn
and C the event that a king is drawn. So we have
n(A) = 13 n(B) = 20 n(C) = 4
n(A B n(B C n(A C 1
5 4
273
Caritas Business College BUSINESS STATISTICS – OAD 221
and n(A B C 1
274
Caritas Business College BUSINESS STATISTICS – OAD 221
The required probability (using Eq. (8.8) is
P ( A B C ) = 13/52 + 20/52 + 4/52 – 5/52 – 4/52 – 1/52 +1/52
= 28/52
= 7/13
(b) Let A be the event that an ace is drawn, B the event that a king is drawn and C
the event that a queen is drawn. So we have
n(A) = 4 n(B) = 4 n(C) = 4
Since A, B and C are mutually exclusive events, the required probability (using Eq. (8.9) is
P ( A B C ) = 4/52 + 4/52 + 4/52
= 12/52
= 3/13
8.4.2 THE COMPLEMENT RULE
The Rule of Complements defines the probability of the complement of an event in terms
of the probability of the original event. Consider event A defined over the sample space S.
The
complement of set A, denoted by A , is a subset, which contains all outcomes, which do not
belong to A (see Figure 8-7).
Figure 8-7 Complement of an Event
In other words A+A =S
so P(A + A ) = P(S)
or P(A) + P( A ) = 1
or P( A ) = 1 - P(A)..............................................(8.10)
275
Caritas Business College BUSINESS STATISTICS – OAD 221
Eq. (8.10) is our Rule of Complements. As a simple example, if the probability of rain
tomorrow is 0.3, then the probability of no rain tomorrow must be 1 - 0.3 = 0.7. If the
probability of drawing a king is 4/52, then the probability of the drawn card's not being a
king is 1 - 4/52 = 48/52.
Example 8-7
Find the probability of the event of getting a total of less than 12 in the experiment of
throwing a die twice.
Solution: Let A be the event of getting a total
12. Then we have,
A = {6,6} and P(A) = 1/36
The event of getting a total of less than 12 is the complement of A, so the required
probability is
P( A ) = 1 - P(A)
P( A ) = 1 – 1/36
P( A ) = 35/36
8.4.3 THE CONDITIONAL PROBABILITY RULE
As a measure of uncertainty, probability depends on information. We often face situations
where the probability of an event A is influenced by the information that another event B has
occurred. Thus, the probability we would give the event "Xerox stock price will go up
tomorrow" depends on what we know about the company and its performance; the
probability is conditional upon our information set. If we know much about the company, we
may assign a different probability to the event than if we know little about the company. We
may define the probability of event A conditional upon the occurrence of event B. In this
example, event A may be the event that the stock will go up tomorrow, and event B may be
a favorable quarterly report.
276
Caritas Business College BUSINESS STATISTICS – OAD 221
Consider two events A and B defined over the sample space S, as shown in Figure 8-8
Figure 8-8 Conditional Probability of Event A
Thus, the probability of event A given the occurrence of event B is
P(A/ B)
n(A B)
n(B)
n(A B)
P(A / B) N
n(B)
N
P(A/ B) P(A B …………(8.11)
)
P(B)
The vertical line in P ( A / B ) is read given, or conditional upon.
Therefore, the probability of event A given the occurrence of event B is defined as the
probability of the intersection of A and B, divided by the probability of event B.
Example 8-8
For an experiment of throwing a die twice, find the probability:
(a) of the event of getting a total of 9, given that the die has shown up points
between 4 and 6 (both inclusive)
(b) of the event of getting points between 4 and 6 (both inclusive), given that a total
of 9 has already been obtained
Solution: Let getting a total 9 be the event A and the die showing points between 4 and
6 (both inclusive) be the event B
Thus, N(S) = 36 and A = {(3,6) (4,5) (5,4) (6,3)}
277
Caritas Business College BUSINESS STATISTICS – OAD 221
B = {(4,4) (4,5) (4,6) (5,4) (5,5) (5,6) (6,4) (6,5) (6,6)}
and P ( A
B ) = {(4,5) (5,4)}
So n(A) = 4 n(B) = 9 n(A B)=2
So the required probabilities are
(a) P(A B)
P(A/ B)
P(B)
P(A/ B) 2 / 36
9 / 36
P(A/ B) 2
9
P(B A
(b) P(B / A) )
P(A)
P(B / A)
2 / 36
4 / 36
P(B / A)
1
8.4.4 THE PRODUCT RULE
The Product Rule (also called Multiplication Theorem) allows us to write the probability
of the simultaneous occurrence of two (or more) events.
In the conditional probability rules
P(A/ B)
P(A B)
P(B)
and P(B / A) P(B A)
P(A)
278
Caritas Business College BUSINESS STATISTICS – OAD 221
A B or B A is the event A and B occur simultaneously. So rearranging the
conditional probability rules, we have our Product Rule
P(A B)
P ( A / B ). P ( B )
and P(A B) P ( B / A ). P ( A …………(8.12)
)
279
Caritas Business College BUSINESS STATISTICS – OAD 221
The Product Rule states that the probability that both A and B will occur simultaneously is
equal to the probability that B (or A) will occur multiplied by the conditional probability that A
(or B) will occur, when it is known that B (or A) is certain to occur or has already
occurred.
Example 8-9
A box contains 10 balls out of which 2 are green, 5 are red and 3 are black. If two balls are
drawn at random, one after the other without replacement, from the box. Find the
probabilities that:
(a) both the balls are of green color
(b) both the balls are of black color
(c) both the balls are of red color
(d) the first ball is red and the second one is black
(e) the first ball is green and the second one is red
Solution: (a) P (G 1 G 2 ) P (G / G 1 ). P (G 1 )
2
1 2
9 x 10
1
45
P ( B1
(b) B 2 ) P ( B 2 / B1 ). P ( B1 )
2 3
9 x 10
1
15
(c) P ( R1 R2 ) P ( R / R1 ). P ( R1 )
2
4 5
9 x 10
2
9
(d) P ( R1 B )
2
280
Caritas Business College BUSINESS STATISTICS – OAD 221
P(B2
/ R1 ). P ( R1 )
281
Caritas Business College BUSINESS STATISTICS – OAD 221
3 5
9 x 10
1
6
(e) P (G 1 R2) P ( R / G 1 ). P (G 1 )
2
5 2
9 x 10
1
9
Example 8-10
A consulting firm is bidding for two jobs, one with each of two large multinational
corporations. The company executives estimate that the probability of obtaining the
consulting job with firm A, event A, is 0.45. The executives also feel that if the company
should get the job with firm A, then there is a 0.90 probability that firm B will also give the
company the consulting job. What are the company's chances of getting both jobs?
Solution: We are given P(A) = 0.45. We also know that P(B / A) = 0.90, and we are looking
for P ( A
B ) , which is the probability that both A and B will occur.
So P(A B) P ( B / A ). P ( A )
P(A B)
0 .90 x 0 .45
0 .405
Independent Events
Two events are said to be independent of each other if the occurrence or non-occurrence
of one event in any trial does not affect the occurrence of the other event in any trial.
Events A and B are independent of each other if and only if the following three conditions
hold: Conditions for the independence of two events A and B:
P(A/B) P(A …………(8.13a)
)
282
Caritas Business College BUSINESS STATISTICS – OAD 221
P(B / A)
P(B …………(8.13b)
)
and P(A B) P ( A ). P ( B …………(8.14)
)
283
Caritas Business College BUSINESS STATISTICS – OAD 221
The first two equations have a clear, intuitive appeal. The top equation says that when A and
B are independent of each other, then the probability of A stays the same even when we
know that B has occurred - it is a simple way of saying that knowledge of B tells us nothing
about A when the two events are independent. Similarly, when A and B are independent,
then knowledge that A has occurred gives us absolutely no information about B and its
likelihood of occurring.
The third equation, however, is the most useful in applications. It tells us that when A and B
are independent (and only when they are independent), we can obtain the probability of the
joint occurrence of A and B (i.e. the probability of their intersection) simply by multiplying the
two separate probabilities. This rule is thus called the Product Rule for Independent
Events.
As an example of independent events, consider the following: Suppose I roll a single die.
What is the probability that the number 5 will turn up? The answer is 1/6. Now suppose that I
told you that I just tossed a coin and it turned up heads. What is now the probability that the
die will show the number 5? The answer is unchanged, 1/6, because events of the die and
the
coin are independent of each other. We see P (6 / H ) P (6 ) , which is the first rule
that
above.
The rules for union and intersection of two independent events can be extended to
sequences of more than two events.
Intersection Rule
The probability of the intersection of several independent events A1, A2, ……is just the
product of separate probabilities i.e.
P ( A1 A2 A3 ) P ( A1 ). P ( A2 ). P ( A3 …………(8.15)
).........
Union Rule
284
Caritas Business College BUSINESS STATISTICS – OAD 221
The probability of the union of several independent events A1, A2, ……is given by the
following equation
P ( A1 A2 A3 ..........) 1 …………(8.16)
P ( A1 ). P ( A2 ). P ( A3
).........
The union of several events is the event that at least one of the events happens.
Example 8-11
A problem in mathematics is given to five students A, B,C, D and E. Their chances of solving
it are 1/2, 1/3, 1/3, 1/4 and 1/5 respectively. Find the probability that the problem will
(a) not be solved
(b) be solved
Solution: (a) The problem will not be solved when none of the students solve it. So the
required probability is:
P ( problem will not be
solved ) P ( A ). P ( B ). P (C ). P ( D ). P ( E )
(1 1 / 2 ).( 1 1 / 3).( 1 1 / 3).( 1 1 / 4 ).( 1 1 / 5 )
2 / 15
(b) The problem will be solved when at least one of the students solve it. So the
required probability is:
P(A B C D E) 1
P ( A ). P ( B ). P (C ). P ( D ). P ( E )
1 2 / 15
13 / 15
8.5 BAYES’ THEOREM
As we have already noted in the introduction, the basic objective behind calculating
probabilities is to help us in making decisions by quantifying the uncertainties involved in the
situations. Quite often, whether it is in our personal life or our work life, decision-making is an
ongoing process. Consider for example, a seller of winter garments, who is interested in
the demand of the product. In deciding on the amount he should stock for this winter, he
285
Caritas Business College BUSINESS STATISTICS – OAD 221
has
286
Caritas Business College BUSINESS STATISTICS – OAD 221
computed the probability of selling different quantities and has noted that the chance
of selling a large quantity is very high. Accordingly, he has taken the decision to stock a
large quantity of the product. Suppose, when finally the winter comes and the season ends,
he discovers that he is left with a large quantity of stock. Assuming that he is in this
business, he feels that the earlier probability calculation should be updated given the new
experience to help him decide on the stock for the next winter.
Similar to the situation of the seller of winter garment, situations exist where we are
interested in an event on an ongoing basis. Every time some new information is available,
we do revise our odds mentally. This revision of probability with added information is
formalised in probability theory with the help of famous Bayes' Theorem. The theorem,
discovered in 1761 by the English clergyman Thomas Bayes, has had a profound impact on
the development of statistics and is responsible for the emergence of a new philosophy of
science. Bayes himself is said to have been unsure of his extraordinary result, which was
presented to the Royal Society by a friend in 1763 - after Bayes' death. We will first
understand The Law of Total Probability, which is helpful for derivation of Bayes' Theorem.
8.5.1 The Law of Total Probability
Consider two events A and B. Whatever may be the relation between the two events, we can
always say that the probability of A is equal to the probability of the intersection of A and B,
plus the probability of the intersection of A and the complement of B (event B ).
P(A) P(A B) P(A B)
or P(A) P ( A / B ). P ( B ) …………(8.17)
P ( A / B ). P ( B
)
The sets B and B form a partition of the sample space. A partition of a space is the
division of the sample space into a set of events that are mutually exclusive (disjoint sets)
and cover
the whole space. Whatever event B may be, either B or B must occur, but not both. Figure
287
Caritas Business College BUSINESS STATISTICS – OAD 221
8-9 demonstrates this situation and the law of total probability.
288
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 8-9 Total Probability of Event A
The law of total probability may be extended to more complex situations, where the sample
space X is partitioned into more than two events. Say, we have partition of the space into a
collection of n sets B1, B2,………Bn .The law of total probability in this situation is:
P(A)
P(A
n
Bi )
i
1
or P(A)
P ( A / B i ). P ( B i
n …………(8.18)
)
i
1
Figure 8-10 shows the partition of a sample space into five events B 1, B2, B3, B4 and B5 ; and
shows their intersections with set A.
Figure 8-10 Total Probability of Event A
We can demonstrate the rule with a more specific example. Let us define A as the event that
an honour card is drawn out of a deck of 52 cards (the honour cards are the aces, kings,
queens, jacks and 10). Letting H, C, D, and S denote the events that the card drawn is a
heart, club, diamond, or spade, respectively, we find that the probability of an honour card is:
289
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 8-11 Total Probability of Event A: An Honour Card
P(A) P(A H) P(A C) P(A D) P(A S)
= 5/52 + 5/52 + 5/52 + 5/52
= 20/52
= 5/13
which is what we know the probability of an honour card to be just by counting 20 honour
cards out of a total of 52 cards in the deck. The situation is shown in Figure 8-11.
As can be seen from the figure, the event A is the set addition of the intersections of A
with each of the four sets H, D, C, and S.
Example 8-12
A market analyst believes that the stock market has a 0.70 probability of going up in the next
year if the economy should do well, and a 0.20 probability of going up if the economy should
not do well during the year. The analyst believes that there is a 0.80 probability that the
economy will do well in the coming year. What is the probability that stock market will go up
next year?
Solution: Let U be the event that the stock market will go and W is the event that the
economy will do well in the coming year.
Then
P (U )
P (U / W ). P (W ) P (U
/ W ). P (W )
( 0 .70 )( 0 .80 ) ( 0 .20 )( 0 .20 )
0 .56 0 .04
. 0 .60
BAYES’ THEOREM
We will now develop the Bayes’ theorem. Bayes' theorem is easily derived from the law of
total probability and the definition of conditional probability.
By definition of conditional probability, we have
290
Caritas Business College BUSINESS STATISTICS – OAD 221
P(B / A) P(B A
) …………(8.19)
P(A)
By product rule, we have
P(B
A) ) P ( A B) P ( A / B ). P ( B …………(8.20)
)
Substituting Eq.(8.19) in Eq.(8.20), we have
P(B / A) P ( A / B ). P ( B
) …………(8.21)
P(A)
By the law of total probability, we have
P ( A ) P ( A / B ). P ( B )
P ( A / B ). P ( B )
Substituting this expression for P(A) in the denominator of Eq.(8.21), we have the Bayes’
theorem
P(B / A) P ( A / B ). P ( B
…………(8.22)
)
P ( A / B ). P ( B ) P ( A / B ). P ( B )
Thus the theorem allows us to reverse the conditionality of events: we can obtain the
probability of B given A from the probability of A given B(and other information).
As we see from the theorem, the probability of B given A is obtained from the probabilities
of B and B and from the conditional probabilities of A given B and A given B .
The probabilities P(B) and P( B ) are called prior probabilities of the events B and B ; the
probability P(B /A) is called the posterior probability of B. It is possible to write
Bayes'
theorem in terms of B and A, thus giving the posterior probability of B , P( B /A). Bayes'
theorem may be viewed as a means of transforming our prior probability of an event B into a
posterior probability of the event B - posterior to the known occurrence of event A.
291
Caritas Business College BUSINESS STATISTICS – OAD 221
The Bayes' theorem can be extended to a partition of more than two sets. This is done by
using the law of total probability involving a partition in sets B 1, B2, ……… Bn .The resulting
form of Bayes' theorem is:
292
Caritas Business College BUSINESS STATISTICS – OAD 221
P( P ( A / B i ). P ( B i
Bi / A) …………(8.23)
)
n
P ( A / B i ). P ( B i )
i
1
The theorem gives the probability of one of the sets in the partition Bi, given the occurrence
of event A.
Example 8-13
An Economist believes that during periods of high economic growth, the Indian Rupee
appreciates with probability 0.70; in periods of moderate economic growth, it
appreciates with probability 0.40; and during periods of low economic growth, the Rupee
appreciates with probability 0.20.During any period of time the probability of high economic
growth is 0.30; the probability of moderate economic growth is 0.50 and the probability of
low economic growth is 0.20. Suppose the Rupee value has been appreciating during the
present period. What is the probability that we are experiencing the period of (a) high, (b)
moderate, and (c) low, economic growth?
Solution: Our partition consists of three events: high economic growth (event H), moderate
economic growth (event M) and low economic growth (event L). The prior probabilities of
these events are:
P(H) = 0.30 P(M) = 0.50 P(L) = 0.20
Let A be the event that the rupee appreciates. We have the conditional probabilities
P(A / H) = 0.70 P(A / M) = 0.40 P(A / L) = 0.20
By using the Bayes’ theorem we can find out the required probabilities
P(H /A), P(M / A) and P(L / A)
(a) P(H /A)
293
Caritas Business College BUSINESS STATISTICS – OAD 221
P(H P ( A / H ). P ( H )
/ A) P ( A / H ). P ( H ) P ( A / L ). P ( L )
P ( A / M ). P ( M )
( 0 .70 )( 0 .30 )
( 0 .70 )( 0 .30 ) ( 0 .40 )( 0 .50 ) ( 0 .20 )( 0 .20 )
0 .467
(b) P(M /A)
P ( A / M ). P ( M )
P(M / A) P ( A / H ). P ( H ) P ( A / L ). P ( L )
P ( A / M ). P ( M )
( 0 .40 )( 0 .50 )
( 0 .70 )( 0 .30 ) ( 0 .40 )( 0 .50 ) ( 0 .20 )( 0 .20 )
0 .444
(c) P(L /A)
P(L / A) P ( A / L ). P ( L )
P ( A / H ). P ( H ) P ( A / M ). P ( M ) P ( A / L ). P ( L )
( 0 .20 )( 0 .20 )
( 0 .70 )( 0 .30 ) ( 0 .40 )( 0 .50 ) ( 0 .20 )( 0 .20 )
0 .089
8.6 SOME COUNTING CONCEPTS
If there are n events and event i can occur in Ni possible ways, then the number of ways in
which the sequence of n events may occur is
N1. N2. N3 .……….Nn............................................................... (8.24)
Suppose that a bank has two branches, each branch has two departments, and each
department has four employees. Then there are (2)(2)(4) choices of employees, and the
probability that a particular one will be randomly selected is 1/(2)(2)(4) = 1/16.
We may view the choice as done sequentially: First a branch is randomly chosen, then a
department within the branch, and then the employee within the department. This is
demonstrated in the tree diagram in Figure 8-12.
286
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 8-12 Tree Diagram
For any positive integer n, we define n factorial as
n(n- 1)(n- 2) ………1..............................................................(8.25)
We denote n factorial by n!. The number n! is the number of ways in which n objects can be
ordered. By definition, 0! = 1.
For example, 5! is the number of possible arrangements of five objects. We have 5! = (5) (4)
(3) (2) (1) = 120. Suppose that five applications arrive at a center on the same day, all
written at different times. What is the probability that they will be read in the order in which
they were written? Since there are 120 ways to order five applications, the probability of a
particular order (the order in which the applications were written) is 1/120.
Permutations are the possible ordered selections of r objects out of a total of n objects. The
number of permutations of n objects taken r at a time is denoted by n P
r
n n!
P …………(8.26)
r
(n r
)!
Suppose that 4 people are to be randomly chosen out of 10 people who agreed to be
interviewed in a market survey. The four people are to be assigned to four interviewers. How
many possibilities are there? The first interviewer has 10 choices, the second 9 choices, the
third 8, and the fourth 7. Thus, there are (10)(9)(8)(7) = 5,040 selections. We can see that
this
is equal to n(n - l)(n - 2) ……… (n - r + 1), which is equal to n n!
!.
Pr
287
Caritas Business College BUSINESS STATISTICS – OAD 221
( n r )!
288
Caritas Business College BUSINESS STATISTICS – OAD 221
If choices are made randomly, the probability of any predetermined assignment of 4
people out of a group of 10 is 1/5,040.
Combinations are the possible selections of r items from a group of n items regardless of the
n
order of selection. The number of combinations is denoted by C r and is read n choose r.
We define the number of combinations of r out of n elements as
n n!
Cr …………(8.27)
r! ( n r )!
Suppose that 3 out of the 10 members of the board of directors of a large corporation are
to be randomly selected to serve on a particular task committee. How many possible
selections
are there? Using Eq. (8.27), we find that the number of combinations is n
n!
C r r! ( n r )! =
10!/(3!7!) = 120.
If the committee is chosen in a truly random fashion, what is the probability that the three-
committee members chosen will be the three senior board members? This is 1 combination
out of a total of 120, so the answer is 1/120 = 0.00833.
8.7 SELF-ASSESSMENT QUESTIONS
1. Explain what do you understand by the term ‘probability’. How is the concept of
probability is relevant to decision making under uncertainty?
2. What are different approaches to the definition of probability? Are these approaches
contradictory to one another? Which of these approaches you will apply for
calculating the probability that:
(a) A leap year selected at random, will contain 53 Monday.
(b) An item, selected at random from a production process, is defective.
(c) Mr. Bhupinder S. Hooda will win the assembly election from Kiloi.
3. With the help of an example explain the meaning of the following:
289
Caritas Business College BUSINESS STATISTICS – OAD 221
(a) Random experiment, and sample space
(b) An event as a subset of sample space
(c) Equally likely events
(d) Mutually exclusive events.
(e) Exhaustive events
(f) Elementary and compound events.
4. A proofreader is interested in finding the probability that the number of mistakes in a
page will be less than 10. From his past experience he finds that out of 3600 pages
he has proofed, 200 pages contained no errors, 1200 pages contained 5 errors, and
2200 pages contained 11 or more errors. Can you help him in finding the required
probability?
5. State and develop the Addition Theorem of probability for:
(a) mutually exclusive events
(b) overlapping events
(c) complementary events
5. Explain the concept of conditional probability with the help of a suitable example.
6. State and develop the Multiplication Theorem of probability for:
(a) dependent events
(b) independent event
7. State the Bayes’ Theoram of probability. Using an appropriate example, develop the
Bayesian probability rule and generalize it.
8. What do you understand by permutations and combinations?
(a) In how many ways we can select three players out of 12 players of the
Indian Cricket team, for playing in the World XI team?
(b) In how many ways can a sub-committee of 2 out of 6 members of the
executive committee of the employees’ association be constituted?
9. What is the probability that a non leap year, selected at random, will contain
290
Caritas Business College BUSINESS STATISTICS – OAD 221
(a) 52 Sundays? (b) 53 Sundays? (c) 54 Sundays?
10. A card is drawn at random from well shuffled deck of 52 cards, find the probability
that
(a) the card is either a club or diamond
(b) the card is not a king
(c) the card is either a face card or a club card.
11. From a well-shuffled deck of 52 cards, two cards are drawn at random.
(a) If the cards are drawn simultaneously, find the probability that these
consists of (i) both clubs, (ii) a king and a queen, (iii) a face card and a 8.
(b) If the cards are drawn one after the other with replacement. Find the
probability that these consists of (i) both clubs, (ii) a king and a queen, (iii) a
face card and a 8.
12. A problem in mathematics is given to four students A, B,C, and D their chances of
solving it are 1/2 , 1/3, 1/4 and 1/5 respectively. Find the probability that the problem
will
(a) be solved
(b) not be solved
13. The odds that A speaks the truth are 3:2 and the odds that B does so are 7:3. In what
percentage of cases are they likely to
(a) contradict each other on an identical point?
(b) agree each other on an identical point?
14. Among the sales staff engaged by a company 60% are males. In terms of
their professional qualifications, 70% of males and 50% of females have a degree in
marketing. Find the probability that a sales person selected at random will be
(a) a female with degree in marketing
(b) a male without degree in marketing
291
Caritas Business College BUSINESS STATISTICS – OAD 221
15. A and B play for a prize of Rs. 10,000. A is to throw a die first and is to win if he
throws 1: If A fails, B it to throw and is to win if he throws 2 or 1. If B fails, A is to
throw again and to win if he throws 3, 2 or 1: and so on. Find their respective
expectations.
16. A factory has three units A, B, and C. Unit A produces 50% of its products, and units
B and C each produces 25% of the products. The percentage of defective items
produced by A, B, and C units are 3%, 2% and 1%, respectively. If an item is
selected at random from the total production of the factory is found defective, what is
the probability that it is produced by:
(a) Unit A (b) Unit B (c) Unit C
8.8 SUGGESTED READINGS
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons.,
New Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
292
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 9: PROBABILITY DISTRIBUTIONS
Topic : Probability Distributions
Objectives: The overall objective of this lesson is to discuss the concept of random
variable and discrete probability distributions. After successful
completion of the lesson the students will be able to appreciate the
usefulness of probability distributions in decision-making and also
identify situations where Binomial and Poisson probability distributions
can be applied.
Structure
9.1 Introduction
9.2 Discrete Probability Distribution
9.3 Bernoulli Random Variable
9.4 The Binomial Distribution
9.5 The Poisson Distribution
9.6 Self-Assessment Questions
9.7 Suggested Readings
9.1 INTRODUCTION
In many situations, our interest does not lie in the outcomes of an experiment as such;
we may find it more useful to describe a particular property or attribute of the outcomes of an
experiment in numerical terms. For example, out of three births; our interest may be in
the
293
Caritas Business College BUSINESS STATISTICS – OAD 221
matter of the probabilities of the number of boys. Consider the sample space of 8 equally
likely sample points.
GGG GGB GBG BGG
GBB BGB BBG BBB
Now look at the variable “the number of boys out of three births”. This number varies
among sample points in the sample space and can take values 0,1,2,3, and it is random –
given to chance.
“A random variable is an uncertain quantity whose value depends on chance.”
A random variable may be…
Discrete if it takes only a countable number of values. For example, number of
dots on two dice, number of heads in three coin tossing, number of defective items,
number of boys in three births and so on.
Continuous if can take on any value in an interval of numbers (i.e. its possible
values are unaccountably infinite). For example, measured data on heights, weights,
temperature, and time and so on.
A random variable has a probability law - a rule that assigns probabilities to different values
of the random variable. This probability law - the probability assignment is called the
probability distribution of the random variable. We usually denote the random variable by
X. In this lesson, we will discuss discrete probability distributions. Continuous probability
distributions will be discussed in the next lesson.
9.2 DISCRETE PROBABILITY DISTRIBUTION
The random variable X denoting “the number of boys out of three births”, we introduced in
the introduction of the lesson, is a discrete random variable; so it will have a discrete
probability distribution. It is easy to visualize that the random variable X is a function of
sample space. We can see the correspondence of sample points with the values of the
random variable as follows:
294
Caritas Business College BUSINESS STATISTICS – OAD 221
BBB GGB GBG BGG
(X=0) (X=1)
GBB BGB BBG BBB
(X=2) (X=3)
The correspondence between sample points and the value of the random variable allows us to
determine the probability distribution of X as follows:
P(X=0) = 1/8 since one out of 8 equally likely points leads to X = 0
P(X=1) = 3/8 since three out of 8 equally likely points leads to X = 1
P(X=2) = 3/8 since three out of 8 equally likely points leads to X = 2
P(X=3) = 1/8 since one out of 8 equally likely points leads to X = 3
The above probability statement constitute the probability distribution of the random variable
X = number of boys in three births. We may appreciate how this probability law is obtained
simply by associating values of X with sets in the sample space. (For example, the set GGB,
GBG, BGG leads to X = 1). We may write down the probability distribution of X in table
format (see Table 9-1) or we may plot it graphically by means of probability Histogram (see
Figure 9-1a) or a Line chart (see Figure 9-1b).
Table 9-1 Probability Distribution of the Number of Boys out of Three Births
No. of Boys X Probability P(X)
0 1/8
1 3/8
2 3/8
3 1/8
295
Caritas Business College BUSINESS STATISTICS – OAD 221
(1, 0.375) (2, 0.375)
(1, 0.375)
P(
P(
(2, 0.375)
(0, 0.125) (3, 0.125)
(0, 0.125) (3, 0.125)
X X
Figure 9-1 Probability Distribution of the Number of Boys out of Three Births
The probability distribution of a discrete random variable X must satisfy the following two
conditions:
1. P(X = x) 0 for all values x
2. PX x 1
all x
These conditions must hold because the P(X = x) values are probabilities. First condition
specifies that all probabilities must be greater than or equal to zero, as we know from Lesson
8.
For the second condition, we note that for each value x, P(x) = P(X = x) is the probability of
the event that the random variable equals x. Since by definition all x means all the values the
random variable X may take, and since X may take on only one value at a time, the
occurrences of these values are mutually exclusive events, and one of them must take
place. Therefore, the sum of all the probabilities P(X = x) must be 1.00.
9.2.1 Cumulative Distribution Function
The probability distribution of a discrete random variable lists the probabilities of occurrence
of different values of the random variable. We may be interested in cumulative probabilities
of the random variable. That is, we may be interested in the probability that the value of
the
296
Caritas Business College BUSINESS STATISTICS – OAD 221
random variable is at most some value x. This is the sum of all the probabilities of the values i
of X that are less than or equal to x.
The cumulative distribution function (also called cumulative probability function) F(X =
x) of a discrete random variable X is
F(X = x) = P(X x) =
Pi
all i x
For example, to find the probability of at most two boys out of three births, we have
F(X = 2) = P(X 2) =
Pi
all i 2
= P(X = 0)+ P(X = 1)+ P(X = 2)
=1/3 + 3/8 + 3/8
= 7/8
9.2.2 Expected Value and Variance of a Discrete Random Variable
The expected value of a discrete random variable X is equal to the sum of all values of
the random variable, each value multiplied (weighted) by its probability.
= E(X) = x.Px
all x
The variance of a discrete random variable is given by
2 = V (X) = E [(X - )2] = (x ) 2 .P(x)
all x
In the same way we can calculate the other summary measures viz. skewness, kurtosis
and moments.
9.2.3 Probability Distributions are Theoretical Distributions
Consider a random variable X that measures the “number of heads” in a three-trial
coin tossing experiment. The probability distribution of X will be
X : 0 1 2 3
P(X) : 1/8 3/8 3/8 1/8
297
Caritas Business College BUSINESS STATISTICS – OAD 221
Now imagine this experiment is repeated 200 times, we may expect ‘no head’ and ‘three
heads’ will each occur 25 times; ‘one head’ and ‘two heads’ each will occur 75 times. Since
these results are what we expect on the basis of theory, the resultant distribution is called a
theoretical or expected distribution.
However, when the experiment is actually performed 200 times, the results, which we may
actually obtain, will normally differ from the theoretically expected results. It is quite possible
that in actual experiment ‘no head’ and ‘three heads’ may occur 20 and 28 times respectively
and ‘one head’ and ‘two heads’ may occur 66 and 86 times respectively. The distribution so
obtained through actual experiment is called the empirical or observed distribution.
In practice, however, assessing the probability of every possible value of a random variable
through actual experiment can be difficult, even impossible, especially when the probabilities
are very small. But we may be able to find out what type of random variable the one at
hand is by examining the causes that make it random. Knowing the type, we can often
approximate the random variable to a standard one for which convenient formulae are
available.
The proper identification of experiments with certain known processes in Probability theory
can help us in writing down the probability distribution function. Two such processes are the
Bernoulli Process and the Poisson Process. The standard discrete probability
distributions that are consequent to these processes are the Binomial and the Poisson
distribution. We will now look into the conditions that characterize these processes, and
examine the standard distributions associated with the processes. This will enable us to
identify situations for which these distributions apply.
Let us first study the Bernoulli random variable, named so in honor of the mathematician
Jakob Bernoulli (1654-1705). It is the building block for other random variables and the
resulting distributions we will study in this lesson.
298
Caritas Business College BUSINESS STATISTICS – OAD 221
9.3 BERNOULLI RANDOM VARIABLE
Suppose an operator uses a lathe to produce pins, and the lathe is not perfect in the sense
that it does not always produce a good pin. Rather, it has a probability p of producing a good
pin and (1 - p) of producing a defective one. Let us denote a good pin as “success” and a
defective pin as “failure”.
Just after the operator produces one pin, it is inspected; let X denote the "number of good
pins produced” i. e. “the number of successes”.
Now analyzing the trial- “inspecting a pin” and our random variable X-“number of
successes”, we note two important points:
The trial-“inspecting a pin” has only two possible outcomes, which are mutually
exclusive. Such a trial, whose outcome can only be either a success or a failure, is a
Bernoulli trial. In other words, the sample space of a Bernoulli trial is
S = {success, failure}
The random variable, X, that measures number of successes in one Bernoulli trial, is a
Bernoulli random variable. Clearly, X is 1 if the pin is good and 0 if it is defective.
It is easy to derive the probability distribution of Bernoulli random variable
X : 0 1
P(X) : p 1-p
If X is a Bernoulli random variable, we may write
X ~ BER (p)
Where ~ is read as “is distributed as” and BER stands for Bernoulli.
A Bernoulli random variable is too simple to be of immediate practical use. But it forms the
building block of the Binomial random variable, which is quite useful in practice. The
binomial random variable in turn is the basis for many other useful cases, such as Poisson
random variable.
299
Caritas Business College BUSINESS STATISTICS – OAD 221
9.4 THE BINOMIAL DISTRIBUTION
In the real world we often make several trials, not just one, to achieve one or more
successes. Let us consider such cases of several trials.
Consider n number of identically and independently distributed Bernoulli random variables
X1, X2 ………, Xn. Here, identically means that they all have the same p, and independently
means that the value of one X does not in any way affect the value of another. For example,
the value of X2 does not affect the value of X3 or X8 and so on. Such a sequence of
identically and independently distributed Bernoulli variables is called a Bernoulli Process.
Suppose an operator produces n pins, one by one, on a lathe that has probability p of
making a good pin at each trial, the sequence of numbers (1 or 0) denoting the good and
defective pins produced in each of the n trials is a Bernoulli process. For example, in the
sequence of nine trials denoted by
001011001
the third, fifth, sixth and ninth are good pins, or successes. The rest are failures.
In practice, we are usually interested in the total number of good pins rather than the
sequence of 1's and 0's. In the example above, four out of nine are good. In the general
case, let X denote the total number of good pins produced in n trials. We then have
X = X1 + X2 +………+ Xn
where all Xi ~ BER(p) and are independent.
The random variable that counts the number of successes in many
independent, identical Bernoulli trials is called a Binomial Random
Variable.
9.4.1 Conditions for a Binomial Random Variable
We may appreciate that the condition to be satisfied for a binomial random variable is that
the experiment should be a Bernoulli Process.
300
Caritas Business College BUSINESS STATISTICS – OAD 221
Any uncertain situation or experiment that is marked by the following three properties is
known as a Bernoulli Process:
There are only two mutually exclusive and collectively exhaustive outcomes in the
experiment i.e. S = {success, failure}
In repeated trials of the experiment, the probabilities of occurrence of these
events remain constant
The outcomes of the trials are independent of one another
The probability distribution of Binomial Random Variable is called the Binomial
Distribution
9.4.2 BINOMIAL PROBABILITY FUNCTION
Now we will develop the distribution of our Binomial random variable. To describe the
distribution of Binomial random variable we need two parameters, n and p we write
X ~ B (n, p)
to indicate that X is Binomially distributed with n number of independent trials and p
probability of success is each trial. The letter B stands for binomial.
Let us analyze the probability that the number of successes X in the n trials is exactly x
(obviously number of failures are n-x) i.e. X = x and x = 0,1,2,…………. n; as n trials are
made, at the best all n can be successes.
Now we know that there are nCx ways of getting x successes out of n trials. We also observe
that each of these nCx possibilities has px(1-p)n-x probability of occurrence corresponding to x
successes and (n-x) failures. Therefore,
P(X = x) = nCx p x (1-p)n-x for x = 0,1,2,………, n
This equation is the Binomial probability formula. If we denote the probability of failure as q
then the Binomial probability formula is
P(X = x) = nCx p x qn-x for x = 0,1,2,………, n
301
Caritas Business College BUSINESS STATISTICS – OAD 221
We may write down the Binomial probability distribution in table format (see Table 9-2)
Table 9-2 Binomial Distribution of
XX=x P(X = x)
n 0 n
0 C0 p q
n 1 n-1
1 C1 p q
… …
n x n-x
x Cx p q
… …
… …
n
n Cn pn q0
Each of the term for x = 0,1,2,………, n correspond to the Binomial expansion of (p + q)n
9.4.3 Characteristics of a Binomial Distribution
1. Expected Value or Mean
The expected value or the mean, denoted by , of a Binomial distribution is computed as
E (X) = = n x.P(x)
x0
An evaluation of will show that
=np
2. Variance
The variance, denoted by 2, of a Binomial distribution is computed as
V (X) =2 = E [(X - )2]
2
= (x
n ) .P(x)
x0
An evaluation of 2 will show that 2 = n p q
3. Moments about the Origin
The rth moment about the origin denoted by m0 , of a Binomial distribution is computed as:
r
0 r
mr = n x .P(x)
x0
302
Caritas Business College BUSINESS STATISTICS – OAD 221
For example, (a) First moment about the origin will be
0
m1 = n x.P(x)
x0
= np
=
(b) Second moment about the origin will be
0 2
m2 = n x .P(x)
x0
= n(n-1)p2 + np
4. Moments about the Mean
The rth moment about the mean denoted by m , of a binomial distribution is computed as:
r
mr = n (x ) .P(x)
r
x0
For example, (a) First moment about the mean will be
1
m1 =
n (x ) .P(x)
x0
=0
(b) Second moment about the mean will be
2
m2 = n (x ) .P(x)
x0
= npq
= 2
(c) Third moment about the mean will be
3
m3 =
n (x ) .P(x)
x0
= npq(q-p)
(d) Fourth moment about the mean will be
303
Caritas Business College BUSINESS STATISTICS – OAD 221
n
= (x ) 4 .P(x)
m4 x0
= 3(npq)2 + npq(1-6pq)
5. Skewness
To bring out the skewness of a Binomial distribution we can calculate, moment coefficient of
skewness, 1
1 = 1
(m )2
= 3
(m
2
)3
m
= 3 3
m
2
npq(q p)
= 3
npq
q p
=
npq
Evaluating 1
q p
= we note:
npq
the Binomial distribution is skewed to the right i.e. has positive skewness when 1 >
0, which is so when p < q
the Binomial distribution is skewed to the left i.e. has negative skewness when 1
< 0, which is so when p > q
the Binomial distribution is symmetrical i.e. has no skewness when 1 = 0, which is
so when p = q
Thus, n being the same, the degree of skewness in a Binomial distribution tends to vanish as
p approaches ½ i.e. as p ½
304
Caritas Business College BUSINESS STATISTICS – OAD 221
for a given value of p, as n increases the Binomial distribution moves to the right,
flattens and spreads out
As n ∝, 1 0, the distribution tends to be symmetrical.
5. Kurtosis
A measure of kurtosis of the Binomial distribution is given by the moment coefficient of
kurtosis 2
2 = 2 – 3
m4
= 3
2
3n2 p 2q 2 npq(1 6 pq)
= -3
2 2 2
n p q
1 6 pq
= npq
16
Evaluating 2 = we note
pq
npq
the Binomial distribution is leptokurtic when 2 > 0, which is so when 6pq <1.
the Binomial distribution is platykurtic when 2 < 0, which is so when 6pq >1.
the Binomial distribution is mesokurtic when 2 = 0, which is so when 6pq =1.
6. Normal approximation of the Binomial distribution
If n is large and if neither of p or q is too close to zero, the Binomial distribution can be
closely approximated by a Normal distribution with standardized variable
Z = X np
npq
7. Poisson approximation of the Binomial distribution
Binomial distribution can reasonably be approximated by the Poisson distribution when n is
infinitely large and p is infinitely small i. e. when
305
Caritas Business College BUSINESS STATISTICS – OAD 221
n ∝ and p 0
Example 9-1
Assuming the probability of male birth as ½, find the probability distribution of number of
boys out of 5 births.
(a) Find the probability that a family of 5 children have
(i) at least one boy
(ii) at most 3 boys
(b) Out of 960 families with 5 children each find the expected number of families with (i)
and (ii) above
Solution: Let the random variable X measures the number of boys out of 5 births. Clearly
X is a binomial random variable. So we apply the Binomial probability function to calculate
the required probabilities.
X ~ B (5, ½)
P(X = x) = nCx p x qn-x for x = 0, 1, 2, 3, 4, 5
The probability distribution of X is given below
X=x : 0 1 2 3 4 5
P(X = x) : 1/32 5/32 10/32 10/32 5/32 1/32
(a) The required probabilities are
(i) P(X 1) = 1- P(X = 0)
= 1- 1/32
= 31/32
(ii) P(X 3) = P(X = 0)+ P(X = 1)+ P(X = 2)+ P(X = 3)
= 1/32 + 5/32 + 10/32 + 10/32
= 26/32
306
Caritas Business College BUSINESS STATISTICS – OAD 221
(b) Out of 960 families with 5 children, the expected number of families with
(i) at least one boy = 960 * P(X 1)
= 960 * 31/32
= 930
(ii) at most 3 boys = 960 * P(X 3)
= 960 * 26/32
= 720
9.5 THE POISSON DISTRIBUTION
Poisson Distribution was developed by a French Mathematician Simeon D Poisson
(1781- 1840). If a random variable X is said to follow a Poisson Distribution, then its
probability
distribution is given by
e x = 0,1,2,………
x
P(X = x) =
x!
Where x is the number of successes
is the mean of the Poisson distribution and
e = 2.71828 (the base of natural logarithms)
The random variable X counts the number of successes in Poisson Process. A
Poisson process corresponds to a Bernoulli process under the following conditions:
the number of trials n, is infinitely large i.e. n ∝
the constant probability of success p, for each trial is infinitely small i.e. p 0
(obviously q 1)
np = is finite
We can develop the Poisson probability rule from the Binomial probability rule under the
above conditions.
307
Caritas Business College BUSINESS STATISTICS – OAD 221
Let us consider a Bernoulli process with n trials and probability of success in any trial
p = , where 0. Then, we know that the probability of x successes in n trials is given
n
by
x n x
P (X = x)
= nCx 1
n n
n! x
n x
= 1
x!(n x)! n n
n[nx 1][n 2]..........[n (x 1)]
n x
= 1
x! n n
x n n n n (x 1)
1 2
= . . x
x! n n ........... 1
n n
n .
x x
1 2 x n
1
= 1 1 ............1 1 1
x! n n n n n
x
1 2 x 1
Now if n ∝, then the terms, 1 ;1 ;............;1 and 1 will all be
n n
n n
tending to 1
and 1 e if n ∝
n
n
Thus we
have
e
P(X = x) = x = 0, 1, 2,………
x
x!
308
Caritas Business College BUSINESS STATISTICS – OAD 221
This equation is the probability distribution function of Poisson distribution.
Thus, we have seen that to describe the distribution of Poisson random variable we need
only one parameter , we write
If X ~ POI ()
309
Caritas Business College BUSINESS STATISTICS – OAD 221
Then P(X = x) = e x = 0, 1, 2,………
x
x!
We may write down the Poisson probability distribution in table format (see Table 9-3)
Table 9-3 Poisson Distribution of
XX=x P(X = x)
-
0 e
1 e- or P(X = 0)
2
2 or P(X = 1)
2
e
2!
… …
… …
x
x e or P(X = x-1)
x! x
… …
… …
Poisson distribution may be expected in situations where the chance of occurrence of any
event is small, and we are interested in the occurrence of the event and not in its non-
occurrence. For example, number of road accidents, number of defective items, number of
deaths in flood or because of snakebite or because of a rare disease etc. In these situations,
we know about the occurrence of an event although its probability is very small, but we do
not know how many times it does not occur. For instance, we can say that two road
accidents took place today, but it is almost impossible to say as to how many times, accident
fails to take place. The reason is that the number of trials is very large here and the nature of
event is of rare type. The Poisson random variable X, counts the number of times a rare
event occurs during a fixed interval of time or space.
9.5.1 Characteristics of a Poisson Distribution
1. Expected Value or Mean
The expected value or the mean, denoted by , of a Poisson distribution is computed as
E (X) = = x.P(x)
all x
310
Caritas Business College BUSINESS STATISTICS – OAD 221
An evaluation of mean will show that it is always itself.
2. Variance
The variance, denoted by 2, of a Poisson distribution is computed as
V (X) =2 = E [(X - )2]
= (x ) 2 .P(x)
all x
An evaluation of 2 will show that
2 =
3. Moments about the Origin
The rth moments about the origin denoted by m0 , of a Poisson distribution is computed as:
r
0
mr = x .P(x)
r
all x
For example, (a) First moment about the origin will be
0
m1 = x.P(x)
all x
=
(b) Second moment about the origin will be
0
m2 = x .P(x)
2
all x
= + 2
4. Moments about the Mean
The rth moments about the mean denoted by m , of a Poisson distribution is computed as:
r
mr = (x ) .P(x)
r
all x
For example, (a) First moment about the mean will be
311
Caritas Business College BUSINESS STATISTICS – OAD 221
m1 = (x ) .P(x)
1
all x
=0
(b) Second moment about the mean will be
2
m2 = (x ) .P(x)
all x
=2
=
(c) Third moment about the mean will be
m3 = (x ) .P(x)
3
all x
=
(d) Fourth moment about the mean will be
4
m4 = (x ) .P(x)
all x
= 32+
5. Skewness
To bring out the skewness we can calculate, moment coefficient of skewness, 1
1 = 1
(m )2
= 3
(m
2
)3
m 3
= 3
m
2
1
=
312
Caritas Business College BUSINESS STATISTICS – OAD 221
Evaluating 1 1
= we note that Poisson distribution is always skewed to the right i.e. has
positive skewness which is so as it is a distribution of rare events.
The degree of skewness in a Poisson distribution decreases as the value of increases.
6. Kurtosis
A measure of kurtosis of the Poisson distribution is given by the moment coefficient of
kurtosis 2
2 = 2 – 3
m4
= 3
2
1
=
Evaluating 2 1
we note that the Poisson distribution is leptokurtic.
=
7. Poisson approximation of the Binomial distribution
Poisson distribution can reasonably approximate Binomial distribution when n is infinitely
large and p is infinitely small i. e. when
n ∝ and p 0
Example 9-2
At a parking place the average number of car-arrivals during a specified period of 15 minutes
is 2. If the arrival process is well described by a Poisson process, find the probability that
during a given period of 15 minutes
(c) no car will arrive
(d) atleast two cars will arrive
(e) atmost three cars will arrive
(f) between 1 and 3 cars will arrive
313
Caritas Business College BUSINESS STATISTICS – OAD 221
Solution: Let X denote the number of cars arrivals during the specified period of 15
minutes. So X ~ POI ()
We apply the Poisson probability function P(X = x) =
e x = 0,1,2,……… to
x
x!
calculate the required probabilities.
2
(a) P(no car will arrive) = P(X = 0) = e
0
2
0!
= 0.1353
(b) P(atleast two cars will arrive) = P(X 2)
=1-[ P(X = 0) + P(X = 1)]
2 2 1
= 1-[ e +e 2 ]
0 1!
2
0!
= 1-[0.1353 + 0.2707]
= 1 – 0.4060
= 0.5940
(c) P(atmost three cars will arrive) = P(X 3)
2
3 e
= x
x 0 2
x!
=P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
= 0.8571
(d) P(between 1 and 3 cars will arrive) = P(1 X 3)
= P(X 3) - P(X = 0)
x 0
2 x
3 e 2
= x!
314
Caritas Business College BUSINESS STATISTICS – OAD 221
2 0 - 0!
e 2
= 0.8571 –0.1353
= 0.7218
315
Caritas Business College BUSINESS STATISTICS – OAD 221
9.6 SELF-ASSESSMENT QUESTIONS
1. Explain what do you understand by random experiment and a random variable.
Briefly explain the following:
a. Discrete and continuous random variables
b. Discrete probability distribution.
2. “Binomial random variable measures the number of successes in a Bernoulli Process”.
Explain this statement. Also develop and generalize Binomial probability rule with
the help of an example.
3. State the important properties of a Binomial distribution. Give examples of some of
the important area where Binomial distribution is used.
4. Under what condition can the Poisson distribution approximate Binomial distribution?
Develop the Poisson probability rule from the Binomial probability rule under these
conditions.
5. List some of the important areas where Poisson distribution is used. Also state
the important properties of a Poisson distribution.
6. On an average a machine produces 20 % defective item find the probability that
a random sample of 4 items consists of
(a) none to four defective items (b) atleast 3 defective items
(c) almost 2 defective items.
Out of 200 samples of 4 items, find the expected number of samples with (a), (b), and
(c) above
7. A gardener knows from his personal experiences that 2% of seedlings fail to
service on transplantation. Find the mean, standard deviation and moment coefficient
of skewness of the distribution of rate of failure to service in a sample of 400
seedlings.
316
Caritas Business College BUSINESS STATISTICS – OAD 221
8. If the sum of mean and variance of a binomial distribution of 5 trials is 9/5, find the
binomial distribution.
9. The mean and variance of a binomial distribution are 2 and 1.5 respectively. Find
the probability of
(a) 2 successes (b) atleast 2 successes (c) at most 2 successes.
10. 150 random samples of 4 units each are inspected for number of defective item.
The results are:
Number of defective items : 0 1 2 3 4
Number of Samples : 28 62 46 10 4
Fit a binomial distribution to the observed data.
11. The probability that a particular injection will have reaction to an individual is
0.002. Find the probability that out of 1000 individuals (a) no, (b) 1, (c) at least 1, and
(d) almost 2; individuals will have reaction from the injection.
12. In a razor blades manufacturing factory, there is small chance of 1/500 for any blade
to be defective. The blades are supplied in packets of 10. Find the approximate
number of packets containing (a) no, (b) 1, and (c) 2 defective blades in a
consignment of 10,000 packets.
13. If P(x = 1) = P(x = 2), for a distribution of Poisson random variable X. Find the
mean of the distribution.
14. The distribution of typing mistakes committed by a typist is given below:
Number of mistakes (X) : 0 1 2 3 4 5
Number of pages (f) : 142 156 69 27 5 1
Fit a Poisson distribution and find the expected frequencies.
317
Caritas Business College BUSINESS STATISTICS – OAD 221
9.7 SUGGESTED READINGS
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons.,
New Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall of
India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New York.
318
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 10: PROBABILITY DISTRIBUTIONS II
Topic : Probability
Objectives: The overall objective of the present lesson is to overview the concept of
continuous random variable and Normal distribution. After successful
completion of the lesson the students will be able to appreciate the
usefulness of normal distribution in decision-making and also identify
situations where normal probability distribution can be applied.
Structure
10.1 Introduction
10.2 Continuous Probability Distribution
10.3 The Normal Distribution
10.4 The Standard Normal Distribution
10.5 The Transformation of Normal Random Variables
10.6 Self-Assessment Questions
10.7 Suggested Readings
10.1 INTRODUCTION
We have learnt that a probability distribution is basically a convenient representation of the
different values a random variable may take, together with their respective probabilities of
occurrence. In the last lesson, we have examined situations involving discrete random
319
Caritas Business College BUSINESS STATISTICS – OAD 221
variables and the resulting discrete probability distributions. Consider the following random
variables that we have taken up in the last lesson:
1. Number of Successes (X1) in a Bernoulli’s Process
2. Number of Successes (X2) in a Poisson Process
In the first case, Binomial random variable X1 could take only finite number of integer values;
0,1,2…n; whereas in the second case, Poisson random variable X2 could take an infinite
number of integer value; 0,1,2,3…...........The random variables X1 and X2 are discrete, in
the
sense that they could be listed in a sequence, finite or infinite. In contrast to these, let us
consider a situation, where the variable of interest may take any value within a given range.
Suppose we are planning for measuring the variability of an automatic bottling process that
fills ½-liter (500 cm3) bottles with cola. The variable, say X, indicating the deviation of the
actual volume from the normal (average) volume can take any real value - positive or
negative; integer or decimal. This type of random variable, which can take an infinite number
of values in a given range, is called a continuous random variable, and the probability
distribution of such a variable is called a continuous probability distribution. The concepts
and assumption inherent in the treatment of such distributions are quite different from those
used in the context of a discrete distribution. In the present lesson, after understanding the
basic concepts of continuous distributions, we will discuss Normal distribution - an important
continuous distribution that is applicable to many real-life processes.
10.2 CONTINUOUS PROBABILITY DISTRIBUTION
Consider our planning for measuring the variability of the automatic bottling process that fills
½-liter (500cm3) bottles with cola. The random variable X indicates ‘the deviation of the
actual volume from the normal (average) volume.’ Let us, for some time, measure our
random variable X to the nearest one cm3.
317
Caritas Business College BUSINESS STATISTICS – OAD 221
F
Figure 10-1 Histograms of the Distribution of X as Measurements is refined to Smaller
and Smaller Intervals of Volume, and the Limiting Density Function f(x)
Suppose Figure10-1a represent the histogram of the probability distribution of X. The
probability of each value of X is the area of the rectangle over the value. Since the rectangle
will have the same base, the height of each rectangle is proportional to the probability. The
probabilities also add to 1.00 as required for a probability distribution.
Volume is a continuous random variable; it can take on any value measured on an interval of
numbers. Now let us imagine the process of refining the measurement scale of X to
the nearest 1/2 cm3, the nearest 1/10 cm3… and so on. Obviously, as the process of refining
the measurement scale continues, the number of rectangles in the histogram increases and
the width of each rectangle decreases. The probability of each value is still measured by the
area of the rectangle above it, and the total area of all rectangles remains 1.00. As
we keep refining our measurement scale, the discrete distribution of X tends to a
continuous probability distribution. The step like surface formed by the tops of the rectangles
in the histogram tends to a smooth function. This function is denoted by f(x) and is called the
probability density function of the continuous random variable X. The density function is
the
318
Caritas Business College BUSINESS STATISTICS – OAD 221
limit of the histograms as the number of rectangles approaches infinity and the width of each
rectangle approaches zero. The density function of the limiting continuous variable X
is shown in Figure 10-1 i.e. the values X can assume between the intervals –2.00 to –3.00
approaches infinity. The probability that X assumes a particular value (Say X = 1.5)
approaches zero. Probabilities are still measured as areas under the curve. The
probability that deviation will be between –1.50 and –1.00 is the area under f(x) between the
points x = -
1.50 and x = -1.00. Let us now make some formal definitions.
A continuous random variable is a random variable that can take on any
value in an interval of numbers.
The probabilities associated with a continuous random variable X are determined by the
probability density function of the random variable. The function, denoted by f(x), has the
following properties:
1. f(x) = 0 for all x
2. The probability that X will be between two numbers a and b is equal to the
area under f(x) between a and b.
b
P(a < X < b) = f ( x ).dx
a
3. The total area under the entire curve of f(x) is equal to 1.00.
P ( f (x 1.00
X ) ).dx
-
When the sample space is continuous, the probability of any single given value is zero. For a
continuous random variable, therefore, the probability of occurrence of any given value is
zero. We see this from property 2, noting that the area under a curve between a point
and itself is the area of a line, which is zero. For a continuous random variable, non-zero
probabilities are associated only with intervals of numbers.
319
Caritas Business College BUSINESS STATISTICS – OAD 221
We define the cumulative distribution function F(x) for a continuous random variable similarly
to the way we defined it for a discrete random variable: F(x) is the probability that X is less
than (or equal to) x.
Thus, the cumulative distribution function of a continuous random variable:
F(x) = P(X = x) = area under f(x) between the smallest possible value of X (often -∝)
and point x
x
=
f ( x).dx
-
The cumulative distribution function F(x) is a smooth, non-decreasing function that
increases from 0 to 1.00.
The expected value of a continuous random variable X, denoted by E(X), and its
variance, denoted by V(X), require the use of calculus for their computation. Thus
E(X )
x. f ( x ).dx
-
V(X) 2
x E ( x) . f ( x).dx
-
10.3 THE NORMAL DISTRIBUTION
The Normal Distribution is the most versatile of all the continuous probability distributions.
It is being widely used in all data-based research in the field of agriculture, trade, business
and industry It is found to be useful in characterizing uncertainties in many real-life
processes, in statistical inferences, and in approximating other probability distributions.
A large number of random variables occurring in practice can be approximated to the normal
distribution.
320
Caritas Business College BUSINESS STATISTICS – OAD 221
A random variable that is affected by many independent causes, and the
effect of each cause is not overwhelmingly large compared to other
effects, closely follow a normal distribution.
The lengths of pins made by an automatic machine; the times taken by an assembly worker
to complete the assigned task repeatedly; the weights of baseballs; the tensile strengths
of a batch of bolts; and the volumes of cola in a particular brand of canned cola - are good
examples of normally distributed random variables. All of these are affected by several
independent causes where the effect of each cause is small. This knowledge helps us in
calculating the probabilities of different events in varied situations, which in turn is useful for
decision-making.
In many real life situations, we face the problem of making statistical inferences about
processes based on limited data. Limited data is basically a sample from the full body of
data on the process. Irrespective of how the full body of data is distributed, it has been found
that the Normal Distribution can be used to characterize the sampling distribution of many of
the sample statistics. (we will see it in next few lessons). This helps considerably in
Statistical Inferences.
Finally, the Normal Distribution can be used to approximate certain probability
distributions. This helps considerably in simplifying the probability calculations.
10.3.1 Probability Density Function
If X is normally distributed with mean and variance 2, we write
X ~ N (μ, σ2)
and the probability density function (x) is given by the formula
2
1 1 x
f (x)
2
e
-∝ < x < +∝
2
321
Caritas Business College BUSINESS STATISTICS – OAD 221
In the equation e is the base of natural logarithm, equal to 2.71828.. .By substituting desired
values for and , we can get any desired density function. For example, a
distribution with mean 100 and standard deviation 5 will have the density function.
2
1 1 x 100
f (x)
-∝ < x < +∝
2 5 2 5
e
This function when plotted (see Figure 10-2) will give the famous bell-shaped mesokurtic
normal curve.
Figure 10-2 A Normal Distribution with = 100 and = 5
Many mathematicians have worked on the mathematics behind the normal distribution and
have made many independent discoveries. In the initial stages, the normal distribution was
Figure
developed by Abraham De Moivre 10-1 Normal
(1667-1754). Curve
His work was later taken up by Pierre S
Laplace (1949-1827). But the discovery of equation for the normal density function is
attributed to Carl Friedrich Gauss (1777-1855),who did much work with the formula. In
science books, this distribution is often called the Gaussian distribution.
We will now examine the properties of the Normal distribution.
10.3.2 Properties of Normal Distribution
1. The normal curve is not a single curve representing only one continuous distribution.
Obviously, it represents a family of normal curves; since for each different value of
322
Caritas Business College BUSINESS STATISTICS – OAD 221
and , there is a specific normal curve different in its positioning on the X-axis and
the extent of spread around the mean. Figure 10-3 shows three different normal
distributions – with different shapes and positions.
Figure 10-3 Three Different Normal Distribution
2. The normal curve is bell-shaped and perfectly symmetric about its mean. As a result
50% of the area lies to the right of mean and balance 50% to the left of mean. Perfect
symmetry, obviously, implies that mean, median and mode coincide in case of a
normal distribution. The normal curve gradually tapers off in height as it moves in
either direction away from the mean, and gets closer to the X-axis.
3. The normal curve has a (relative) kurtosis of 0, which means it has average
peakedness and is mesokurtic.
4. Theoretically, the normal curve never touches the horizontal axis and extends to
infinity on both sides. That is the curve is asymptotic to X-axis.
5. If several independent random variables are normally distributed, then their sum will
also be normally distributed. The mean of the sum will be the sum of all the
individual means, and by virtue of the independence, the variance of the sum will be
the sum of all the individual variances.
If X1, X2,…............Xn are independent normal variables, the their sum S will also be a
normal variable with
323
Caritas Business College BUSINESS STATISTICS – OAD 221
E(S) = E(X1) + E(X2) +…............E(Xn)
and V(S) = V(X1) + V(X2) +…............V(Xn)
6. If a normal variable X under goes a linear change in scale such as Y = aX + b, where
a and b are constants and a 0; the resultant Y variable will also be normally
distributed with mean = a E(X) + b and Variance = a2 V(X)
We can combine the above two properties.
If X1, X2,…............Xn are independent random variables that are normally distributed, then the
random variable Q defined as
Q = a1X1+ a2X2 + … anXn + b will also be normally distributed with
E(Q) = a1E(X1) + a2E(X2) +…............anE(Xn) + b
and V(Q) = a 2 V(X1) + a 2 V(X2) +….............a 2 V(Xn)
1 2 n
Let us see the application of this result with the help of an example.
Example 10-1
A cost accountant needs to forecast the unit cost of a product for the next year. He notes
that each unit of the product requires 10 labor hours and 5 kg of raw material. In addition,
each unit of the product is assigned an overhead cost of Rs 200. He estimates that the cost
of a labor hour next year will be normally distributed with an expected value of Rs 45 and a
standard deviation of Rs 2; the cost of raw material will be normally distributed with an
expected value of Rs 60 and a standard deviation of Rs 3. Find the distribution of the unit
cost of the product. Find its expected value and variance.
Solution: Since the cost of labor L may not influence the cost of raw material M, we can
assume that the two are independent. This makes the unit cost of the product Q a random
variable. So if
L ~ N (45, 22) and M ~ N (60, 32)
Then, Q = 10L + 5M + 200 will follow normal distribution with
324
Caritas Business College BUSINESS STATISTICS – OAD 221
Mean = E(Q) = 10E(L) + 5E(M) + 200
= 10(45) + 5(60) + 200
= 950
Variance = V(Q) = 102V(L) + 52V(M)
= 100(4) + 25(9)
= 625
So Q ~ N (950, 252)
7. Same important area relationships under normal curse are
Area between μ - 1σ and μ + 1σ is about
0.6826 Area between μ - 2σ and μ + 2σ is
about 0.9544 Area between μ - 3σ and μ + 3σ
is about 0.9974 Area between μ – 1.96σ and μ
+ 1.96σ is 0.95 Area between μ – 2.58σ and μ
+ 2.58σ is 0.99
10.4 THE STANDARD NORMAL DISTRIBUTION
There are infinitely many possible normal random variables and the resulting normal curves
for different values of μ and σ2. So the range probability P(a < X < b) will be different for
different normal curves. We can make use of integral calculus to compute the required
range probability
b
P(a < X < b) = f ( x ).dx
a
It may be appreciated that we can simplify this process of computing range probabilities to a
great extent by tabulating the range probabilities. Since it is not practicable and indeed
impossible to have separate probability tables for each of the infinitely many possible normal
curves, we select one normal curve to serve as a standard. Probabilities associated with the
range of values of this standard normal random variable are tabulated. A special
325
Caritas Business College BUSINESS STATISTICS – OAD 221
transformation then allows us to apply the tabulated probabilities to any normal random
variable. The standard normal random variable is denoted by a special name, Z (rather
than the general name X we use for other random variables).
We define the standard normal random variable Z as the normal random
variable with mean = 0 and standard deviation = 1.
We say
Z ~ N (0,12)
10.4.1 Standard Area Tables
The probabilities associated with standard normal distribution are tabulated in two ways –
say Type I and Type II tables, as shown in Figure 10-4. Type I Tables give the area between
0 and any other z value, as shown by vertical hatched area in Figure 10-4a. The hatched
area shown in figure is P (0 < Z < z).
P(0 < Z < z)P(Z > z)
Figure 10-4 Standard Area Tables
Type II Tables give the area towards the tail–end of the standard normal curve beyond the
ordinate at any particular z value. The hatched area shown in Figure 10-4b is P (Z >
z).
As the normal curve is perfectly symmetrical, the areas given by Type 1 Tables
when subtracted from 0.5 will provide the same areas as given by Type II Tables and vice-
versa.
i.e P (0 < Z < z) = 0.5 - P (Z > z).
326
Caritas Business College BUSINESS STATISTICS – OAD 221
10.4.2 Finding Probabilities of the Standard Normal Distribution
We will now illustrate the use of standard normal area tables for calculating the range
probabilities. Probability of intervals is areas under the density curve (z) over the intervals
in question.
Example 10-2
Find the probability that the value of the standard normal random variable will be…
(a) between 0 and 1.74 (b) less than -1.47
(c) between 1.3 and 2 (d) between -1 and
2
Solution: (a) P(Z is between 0 and 1.74)
That is, we want P(0 < Z < 1.74). In Figure 10-4a, substitute 1.74 for the point z on the
graph. We are looking for the table area in the row labeled 1.7 and the column labeled 0.04.
In the table, we find the probability 0.4591.Thus
P (0 < Z < 1.74) = 0.4591
(b) P(Z is less than -1.47)
That is, we want P(Z < -1.47). By the symmetry of the normal curve, the area to the left of -
1.47 is exactly equal to the area to the right of 1.47. We find
P(Z < -1.47) = P(Z >1.47)
= 0.5000 - 0.4292
= 0.0808
(c) P(Z is between 1.3 and 2)
That is, we want P(1.3 < Z < 2). The required probability is the area under the curve between
the two points 1.3 and 2. The table gives us the area under the curve between 0 and 1.3,
and the area under the curve between 0 and 2. Areas are additive; therefore,
P(1.30 < Z< 2) = TA(for 2.00) - TA(for 1.30)
= P(0 < Z < 2) - P(0 < Z < 1.3)
327
Caritas Business College BUSINESS STATISTICS – OAD 221
= 0.4772 - 0.4032
= 0.0740
(d) P(Z is between -1 and 2)
That is, we want P(-1< Z < 2). The required probability is the area under the curve between
the two points -1 and 2. The table gives us the area under the curve between 0 and 1, and
the area under the curve between 0 and 2. Areas are additive; therefore,
P(-1 < Z< 2) = P(-1 < Z < 0) + P(0 < Z < 2)
= P(0 < Z < 1) + 0.4772
= 0.3413 + 0.4772
= 0.8185
In cases, where we need probabilities based on values with greater than second-decimal
accuracy, we may use a linear interpolation between two probabilities obtained from the
table.
Example 10-3
Find P(0 ≤ Z ≤ 1.645)
Solution: P(0 ≤ Z ≤ 1.645) is found as the midpoint between the two probabilities P(0 ≤ Z ≤
1.64) and P(0 ≤ Z ≤ 1.65). So
P(0 ≤ Z ≤ 1.645) = ½[P(0 ≤ Z ≤ 1.64) + P(0 ≤ Z ≤ 1.65)]
= ½[0.4495 + 0.4505]
= 0.45
10.4.3 Finding Values of Z Given a Probability
In many situations, instead of finding the probability that a standard normal random variable
will be within a given interval; we may be interested in the reverse: finding an interval with a
given probability. Consider the following examples.
Example 10-4
328
Caritas Business College BUSINESS STATISTICS – OAD 221
Find a value z of the standard normal random variable such that the probability that
the random variable will have a value between 0 and z is 0.40.
Solution: We look inside the table for the value closest to 0.40. The closest value we find to
0.40 is the table area 0.3997. This value corresponds to 1.28 (row 1.2 and column .08).
So for P(0 < Z < z) = 0.40; z = 1.28
Example 10-5
Find the value of the standard normal random variable that cuts off an area of 0.90 to its left.
Solution: Since the area to the left of the given point z is greater than 0.50, z must be on the
right side of 0. Furthermore, the area to the left of 0 all the way to -∝ is equal to 0.50.
Therefore, TA = 0.90 - 0.50 = 0.40. We need to find the point z such that TA = 0.40.
We find that for TA = 0.40; z =1.28.
Thus z =1.28 cuts off an area of 0.90 to the left of standard normal curve.
Example 10-6
Find a 0.99 probability interval, symmetric about 0, for the standard normal random variable.
Solution: The required area between the two z values that are equidistant from 0 on
either side is 0.99. Therefore, the area under the curve between 0 and the positive z value
is TA = 0.99/2 = 0.495. We now look in our normal probability table for the area closest to
0.495. The area 0.495 lies exactly between the two areas 0.4949 and 0.4951,
corresponding to z = 2.57 and z = 2.58, Therefore, a simple linear interpolation between
the two values gives us z =
2.575. The answer, therefore, is z = ± 2.575.
So for P(-z < Z< z) = 0.99; z = 2.575
10.5 THE TRANSFORMATION OF NORMAL RANDOM VARIABLES
The importance of the standard normal distribution derives from the fact that any normal
random variable may be transformed to the standard normal random variable. If we want
to
329
Caritas Business College BUSINESS STATISTICS – OAD 221
transform X, where X ~ N (μ, σ 2), into the standard normal random variable Z ~ N (0, 12), we
can do this as follows:
X
Z
We move the distribution from its center of μ to a center of 0. This is done by subtracting μ
from all the values of X. Thus, we shift the distribution μ units back so that its new center is
0. To make the standard deviation of the distribution equal to 1, we divide the random
variable by its standard deviation σ. The area under the curve adjusts so that the total
remains the same. All probabilities (areas under the curve) adjust accordingly. Thus, the
transformation from X to Z is achieved by first subtracting μ from X and then dividing the
result by σ.
Example 10-7
If X ~ N (50, 10 2), find the probability that the value of the random variable X will be greater
than 60
Solution
: X 60
P(X > 60) = P( >
)
10
= P( Z > 60 50 )
10
= P( Z >1)
= P( Z > 0) - P(0 < Z <1)
= 0.5000 - 0.3413
= 0.1587
Example 10-8
The weekly wage of 2000 workmen is normally distribution with mean wage of Rs 70 and
wage standard deviation of Rs 5. Estimate the number of workers whose weekly wages are
330
Caritas Business College BUSINESS STATISTICS – OAD 221
(a) between Rs 70 and Rs 71 (b) between Rs 69 and Rs
73
(c) more than Rs 72 (d) less than Rs 65
Solution: Let X be the weekly wage in Rs, then
X ~ N (70, 5 2)
(a) The required probability to be calculated is P(70 < X < 71)
70 X
So P(70 < X < 71) = P( < 71
< )
70 70
= P( <Z<
5 71 70
5 )
= P(0 < Z < 0.2)
= 0.0793
So the number of workers whose weekly wages are between Rs 70 and Rs 71
= 2000 x 0.0793
= 159
(b) The required probability to be calculated is P(69 < X < 73)
69 X
So P(69 < X < 73) = P( < 73
< )
69 70
= P( <Z<
5 73 70
5 )
= P(-0.2 < Z < 0.6)
= P(-0.2 < Z < 0)+ P(0 < Z < 0.6)
= P(0 < Z < 0.2)+ P(0 < Z < 0.6)
= 0.0793 + 0.2257
= 0.3050
So the number of workers whose weekly wages are between Rs 69 and Rs 73
331
Caritas Business College BUSINESS STATISTICS – OAD 221
= 2000 x 0.3050
= 610
332
Caritas Business College BUSINESS STATISTICS – OAD 221
(c) The required probability to be calculated is P(X > 72)
X
So P(X > 72) = P( 72
>
)
= P(Z > 72 70 )
5
= P(Z > 0.4)
= 0.5 - P(0 < Z < 0.4)
= 0.5 – 0.1554
= 0.3446
So the number of workers whose weekly wages are more than Rs 72
= 2000 x 0.3446
= 689
(d) The required probability to be calculated is P(X < 65)
X
So P( X < 65) = P( 65
<
)
= P( Z < 65 70 )
5
= P(Z < -1.0)
= P(Z >1.0)
= P(Z >0) - P(0 < Z < 1.0)
= 0.5 - 0.3413
= 0.1567
So the number of workers whose weekly wages are less than Rs 65
= 2000 x 0.1567
= 313
333
Caritas Business College BUSINESS STATISTICS – OAD 221
10.5.1 The Inverse Transformation
The transformation Z X takes us from a random variable X with mean μ, and standard
deviation σ to the standard normal random variable. We also have an opposite, or inverse,
transformation, which takes us from the standard normal random variable Z to the random
variable X with mean μ and standard deviation σ. The inverse transformation is given as
X Z
We use the inverse transformation when we want to get from a given probability, the value
or values of a normal random variable X.
Example 10-9
The amount of fuel consumed by the engines of a jetliner on a flight between two cities is a
normally distributed random variable X with mean = 5.7 tons and standard derivation =
0.5 tons. Carrying too much fuel is inefficient as it slows the plans. If, however, too little fuel
is loaded on the plane, an emergency landing may be necessary. What should be the
amount of fuel to load so that there is 0.99 probability that the plane will arrive at its
destination without emergency landing?
Solution: Given that X ~ N (5.7, 0.5 2),
We have to find the value x such that
P(X < x) = 0.99
X
or
P( < z) = 0.99
or P(Z < z) = 0.99
= 0.5 + 0.49
= 0.5 + P(0 < Z < z)
From the table, value of z is 2.33
So x z
334
Caritas Business College BUSINESS STATISTICS – OAD 221
x 5.7 2.33x0.5
x 6.865
Therefore, the plane should be loaded with 6.865 tons of fuel to give 0.99 probability that the
fuel will last throughout the flight.
Example 10-10
Monthly sale of beer at a bar is believed to be approximately normally distributed with mean
2450 units and standard 400 units. To determine the level of orders and stock, the
management wants to find two values symmetrically on either side of mean, such that the
probability that sales of beer during the month will be between the two values is
(a) 0.95 (b) 0.99
Find the required values.
Solution: Let X be the monthly sale of beer, then
X ~ N (2450, 400 2),
(a) We have to find the values x1 and x2 such that
P(x1 < X < x2) = 0.95
x1 X x2
or P( < < ) =0.95
or P( z1 < Z < z2 ) =0.95
We know P(-1.96 < Z < 1.96) = 0.95
So z1 = -1.96 and z2 = 1.96
Using the inverse transformation,
x1 x2 z2
and
z1
x1 2450 x2 2450 1.96400
1.96400
x2 3234
x1 1666
335
Caritas Business College BUSINESS STATISTICS – OAD 221
Therefore, the management may be 95% sure that sales in any given month will be
between 1666 and 3234 units.
(b) We have to find the values x1 and x2 such that
P(x1 < X < x2) = 0.99
x1 X x2
or P( < < ) =0.99
or P( z1 < Z < z2 ) =0.99
We know P(-2.58 < Z < 2.58) = 0.99
So z1 = -2.58 and z2 =
2.58 Using the inverse transformation,
x1 and x2 z2
z1
x1 2450 x2 2450 2.58400
2.58400
x2 3482
x1 1418
Therefore, the management may be 99% sure that sales in any given month will be
between 1418 and 3482 units.
We can summarize the procedure of obtaining values of a normal random variable, given
a probability, as:
draw a picture of the normal distribution in question and the standard normal
distribution
in the picture, shade in the area corresponding to the probability
use the table to find the z value (or values) that gives the required probability
use the transformation from Z to X to get the appropriate value (or values) of
the original normal random variable
336
Caritas Business College BUSINESS STATISTICS – OAD 221
10.6 SELF-ASSESSMENT QUESTIONS
1. Define continuous probability distribution. State the properties of the probability
density function of a continuous random variable.
2. (a) Define normal random variable. State the probability density function of a
normal random variable.
(b) List down important properties of a normal curve.
3. Discus the role of normal distribution in statistical theory.
4. What do you mean by standard normal variable? Bring out the need for having a
standard normal curve.
5. Find the probability that a standard normal variable will have a value
(a) less than –10 (b) between -0.01 and 0.05
6. A sensitive measuring device is calibrated so that errors in the measurements it
provides are normally distributed with mean 0 and variance 1.00. Find the probability
that a given error will be between -2 and 2.
7. The deviation of a magnetic needle from the magnetic pole in a certain area
in northern Canada is a normally distributed random variable with mean 0 and
standard deviation 1.00. What is the probability that the absolute value of the
deviation from the north pole at a given moment will be more than 2.4?
8. Find two values of the standard normal random variable, z and -z, such that
(a) the two corresponding "tail areas" of the distribution add to 0.01.
(b) each tail have an area of 0.05
9. Let X be a normally distributed random variable with mean μ = 16 and standard
deviation σ = 3. Find
(a) P(10 < X< 18) (b) P(16 < X< 18) (c) P(X > 14)
337
Caritas Business College BUSINESS STATISTICS – OAD 221
10. For a normally distributed random variable with mean -44 and standard deviation 16,
find the probability that the value of the random variable will be
(a) above 0 (b) -10 (c) below 0
11. A normal random variable has mean 0 and standard deviation 4. Find the probability
that the random variable will be…
(a) above 2.5 (b) between 2 and 3 (c) below 1
12. The time it takes an international telephone operator to place an overseas phone call
is normally distributed with mean 45 seconds and standard deviation 10 seconds.
(a) What is the probability that my call will go through in less than 1 minute?
(b) What is the probability that my call will get through in less than 40 seconds?
(c) What is the probability that I will have to wait more than 70 seconds for my
call to go through?
13. The number of votes cast in favor of a controversial proposition is believed to be
approximately normally distributed with mean 8,000 and standard deviation 1,000.
The proposition needs at least 9,322 votes in order to pass. What is the
probability that the proposition will pass? (Assume numbers are on a continuous
scale.)
14. A manufacturing company regularly consumes a special type of glue purchased
from a foreign supplier. From past experience, the materials manager notes that the
company’s demand for glue during the uncertain lead-time is normally distributed
with a mean of 187.6 gallons and a standard deviation of 12.4 gallons. The company
follows a policy of placing the order when the glue stock falls to a predetermined
value, called “re-order point”. It the demand during lead-time exceeds the reorder
level, the glue would go ‘stock-out’ and production process would have to stop.
(a) If the re-order point is kept at 187.6 gallons, what is the probability that a
stock-out condition would occur?
338
Caritas Business College BUSINESS STATISTICS – OAD 221
(b) If the reorder point is kept at 200 gallons, what is the probability that a stock-
out condition would occur?
(c) If the company wants to be 95% confident that the stock-out condition will
not occur, what should be the reorder point? The reorder point minus the
mean demand during lead-time is known as the "safety stock." What is the
safety stock in this case?
(d) If the company wants to be 99% confident that the stock-out condition will not
occur, what should be the reorder point? What is the safety stock in this
case?
15. If X is a normally distributed random variable with mean 125 and standard deviation
44, find a value x such that the probability that X will be less than x is 0.66.
16. For a normal random variable with mean 10.5 and standard deviation 0.4, find a point
of the distribution such that there is a 0.95 probability that the value of the random
variable will be above it.
17. For a normal random variable with mean 29,500 and standard deviation 410, find a
point of the distribution such that the probability that the random variable will exceed
this value is
(a) 0.03 (b) 0.25
18. Find two values of the normal random variable with mean 80 and standard deviation
5 lying symmetrically on either side of the mean and covering an area of 0.98
between them.
19. For X~ N(32, 72), find two values x1 and x2, symmetrically lying on each side of the
mean, with
(a) P(x1 < X< x2) = 0.99 (b) P(x1 < X < x2) = 0.95
20. The results of a given selection test exercise are summarized as
(i) cleared with distinction = 10%
339
Caritas Business College BUSINESS STATISTICS – OAD 221
(ii) cleared without distinction = 60%
(iii) those who failed = 30%.
A candidate gets failed if he/she obtains less than 40% marks, while one must
obtain at least 75% marks to pass with distinction. Determine the mean and standard
deviation of the distribution of marks, assuming the same to be normal.
21. The demand for gasoline at a service station is normally distributed with mean
27,009 gallons per day and standard deviation 4,530. Find two values that will give a
symmetric 0.95 probability interval for the amount of gasoline demanded daily.
22. The percentage of protein in a certain brand of dog food is a normally distributed
random variable with mean 11.2 % and standard deviation 0.6 %. The manufacturer
would like to state on the package that the product has a protein content of at least
x1
% and no more than x %. He wants the statement to be true for 99% of the packages
sold. Determine the values x1 and x2.
10.7 SUGGESTED READINGS
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons.,
New Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
340
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 11: SAMPLING AND SAMPLING METHODS
Topic : Sampling Techniques
Objective: After going through this chapter, you will be able to understand: various terms
associated with sampling; various methods of probability and non-probability
sampling and how to determine sample size.
Structure
11.1. Census Vs. sampling method
11.2. Definitions
11.3. Probability samples vs. non-probability samples
11.4. Probability sampling methods
11.5. Non-probability sampling methods
11.6. Determination of sample size
11.7. Self-test questions
11.8. Suggested readings
11.1. Census Vs. sampling method
Sample is a part of the population from which it is selected. The process of selecting a
sample is known as sampling. Thus, the sampling theory is a study of relationship that exists
between the population and the samples drawn from the population. The complete
enumeration, popularly known as census, may not be feasible either due to non-availability
of time or because of high cost involved. Therefore, it becomes essential to draw inferences
for the population on the basis of sample information. Thus, sampling helps us to get as
much information as possible of the whole universe. The sampling also helps us in
determining the reliability of the estimates. This can be done by drawing samples from the
same parent population and comparing the results obtained from different samples.
In a survey of the entire population, data is collected from every elementary unit of the
population. Suppose, one is studying the wage structure of the coal mining industry in the
country, then one approach is to collect the data on wages of every worker in the coal
industry. From this data, one can calculate the various characteristics of the population,
such as average wage, the range and the variance, etc. This is referred as census survey.
The advantages of the census approach are
every unit of the population is considered and the respective data on the various
characteristics are compiled,
the analysis made on the basis of census data is very accurate and reliable,
and
341
Caritas Business College BUSINESS STATISTICS – OAD 221
in one time studies of special importance, only census method is adopted in
order to get accurate and reliable data. The data collected by this method
becomes a data base for all future studies. This is one of the reasons why
population data are collected once in a decade by the census method.
Although there are many advantages with the census method, the cost, effort and the
time required to conduct census survey is very large, unless the population is very small,
and in many cases it is so prohibitive that one rarely uses this method in surveys.
Sampling involves an examination of a small portion of the elementary units in a population.
Although, a census operation gives a more reliable data, sampling method is more desired
when
1.1. the population is very large, i.e., infinite and it would be impossible to
conduct census surveys;
2.1. when quick results are required it would be appropriate to conduct sample
surveys rather than census surveys;
3.1. in studies involving destruction of the elementary units under study, it
would only be appropriate to go for sample testing. Items such as light
bulbs and ammunition often must be destroyed as a part of testing
process;
4.1. cost of conducting surveys would be very prohibitive in census method,
and therefore, it is advisable to carry out a sample survey, and lastly;
and
5.1. some times accuracy may be lost because of the large size of the
population. Sampling involves a small portion of the population and
therefore, would involve very few people for conducting surveys and for
data collection and compilation. This would not be so in the census
method and the chances of committing errors would increase.
As the sampling involves less time and money, it would be possible to give attention to
different characteristics of the elementary units. A sample using same money and time can
produce a detailed study of lesser number of units. The process of sampling involves
selecting a sample, collecting all relevant information, and finally drawing conclusions about
the population from which the sample has been drawn.
342
Caritas Business College BUSINESS STATISTICS – OAD 221
11.2. Definitions
The surveys are concerned with the attributes of certain entities, such as business
enterprises, human beings, etc. The attributes that are the object of the study are known as
characteristics and the units possessing them are called the elementary units.
The aggregate of elementary units to which the conclusions of the study apply is termed
as population/universe, and the units that form the basis of the sampling process are
called sampling units. The sampling unit may be an elementary unit.
The sample is defined as an aggregate of sampling units actually chosen in obtaining a
representative subset from which inferences about the population are drawn. The frame— a
list or directory, defines all the sampling units in the universe to be covered. This frame is
either constructed for the purpose of a particular survey or may consist of previously
available description of the population; the latter is the commonly used method. For
example, telephone directory can be used as a frame for conducting opinion surveys in a
city or locality.
In order that, sampling results reflect the characteristics of the population, it is necessary
that the sample selected for study should be
1.1. Truly representative, i.e., the selected sample truly represent the universe
so that the results can be generalised;
2.1. Adequate, i.e., the size of the sample or the sample size should be
adequate enough to represent the various characteristics of the universe;
3.1. Independent, i.e. the elementary units selected should be independent of
one another and all units of the population should have the same chance
of being selected in the sample; and lastly
4.1. Homogeneous, i.e., there should not be any basic difference between the
characteristics of the units in the sample and that of the population.
This means that if two or more samples are drawn from the same
population, the results should be more or less identical.
11.3. Probability samples vs. non-probability samples
A probability sample is one for which the inclusion or exclusion of any individual element of
the population depends upon the application of probability methods and not on a personal
judgement. It is so designed and drawn that the probability of inclusion of an element is
known. The essential feature of drawing such a sample is the randomness. As against the
probability sample, we have a variety of other samples, termed as judgement samples,
purposive samples, quota samples, etc. These samples have one common distinguishing
feature: personal judgement rather than the random procedure to determine the composition
of what is to be taken as a representative sample. The judgement affects the choice of the
individual elements. All such samples are non-random, and no objective measure of
precision may be attached to the results arrived at.
343
Caritas Business College BUSINESS STATISTICS – OAD 221
In a probability sampling, it is possible to estimate the error in the estimates and they can be
minimized also. It is also possible to evaluate the relative efficiency of the various probability
sampling designs. Probability sampling does not depend upon the detailed information about
population for its effectiveness. However, probability sampling requires a high level of skill
and experience for its use. It also requires sufficient time and money to execute.
Non-probability sampling is a procedure of selecting a sample without the use of probability
or randomisation. It is based on convenience, judgement, etc. The major difference between
the two approaches is that it is possible to estimate the sampling variability in the case of
probability sampling while it is not possible to estimate the same in the non-probability
sampling. The classification of various probability and non-probability methods are shown in
Fig. 11.1.
POPULATION
SAMPLING
Probability Samples Non probability Samples
•Simple Random Sampling •Convenience Sampling
•Stratified Random Sampling •Quota Sampling
•Cluster Sampling or •Judgement Sampling
Multistage Sampling
•Systematic Sampling
Fig. 11.1. Classification of sampling schemes
11.4. Probability Sampling Methods
The various probability sampling methods are described as under:
(a) Simple random sampling method
In simple random sampling, drawing of elements from the population is random and the
choice of an element is made in such a way that every element has the same probability of
being chosen. When the sample is so selected, every possible set of elements has the same
chance of being drawn. With N, population size, fairly large, the number of such possible
sets of size n is of course very large. This number is given by NCn. Of course, it is
unnecessary in a specific case to compute the number of possible sets of stated size that
might be drawn from a given population, but the process of sample selection should be such
that the probability of selection is the same for every such set.
The objective is to achieve randomness in drawing the individual elements of a sample for
ensuring that all possible samples have the same chance of being selected. If we are to
draw from a population containing N elementary units, the elementary unit also being a
sampling unit, it is necessary that each of the N units should be individually numbered or
otherwise distinctively designed. One of the approaches for drawing random sample of size
n from a population of N units is to draw n cards from N cards which are numbered from 1
to N and mixed thoroughly. The sample size n, thus drawn, would constitute a simple
random sample (SRS). Another popular method of selecting a random sample is by lottery
method. In this method all the elements are named or numbered on a small slip of paper of
identical shape and size. These slips are folded identically and mixed up well in a container.
Number of slips of desired sample size is selected blindly from this container. Thus, the
selection of elementary units depends purely on chance and no personal bias exists. We
shall illustrate this method of selection of a sample with the following example: Suppose the
warden of a student’s hostel with 200 occupants wants to constitute a welfare committee
with the
344
Caritas Business College BUSINESS STATISTICS – OAD 221
members randomly selected. The lottery method of selecting these five members from a
group of 200 would be first to prepare 200 slips of identical shape and size and write the
name of each student on a slip. Fold these 200 slips identically and mix them well in a
container. Then select five folded slips, from the container at random. The five students
so selected would constitute a welfare committee of the hostel.
There are, however, some difficulties in these procedures. For, if N is large, the task
becomes physically difficult. So it is desirable to use better methods for ensuring
randomness. One such method is the use of random number tables.
Use of random number tables
If the N elements of a total population are numbered serially from 1 to N, a random sample
may be most readily and reliably drawn by using a table of random numbers. Such tables
enable us to select n numbers at random from the full list of serial numbers from 1 to N. In a
random number table, digits in each column are in random order and so are the digits in
each row. As the arrangement is random in all directions, it makes no difference where we
begin in our selection of random numbers from such a table. However, the column
arrangement is generally found more convenient for references.
Several random number tables are available for use. These numbers have been
adequately tested for randomness. Among them, the most popular ones are:
1.1. Tippett’s (1927) 10,400 sets of four-digited random numbers;
2.1. Fisher and Yates (1938) table of random numbers with 1,500 sets of ten-
digited random numbers; and
3.1. Rand Corporation (1955) table of random numbers of 2,00,000 sets of
five-digited random numbers.
Tippet’s table of random numbers is most popularly used in practice. Given below are
the first forty sets from Tippet’s table as an illustration of the general appearance of
random numbers:
2952 6641 3992 9792 7969 5911 3170 5624
4167 9524 1545 1396 7203 5356 1300 2693
2670 7483 3408 2762 3563 1089 6913 7691
0560 5246 1112 6107 6008 8125 4233 8776
2754 9143 1405 9025 7002 6111 8816 6446
Tippett’s numbers have been subjected to numerous tests and used in many
investigations and their randomness has been well established for all practical purposes.
An example to illustrate how Tippett’s table of random numbers may be used is given
below.
Suppose ten numbers from out of 0 and 80 are required. We start anywhere in the table
and write down the numbers in pairs. The table can be read horizontally, vertically,
diagonally or in any methodical way. Starting with the first and reading horizontally first we
obtain 29, 52, 66, 41, 39, 92, 97, 92, 79, 69, 59, 11, 31, 70, 56, 24, 41, 67 and so on.
Ignoring the numbers
greater than 80, we obtain for one purpose ten random numbers, namely 29, 52, 66, 41, 39,
79, 69, 59, 11 and 31.
345
Caritas Business College BUSINESS STATISTICS – OAD 221
The sampling procedure described above is quite satisfactory for a small population. With a
large population, the process of identification of numbers to each elementary sampling unit
becomes very prohibitive with respect to both time and money. Moreover, the population is
often geographically spread out or composed of clearly identified strata possessing unique
characteristics. Whenever any of the above situations arise, alternative sampling schemes
that are sophisticated combinations of simple random sampling provide significantly better
results for the same expenditure and time. As a result, the simple random sampling method
is not very frequently used in practice. However, the simple random sampling scheme is the
basis of any other probabilistic sampling schemes.
(b) Stratified random sampling method
In simple random sampling, the population to be sampled is treated as homogeneous and
the individual elements are drawn at random from the whole universe. However, it is often
possible and desirable to classify the population into distinctive classes or strata and then
obtain a sample by drawing at random the specified number of sampling units from each of
the classes thus constructed. This may be desirable because of our interest in the distinct
classes of the universe as a whole.
In stratified random sampling, the population is sub-divided into strata before the sample is
drawn. Strata are so designed that they should not overlap. A sample of specified size is
drawn at random from the sampling units that make up each stratum. If a given stratum is of
our interest, the corresponding sub-sample provides the basis for estimates concerning the
attributes of the population stratum, or sub-universe from which it is drawn. The total of sub-
samples constitutes the aggregate sample on which estimates of attributes of the entire
population are based.
Stratified samples may be either proportional or non-proportional. In a proportional stratified
sampling, the number of elements to be drawn from each stratum is proportional to the size
of that stratum compared with the population. For example, if a sample size of 500
elementary units have to be drawn from a population with 10,000 units divided in four strata
in the following way:
Population size Sample size
Stratum I = 2000 500 × 0.2 = 100
Stratum II = 3000 500 × 0.3 = 150
Stratum III = 4000 500 × 0.4 = 200
Stratum IV = 1000 500 × 0.1 = 50
Total 10000 500
Thus, the elements to be drawn from each stratum would be 100, 150, 200 and 50
respectively. Proportional stratification yields a sample that represents the population with
respect to the proportion in each stratum in the population. Proportional stratified sampling
yields satisfactory results if the dispersion in the various strata is of proportionately the same
magnitude. If there is a significant difference in dispersion from stratum to stratum, sample
estimates will be much more efficient if non-proportional stratified random sampling is used.
Here, equal numbers of elements are selected from each stratum regardless of how the
stratum is represented in the population.
Thus, in the earlier example, an equal number, i.e., 125, of elementary units will be drawn to
constitute the sample.
A sample drawn by stratified random sampling scheme ensures a representative sample as
the population is first divided into various strata and then a sample is drawn from each
stratum.
Stratified random sampling also ensures greater accuracy and it is maximum if each
stratum is formed in such a way that it consists of uniform or homogeneous items.
Compared with a simple random sample, a stratified sample can be more concentrated
geographically, i.e., the elementary units from different strata may be selected in such a
way that all of them are
346
Caritas Business College BUSINESS STATISTICS – OAD 221
located in one geographical area. This would also reduce both time and cost involved in
data collection. However, care should be exercised in dividing the population into various
strata. Each stratum must contain, as far as possible, homogeneous units, as otherwise the
reliability of the results would be lost.
In conclusion, stratification is an effective sampling device to the extent that it creates
classes that are more homogeneous than the total. When this can be done, the classes are
distinguished that differ among themselves in respect of a stated characteristic. Stratification
may be futile if classes do not differ among themselves. Thus, there should be homogeneity
within classes and heterogeneity between classes.
(c) Cluster sampling or multistage sampling
Under this method, the random selection is made of primary, intermediate and final (or the
ultimate) units from a given population or stratum. There are several stages in which the
sampling process is carried out. At first, the first stage units are sampled by some suitable
method, such as simple random sampling. Then, a sample of second stage unit is selected
from each of the selected first stage units, again by some suitable method which may be
same as or different from the method employed for the first stage units. Further stages may
be added as required. The procedure may be illustrated as follows:
Suppose we want to take a sample of 5,000 households from the State of Haryana. At the
first stage, the state may be divided into a number of districts and a few districts are selected
at random. At the second stage, each district may be sub-divided into a number of villages
and a sample of villages may be taken at random. At the third stage, a number of
households may be selected from each of the villages selected at second stage. To take
another example supposes in a particular survey, we wish to take a sample of 10,000
students from a University. We may take colleges at the first stage, then draw departments
at the second stage, and choose students as the third and last stage.
Merits: Multi-stage sampling introduces flexibility in the sampling method which is lacking
in the other methods. It enables existing divisions and sub-divisions of the population to be
used as units at various stages, and permits the field work to be concentrated and yet
large area to be covered.
Another advantage of this method is that sub-division into second stage units need be
carried out for only those first stage units which are included in the sample. It is, therefore,
particularly valuable in surveys of under-developed areas where no frame is generally
sufficiently detailed and accurate for subdivision of the material into reasonably small
sampling units.
Limitations: However, a multi-stage sample is in general less accurate than a sample
containing the same number of final stage units which have been selected by some
suitable single stage process.
(d) Systematic sampling
Another sampling form, simple in design and execution, may be employed when the
members of population to be sampled are arranged in order, the order corresponding to
consecutive numbers. The arrangements of names in a telephone directory or income-tax
returns in the income tax department are the illustrations of such orderings. A sample of
suitable size is obtained by taking every unit say, seventh unit of the population, one of the
first seven units in this ordered arrangement is chosen at random and the sample is
completely by selecting every seventh unit from the rest of the list. If the first unit selected is
the fifth, the researcher will include in his sample 12th, 19th, 26th, 33rd, etc. We can generalize
the approach as follows: if the requirements of the survey call for the inclusion of one unit
out of every m units in the population, a unit is chosen at random from the first m units,
thereafter, every mth unit in the population when arranged in order, is included in the
sample.
347
Caritas Business College BUSINESS STATISTICS – OAD 221
This mode of selection is called systematic sampling, m is generally referred to as the
N
sampling ratio, i.e., the ratio of the population size to the sample size. Symbolically m = .
n
where N is the population size and n is the sample size. While calculating the value of m,
we may get a fractional value. In such cases, it is rounded off to the nearest digit.
Which sampling scheme to select
In sampling, one scheme is said to be more efficient than another when the sample
estimates developed by the scheme tend to cluster more closely around the population
parameter being estimated. An estimator of the population parameter should possess the
following characteristics:
1.1. It should be unbiased: An estimator is unbiased when the expected
(average) value of the sample statistic is equal to the population
parameter being estimated.
2.1. It should be efficient: Efficiency is with respect to sample size and it means
that the sample estimates should be clustered as closely possible to the
population parameter being estimated for a given sample size. For
example, when the population is normally distributed, both the sample
mean and the median are unbiased estimators of the population mean.
However, for any given sample size, the sample means cluster more
closely around the population mean than do the sample medians. Thus,
both mean and the median are the unbiased estimators of the population
mean. However, the sample mean is the unbiased efficient estimator of
the population mean.
In stratified random sampling, where stratification is meaningful, a stratified random sample
will be more efficient than a simple random sample of the same size. A sampling design is
considered efficient with respect to cost if the sample estimates cluster more closely
around the population parameter being estimated than they would for any alternative
sampling scheme involving equivalent rupee expenditure.
It should be consistent: An estimator is considered to be consistent if the sample
estimates cluster more and more closely around the population
parameter being estimated as the sample size increases.
11.5. Non-Probability Sampling Methods
There are three methods of sampling in this category. These are explained as follows:
348
Caritas Business College BUSINESS STATISTICS – OAD 221
1. Convenience sampling
In this scheme, a sample is obtained by selecting ‘convenient’ population elements. For
example, a sample selected from the readily available sources or lists such as telephone
directory or a register of the small scale industrial units, etc. will give us a convenient
sample. In these cases, even if a random approach is used for identifying the units, the
scheme will not be considered as simple random sampling. For example, if one studies the
wage structure in a close by textile industry by interviewing a few selected workers, then the
scheme adopted here is convenient sampling. The results obtained by convenience
sampling method can hardly be said to be representative of the population parameters.
Therefore, the results obtained are generally biased and unsatisfactory. However,
convenient sampling approach is generally used for making pilot studies, particularly for
testing a questionnaire and to obtain preliminary information about the population.
2. Quota sampling
In this method of sampling, the basic parameters which describe the population are
identified first. Then the sample is selected which conform to these parameters. Thus, in a
quota sample, quotas are fixed according to these parameters, and each field investigator is
assigned with quotas of the number of units to be interviewed. Within the preassigned
quotas, the selection of the sample elements depends on the personal judgement. For
example, if one is studying the consumer preferences for ice creams among children and
college going students and supposes it is fixed to interview 250 individuals from each
category. If the city has five colleges, one decides to fix up a quota of 50 students to be
interviewed from each college. It entirely depends upon the interviewer who will constitute
this sub-sample of 50 students in a college— they may be the first 50 students who visit the
ice cream parlour or may be the 50 students who visit the parlour between 4 p.m. and 6
p.m., etc.
Quota sampling method has the advantage that the sample will conform to the selected
parameters of the population. The cost and time involved in getting information from the
sample will be relatively less for a quota sample but there are many weaknesses too. Some
of these are:
1.1. It is difficult to validate the information gathered on the elementary units,
2.1. It may be difficult to specify the characteristics of the population and
therefore it may be hard to identify it,
3.1. Even when the sample does conform to the characteristics used in the
quotas, the sample may be distorted on other factors of importance in the
study. For example, interviewing first 50 students or the last 50 students
visiting the ice cream parlour can make a lot of difference particularly
about their purchasing capacity, tastes, etc. This may completely distort
the results.
Quota sampling method is generally used in public opinion studies, election forecast polls,
as there is not sufficient time to adopt a probability sampling scheme.
349
Caritas Business College BUSINESS STATISTICS – OAD 221
3. Judgement sampling
Judgement sampling method can also be called as sampling by opinion. In this method,
someone who is well acquainted with the population decides which members (elementary
units) in his or her judgement would constitute a proper cross-section representing the
parameters of relevance to the study. This method of sampling is generally used in studies
involving performance of personnel. For example, if one is studying the performance of sales
staff in a marketing organisation, the people here are classified into top grade, medium
grade and low grade performers. Having specified qualities that are important in the study,
the expert (possibly here the Vice-President-sales) indicates the people who, in his or her
knowledge, would be representative of each of the three categories mentioned earlier. This,
of course, is not a scientific method, but in the absence of better evidence, such a judgement
method may have to be used.
11.6. Determination of sample size
We prefer samples to complete enumeration because of convenience and reduced cost of
data collection. However, in sampling, there is a likelihood of missing some useful
information about the population. For a high level of precision, we need to take a larger
sample. How large should be the sample and what should be the level of precision? In
specifying a sample size, care should be taken such that (i) neither so few are selected so
as to render the risk of sampling error intolerably large, nor (ii) too many units are included,
which would raise the cost of the study to make it inefficient. It is, therefore, necessary to
make a trade-off between
(i) increasing sample size, which would reduce the sampling error but increase the cost, and
(ii) decreasing the sample size, which might increase the sampling error while decreasing
the cost. Therefore, one has to make a compromise between obtaining data with greater
precision and with that of lower cost of data collection. Several factors need to be
considered before determining the sample size.
The first and the foremost is the size of the error that would be tolerable for the purposes of
decision-making. The second consideration would be the degree of confidence with the
results of the study, i.e., if one wants to be 100 per cent confident of the results, the entire
population must be studied. However, this is generally too impractical and costly. Therefore,
one must accept something less than 100 per cent confidence. In practice, the confidence
limits most often used are 99 per cent, 95 per cent and 90 per cent. Most commonly used
confidence limit is 95 per cent. This means that there is a 5 per cent risk that the true
population statistic is outside the range of possible error specified by the confidence interval.
This 5 per cent risk appears to be acceptable in most of the decisions. Thus, for 95 per cent
level of confidence, Z value is 1.96. The Z value can be obtained from normal probability
distribution for a specified level of confidence. For determining the sample size, we make
use of the following relationship:
= standard error of the estimate =
x n
x can be calculated if we know the upper and lower confidence limits. Let these limits be
Y, then
Zx =Y
Where Z is the value of the normal variate for a given confidence level. The procedure
has been explained using the illustration given below:
Illustration 11.1. A state cooperative department is performing a survey to determine the
annual salary earned by managers numbering 3000 in the cooperative sector within the
state. How large a sample size it should take in order to estimate the mean annual earnings
within
350
Caritas Business College BUSINESS STATISTICS – OAD 221
plus and minus 1,000 and at 95 per cent confidence level? The standard deviation of
annual earnings of the entire population is known to be Rs. 3,000.
Solution. As the desired upper and lower limit is Rs. 1,000, i.e., we want to estimate
the annual earnings within plus and minus Rs. 1,000.
z = 1,000
x
As the level of confidence is 95 per cent, the Z value is 1.96
1.96 = 1,000
x
1‚000
= = 510.20
x 1.96
The standard error is given by / n where is the population standard deviation
x
n = 510.20
3000
i.e., = 510.20
n
3000
i.e., n = = 5.88
510
This gives n = 34.57
Therefore, the desired sample size is about 35.
11.6.1. Sample size for stratified sampling
Once the strata have been established, we are interested in the size of the stratified
random sample. The size will depend upon whether the proportional or disproportional
(optimal) sample is being taken.
A proportional stratified sample is one in which the sample units in a given stratum are
allocated in proportion to the relative size of the stratum. The following formula is used for
calculation of the proportional sample for each stratum
Ni
ni = ×n
N
Where ni = number of sample units from stratum i, N = the total number of units in the
population, Ni = the total number of units in the stratum i, n = sample size desired.
The standard error of mean is
k
x = 2 2
wi i /ni
i=1
where wi = the weight of stratum i = Ni/N, i = the standard deviation of the ith stratum, k =
the total number of strata. In case of disproportionate stratified sampling, the proportion of
units in the sample stratum is not equal to the proportion of the population. The formula for
sample allocation in this case is
wiin
n i= k
wii
1
Thus, the disproportional stratified sample is more desirable if standard deviation (i) of
each stratum is known. The standard error of the mean of a disproportionate stratified
sample is
351
Caritas Business College BUSINESS STATISTICS – OAD 221
k
(wii)2
x 1
= ni
It may be observed that the standard error for stratified sample is smaller than for simple
random sample, i.e., much smaller samples may be utilized when the population has
been stratified.
Illustration 11.2. In a market area, shops are divided into two categories, viz., those that
have daily turnover of more than Rs. 2000 and those that have daily turnover of less than
Rs. 2000 for the study of estimating the total sales in the area. The total number of shops in
the first stratum are 420 and in the second stratum 180. A sample of 50 was selected, the
standard deviation has been found to be 70 for first stratum and 95 for second stratum. What
size of stratified random sample should be taken under proportional and disproportional
stratified sampling?
Solution. Under the proportional stratified sampling, the sample size is given
Ni
by ni = ×n
N
420
and, therefore n1 = × 50 = 35
6
180
and n2 = × 50 = 15
6
2i2
The standard error ( x ) =
wi ni
2 (70)2 (0.3)2 × (95)2
= (0.7)
× + 15
35
= 122.75 = 11.079
For disproportionate sampling, the sample size is given by:
wiin
ni w ii
0.7 × 70 × 50 2450
n1 = 0.7 × 70 + 0.3 × 95= 77.5 = 32.0
0.3 × 95 × 50 1425
and n2 = 0.7 × 70 + 0.3 × 95= 77.5 = 18.0
The standard error is given by
k
(wii)2
x = (0.7 × 70 + 0.3 × 95)2
1
ni = 50
= 120.125 = 10.96
11.6.2. Cost as a factor in the determination of the sample size
Another consideration in determining the sample size is the cost. Management may
reduce the level of confidence in an attempt to reduce the cost of sampling. An illustration
will clarify how cost of sampling can be reduced by reducing the sample size.
Illustration 11.3. In a market area there are 600 shops. A researcher wishes to estimate
number of customers visiting these shops per day. The researcher wants to estimate the
sampling error in the number of customers visiting is no larger than ± 10 with probability of
0.95. The previous studies indicated that the standard deviation is 85 customers. If the cost
352
Caritas Business College BUSINESS STATISTICS – OAD 221
per interview is Rs. 20 (this includes field work, supervision of interviewers, coding, editing
and tabulation of results and report writing, etc.), calculate the total cost involved.
Researcher is willing to sacrifice some accuracy in order to reduce cost. If he settles for an
estimate with
0.90 probability, how much reduction in cost can be achieved?
Solution. For 95 per cent confidence levels,
Z x = Y
i.e., 1.96 = 10.0
x
10
x=
1.96
Now, is given by / n and therefore, the sample size will be determined by the equation
x
10
n = 1.96
Since = 85, we have
8 10
5 = 1.96
n
n = 277.6
Thus, if the sample is taken as 278, the total cost involved will be 278 × 20 = Rs. 5560. As
this cost is considered to be on the higher side by the researcher and in order to reduce the
cost, the researcher has now settled to 90 per cent confidence level. At 90 per cent
confidence level, the sample size can be calculated as follows:
Z x = 10
1.65 x = 10
1.65 x = 10
10
or x = 1.65
10
n = 1.65
85
i.e., 10
n = 1.65
n= 196.7
The cost of survey for this sample size will be 197 × 20 = Rs. 3940. Thus, we have
observed that by reducing the confidence level from 95 per cent to 90 per cent, the
researcher would reduce the cost from Rs. 5560 to Rs. 3940. The researcher may not like
to reduce the confidence level further and so further cost reduction may not be desirable.
11.7. Self-Test Questions
1. Describe the various methods of drawing a sample. Which one would
you suggest and why?
2. Describe the importance of sampling. Critically examine the merits of probability
sampling and non-probability sampling methods.
3. Specify and explain the factors that make sampling preferable to a complete census
in a statistical investigation.
353
Caritas Business College BUSINESS STATISTICS – OAD 221
4. How would you determine the sample size for stratified sampling? Explain
with the help of a suitable example.
5. To determine the effectiveness of the advertising campaign of a new VCR,
management would like to know what percentage of the household are
aware of the new brand. The advertising agency thinks that this figure is
as high as 70 per cent. The management would like a 95% confidence
interval and a margin of error no greater than plus or minus 2 per cent. (a)
What sample size should be used for this study? (b) Suppose that
management wanted to be 99 per cent confident but could tolerate an error
of plus or minus 3 per cent. How would the sample size change?
11.8. Suggested Readings
1. Statistical Methods by S.P. Gupta. Sultan Chand and Sons, New
Delhi.
2. Statistics for MBA by T.R. Jain and Dr. S.C. Aggarwal. VK (India)
Enterprises, New Delhi. First edition.
3. Business Statistics by Shenoy and Shenoy.
354
Caritas Business College BUSINESS STATISTICS – OAD 221
Module 12: SAMPLING DISTRIBUTIONS
Topic : Sampling Distributions
Objectives: The present lesson is an attempt to overview the concept of sampling
distributions. After successful completion of the lesson the students
will be able to understand the meaning and the need of studying
sampling distribution of a sample statistic.
Structure
12.1 Introduction
12.2 Sampling Distribution of the Mean
12.3 Central Limit Theorem
12.4 Sampling Distribution of the Proportion
12.5 Sampling Distribution of the Difference of Sample Means
12.6 Sampling Distribution of the Difference of Sample Proportions
12.7 Small Sampling Distributions
12.8 Sampling Distribution of the Variance
12.9 F Distribution
12.10 t-Distribution
12.11 Self-Assessment Questions
12.12 Suggested Readings
12.1 INTRODUCTION
Having discussed the various methods available for picking up a sample from a
population, we would naturally be interested in drawing statistical inferences - making
generalizations
355
Caritas Business College BUSINESS STATISTICS – OAD 221
about the population on the basis of a sample drawn from it. The generalizations to be made
about the population are usually either by way of
estimating the unknown population parameters, or
testing appropriate hypotheses stated in relation to population parameters in the light
of sample data
These generalizations, together with the measurement of their reliability, are made in terms
of the relationship between the values of any sample statistic and those of the
corresponding population parameters. Population parameter is any number computed (or
estimated) for the entire population viz. population mean, population median, population
proportion, population variance and so on. Population parameter is unknown but fixed,
whose value is to be estimated from the sample statistic that is known but random. Sample
Statistic is any numbers computed from our sample data viz. sample mean, sample
median, sample proportion, sample variance and so on.
It may be appreciated that no single value of the sample statistic is likely to be equal to the
corresponding population parameter. This owes to the fact that the sample statistic being
random, assumes different values in different samples of the same size drawn from the
same population.
Referring to our earlier discussion on the concept of a random variable in the lessons on
Probability Distributions, it is not difficult to see that any sample statistics is a random
variable and, therefore, has a probability distribution better known as the Sampling
Distribution of the statistic.
The sampling distribution of a statistic is the probability distribution of
all possible values the statistic may take when computed from random
samples of the same size drawn from a specified population.
3
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 12-1 Sampling Distribution of a Statistic
In reality, of course we do not have all possible samples and all possible values of
the statistic. We have only one sample and one value of the statistic. This value is
interpreted with respect to all other outcomes that might have happened, as represented by
the sampling distribution of the statistic. In this lesson, we will refer to the sampling
distributions of only the commonly used sample statistics like sample mean, sample
proportion, sample variance etc., which have a role in making inferences about the
population.
Why We Study Sampling Distributions?
Sample statistics form the basis of all inferences drawn about populations. Thus, sampling
distributions are of great value in inferential statistics. The sampling distribution of a sample
statistic possess well-defined properties which help lay down rules for making
generalizations about a population on the basis of a single sample drawn from it. The
variations in the value of sample statistic not only determine the shape of its sampling
distribution, but also account for the element of error in statistical inference. If we know the
probability distribution of the sample statistic, then we can calculate risks (error due to
356
Caritas Business College BUSINESS STATISTICS – OAD 221
chance) involved in making generalizations about the population. With the help of the
properties of sampling distribution of a sample statistic, we can calculate the probability that
the sample statistic assumes a particular value (if it is a discrete random variable) or has a
value in a given interval. This ability to calculate the probability that the sample statistic lies
in a particular interval is the most important factor in all statistical inferences. We will
demonstrate this by an example.
Suppose we know that 40% of the population of all users of hair oil prefers our brand to the
next competing brand. A "new improved" version of our brand has been developed and
given to a random sample of 100 users for use. If 55 of these prefer our "new improved"
version to the next competing brand, what should we conclude? For an answer, we would
like to know the probability that the sample proportion in a sample of size 100 is as large as
55% or higher when the true population proportion is only 40%, i.e. assuming that the new
version is no better than the old. If this probability is quite large, say 0.5, we might conclude
that the high sample proportion viz. 55% is perhaps because of sampling errors and the new
version is not really superior to the old. On the other hand, if this probability works out to a
very small figure, say 0.001, then rather than concluding that we have observed a rare event
we might conclude that the true population proportion is higher than 40%, i.e. the new
version is actually superior to the old one as perceived by members of the population. To
calculate this probability, we need to know the probability distribution of sample proportion
i.e. the sampling distribution of the proportion.
12.2 SAMPLING DISTRIBUTION OF THE MEAN
Suppose we have a simple random sample of size n, picked up from a population of size N.
We take measurements on each sample member in the characteristic of our interest
and
denote the observation x1 , x2 ,. respectively. The sample mean for this sample is
as ...............
xn
defined as:
357
Caritas Business College BUSINESS STATISTICS – OAD 221
x1 x2 . . xn
X n
If we pick up another sample of size n from the same population, we might end up with a
totally different set of sample values and so a different sample mean. Therefore, there are
many (perhaps infinite) possible values of the sample mean and the particular value that we
obtain, if we pick up only one sample, is determined only by chance. In other words, the
sample mean is a random variable. The possible values of this random variable depends
on the possible values of the elements in the random sample from which sample mean is to
be computed. The random sample, in turn, depends on the distribution of the population
from
which it is drawn. As a random variable, X has a probability distribution. This probability
distribution is the sampling distribution of X .
The sampling distribution of X is the probability distribution of all possible
values the random variable X may take when a sample of size n is
taken from a specified population.
To observe the distribution of X empirically, we have to take many samples of size n and
determine the value of X for each sample. Then, looking at the various observed values
of
X , it might be possible to get an idea of the nature of the distribution. We will derive the
distribution of X in three cases:
(a) Sampling from infinite populations
(b) Sampling with replacement from finite populations
(c) Sampling without replacement from finite populations
12.2.1 Sampling from Infinite Populations
Let us assume we have a population, with mean and variance 2, which is infinitely large.
If we take a sample of size n with individual values x1, x2 ,......x , then
n
358
Caritas Business College BUSINESS STATISTICS – OAD 221
x1 x2 . xn
Sample Mean X
n
here
x1 representing the first observed values in the sample, is a random variable since it may
take any of the population values. Similarly x2 , representing the second observed value in
sample is also a random variable since it may take any of the population values. In other
words, we can say
x , representing the ith observed value in the sample is a random
that i
variable.
Now when the population is infinitely large, whatever is the value of x1 , the distribution of
x2 is not affected by it. This is true for any other pair of random variables as well. In other
words; x , x ,. are independent random variables and all are picked up from the same
1 2
...............
xn
population.
So E x = and Var x = σ2 for i =1, 2,3,………n
i
Finally, we have
= E X
x1 x2 ... xn
=E
x
n
x1 x2
=E xn E [as E(A + B) = E(A) + E(B)]
E
n n n
1 1 1
Ex ......
Ex [as E(nA) = n E(A)]
= E x1 2 2
n n n
1 1 1
= + +.…..+
n n n
=μ
and
2 x1 x2 . . xn
= Var( X ) = Var
x
n
359
Caritas Business College BUSINESS STATISTICS – OAD 221
x1 x2 xn
= Var Var Var
n n n
[as Var(A + B) = Var (A) + Var (B)]
360
Caritas Business College BUSINESS STATISTICS – OAD 221
1 1 1
= Var(x ) Var(x ) ...... Var(x
[as Var(nA) = n2 Var(A)]
)
2 1 2 2 2 n
n n n
1 1
= 2 2 ...... 1 2
2
n
2 2
n n
2
=
n
So, = SD( X ) =
x n
12.2.2 Sampling With Replacement from Finite Populations
The above results have been obtained under the assumption that the random variables
x1 , x2 ,. are independent. This assumption is valid when the population is infinitely
...............
xn
large. It is also valid when the sampling is done with replacement, so that the population is
back to the same form before the next sample member is picked up. Hence, if the sampling
is done with replacement, we would again have:
= E X = μ 2
or = SD( X ) =
x x
2
= Var( X ) =
and n x n
12.2.3 Sampling Without Replacement from Finite Populations
When sampling without replacement from a finite population, the probability distribution of
the second random variable depends on what has been the outcome of the first pick and so
on. In other words, the n random variables representing the n sample members do not
remain
independent, the expression for the variance of X changes. The results in this case will be:
= E X = μ
x
Nn 2
Nn
and 2
= Var( X ) = . or = S.D( X ) = .
x
n N1 x n N1
By comparing these expressions with the ones derived above we find that the variance of
X is the same but further multiplied by a
361
Caritas Business College BUSINESS STATISTICS – OAD 221
factor Nn
. This factor is, therefore, known as
N 1
the finite population multiplier or the correction factor.
362
Caritas Business College BUSINESS STATISTICS – OAD 221
In practice, almost all the samples are picked up without replacement. Also, most
populations are finite although they may be very large and so the variance of the mean
should theoretically be found by using the expression given above. However, if the
population size
(N) is large and consequently the sampling ratio (n/N) small, then the finite population
multiplier is close to 1 and is not used, thus treating large finite populations as if they were
infinitely large. For example, if N = 100,000 and n = 100, the finite population multiplier will
be 0.9995, which is very close to 1 and the variance of the mean would, for all practical
purposes, be the same whether the population is treated as finite or infinite. As a rule of that,
the finite population multiplier may not be used if the sampling ratio (n/N) is smaller than
0.05.
Above discussion on the sampling distribution of mean, presents two very important
results, which we shall be using very often in statistical estimation and hypotheses testing.
We have seen that the expected value of the sample mean is the same as the population
mean.
Similarly, that the variance of the sample mean is the variance of the population divided
by the sample size (and multiplied by the correction factor when appropriate).
The fact that the sampling distribution of X has mean μ is very important. It means that, on
the average, the sample mean is equal to the population mean. The distribution of the
statistic
is centered on the parameter to be estimated, and this makes the statistic X a good
estimator of μ. This fact will become clearer in the next lesson, where we will discuss
estimators and their properties. The fact that the standard deviation of X is means
n
that as the sample
size increases, the standard deviation of X decreases, making X more likely to be close to
μ. This is another desirable property of a good estimator, to be discussed in the next lesson.
363
Caritas Business College BUSINESS STATISTICS – OAD 221
If we take a large number of samples of size n, then the average value of the sample means
tends to be close to the true population mean. On the other hand, if the sample size
is
364
Caritas Business College BUSINESS STATISTICS – OAD 221
increased then the variance of X gets reduced and by selecting an appropriately large
value of n, the variance of X can be made as small as desired.
The standard deviation of X is also called the standard error of the mean. It indicates the
extent to which the observed value of sample mean can be away from the true value, due to
sampling errors. For example, if the standard error of the mean is small, we may be
reasonably confident that whatever sample mean value we have observed cannot be very
far away from the true value.
Before discussing the shape of the sampling distribution of mean, let us verify the above
results empirically, with the help of a simple example.
Consider a discrete uniform population consisting of the values 1, 2, and 3. If the random
variable X represents these population values, its mean is
Xi 6
μ= = =2
N 3
and variance is 2
X 2
i (1 2)2 (2 2)2 (3 2) 2
σ =2
= =
N 3 3
(a) Sampling with Replacement
If random samples of size n = 2 are drawn with replacement from this population, we will
have Nn = 32 = 9 possible samples. These are shown in Box 12-1 along with the
corresponding sample mean values, which vary from 1 to 3.
The resulting distribution of X is given below:
X : 1 1.5 2 2.5 3
P( X ) : 1/9 2/9 3/9 2/9 1/9
Box 12-1
Sample No. 1 Sample No. 2 Sample No. 3
365
Caritas Business College BUSINESS STATISTICS – OAD 221
(1,1) (1,2) (1,3)
X=1 X = 1.5 X=2
Sample No. 4 Sample No. 5 Sample No. 6
(2,1) (2,2) (2,3)
X = 1.5 X=2 X = 2.5
Sample No. 7 Sample No. 8 Sample No. 9
(3,1) (3,2) (3,3)
X=2 X = 2.5 X=3
Now we can find out the mean and variance of the sampling distribution, the necessary
calculations are given in Table 12-1.
Table 12-1 Calculations and 2
for
x x
PX X .PX PX .[ X EX ]2
X
1 1/9 1/9 1/9
1.5 2/9 3/9 2/36
2 3/9 6/9 0
2.5 2/9 5/9 2/36
3 1/9 3/9 1/9
2
PX 1 X .PX 2 PX .[ X EX ] 1/
3
So the mean of the sampling distribution,
= E X
x
= X .PX = 2 = μ
and the variance of the sampling distribution,
2
= Var( X )
x
= PX .[ X EX 2/3 2
]2 =1/3 = =
2
n
(b) Sampling without Replacement
366
Caritas Business College BUSINESS STATISTICS – OAD 221
If random samples of size n = 2 are drawn without replacement from this population, we will
have Pn = 3P2 = 6 possible samples. These are shown in Box 12-2 along with the
N
corresponding sample mean values, which vary from 1.5 to 2.5.
Box 12-2
Sample No. 1 Sample No. 2 Sample No. 3
(1,2) (1,3) (2,1)
X = 1.5 X=2 X = 1.5
Sample No. 4 Sample No. 5 Sample No. 6
(2,3) (3,1) (3,2)
X = 2.5 X=2 X = 2.5
The resulting distribution of X is given below:
X : 1.5 2 2.5
P( X ) : 2/6 2/6 2/6
Now we can find out the mean and variance of the sampling distribution, the necessary
calculations are given in Table 12-2.
Table 12-2 Calculations and 2
for
x x
PX X .PX PX .[ X EX ]2
X
1.5 2/6 3/6 2/24
2 2/6 4/6 0
2.5 2/6 5/6 2/24
PX 1 X .PX 2 PX .[ X EX ] 1/
2
So the mean of the sampling distribution,
= E X
x
= X .PX = 2 = μ
and the variance of the sampling distribution,
2x
= Var( X )
367
Caritas Business College BUSINESS STATISTICS – OAD 221
= PX .[ X EX 2/33 2
Nn
]2 = 1/6 = = .
. 2 n N 1
2
3
1
Now if we compare the shapes of the parent population and the resulting sampling
distribution of mean, we find that although our parent population is uniformly distributed, the
sampling distribution of mean is symmetrically distributed as shown in Figure 12-2.
If we increase the sample size n we observe an interesting and important fact. As n increases
the possible values X can assume increases, so the number of rectangles increases
the probability that X assumes a particular value decreases i.e. the width of
rectangles decreases
Probabili
Parent Population Sampling Distribution of Mean
n=2 n=5
Figure 12-2 Parent Population and Sampling Distribution of Mean for n
= 2 and n = 5
In the limiting case when the sample size n increases infinitely, the particular values X can
assume approaches infinity and the probability that X assumes a particular value
approaches to zero. In other words, the limiting distribution of X is normal distribution.
Thus as n → ∝ X ~ N (μ, n )2
f( X )
Aa
Aaaa
Aaaaaaaa
aa aa
365
Caritas Business College BUSINESS STATISTICS – OAD 221
f (X)
Total Area
P(a < X < b)
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 12-3 Limiting Distribution of X
12.3 THE CENTRAL LIMIT THEOREM
The result we just stated - the limiting distribution of X is the normal distribution - is one of
the most important results in statistics. It is popularly known as the central limit theorem.
When sampling is done from a population with mean μ and standard
deviation σ, the sampling distribution of the sample mean X tends to a
nor- mal distribution with mean μ and standard deviation as the
n
sample
size n increases.
2
For "Large Enough" n: X ~ N (μ, n )
The central limit theorem is remarkable because it states that the distribution of the sample
mean X tends to a normal distribution regardless of the distribution of the population from
which the random sample is drawn. The theorem allows us to make probability statements
about the possible range of values the sample mean may take. It allows us to
compute
probabilities of how far away X may be from the population mean it estimates. We will
extensively use the central limit theorem in the next two lessons about statistical estimation
and testing of hypotheses.
3
Caritas Business College BUSINESS STATISTICS – OAD 221
Figure 12-4 Sampling Distributions of X for different Sample Sizes
The central limit theorem says that, in the limit, as n goes to infinity (n → ∝), the distribution
of X becomes a normal distribution (regardless of the distribution of the population). The
rate at which the distribution approaches a normal distribution does depend, however, on the
shape of the distribution of the parent population:
if the population itself is normally distributed, the distribution of X is normal for any
sample size n
if the population distributions are very different from a normal distribution, a
relatively large sample size is required to achieve a good normal approximation for
the distribution of X
Figure 12-4 shows several parent population distributions and the resulting sampling
distributions of X for different sample sizes.
Since we often do not know the shape of the population distribution, it would be useful to
have some general rule of thumb telling us when a sample is “Large Enough” that we may
apply the central limit theorem:
In general, a sample of 30 or more elements is considered “Large Enough”
for the central limit theorem to be applicable.
367
Caritas Business College BUSINESS STATISTICS – OAD 221
We emphasize that this is a general, and somewhat arbitrary, rule. A larger minimum sample
size may be required for a good normal approximation when the population distribution is
very different from a normal distribution. By the same reason, a smaller minimum sample
size may suffice for a good normal approximation when the population distribution is close to
a normal distribution.
Figure 12-5 Population Distribution and the Sampling Distribution of X
Figure 12-5 should help clarify the distinction between the population distribution and the
sampling distribution of X . The figure emphasizes the three aspects of the central limit
theorem:
1. When the sample size is large enough, the sampling distribution of X is
normal
2. The expected value of X is μ
n
3. The standard deviation of X is
The last statement is the key to the important fact that as the sample size increases,
the variation of X about its mean μ decreases. Stated another way, as we buy more
information
(take a larger sample), our uncertainty (measured by the standard deviation) about the
parameter being estimated decreases.
368
Caritas Business College BUSINESS STATISTICS – OAD 221
The History of the Central Limit Theorem
What we call the central limit theorem actually comprises several theorems developed over
the years. The first such theorem was the discovery of the normal curve by Abraham De
Moivre in 1733, when he discovered the normal distribution as the limit of the binomial
distribution. The fact that the normal distribution appears as a limit of the binomial distribu-
tion as n increases is a form of the central limit theorem. Around the turn of the twentieth
century, Liapunov gave a more general form of the central limit theorem, and in 1922
Lindeberg gave the final form we use in applied statistics. In 1935, W Feller gave the proof
of the necessary condition of the theorem.
Let us now look at an example of the use of the central limit theorem.
Example 12-1
ABC Tool Company makes Laser XR; a special engine used in speedboats. The company’s
engineers believe that the engine delivers an average power of 220 horsepower, and that
the standard deviation of power delivered is 15 horsepower. A potential buyer intends to
sample 100 engines (each engine to be run a single time). What is the probability that
the sample
mean X will be less than 217 horsepower?
Solution: Given that:
Population mean = 220
horsepower Population standard deviation = 15
horsepower Sample size n = 100
Here our random variable X is normal (or at least approximately so, by the central limit
theorem as our sample size is large).
369
Caritas Business College BUSINESS STATISTICS – OAD 221
2
X ~ N (μ, n )
2
15 2100 )
or X ~ N (220,
X
So we can use the standard normal variable Z = to find the required probability,
n
P( X < 217) = P(Z < 217 220 )
15100
= P(Z < -2)
= 0.0228
So there is a small probability that the potential buyer’s tests will result in a sample mean
less than 217 horsepower.
12.4 SAMPLING DISTRIBUTION OF THE PROPORTION
Let us assume we have a binomial population, with a proportion p of the population
possesses a particular attribute that is of interest to us. This also implies that a proportion q
(=1-p) of the population does not possess the attribute of interest. If we pick up a sample
of size n with
replacement and found x successes in the sample, the sample proportion of success ( p ) is
given by
x
p=
n
x is a binomial random variable, the possible value of this random variable depends on the
composition of the random sample from which p is computed. The probability of x
successes in the sample of size n is given by a binomial probability distribution, viz.
P( x) = nCx p x qn-x
x
Since p = and n is fixed (determined before the sampling) the distribution of the number
n
of successes (x) leads to the distribution of p .
370
Caritas Business College BUSINESS STATISTICS – OAD 221
The sampling distribution of p is the probability distribution of all possible
values the random variable p may take when a sample of size n is
taken from a specified population.
The expected value and the variance of x i.e. number of successes in a sample of size n is
known to be:
E(x) = n p
Var (x) = n p q
Finally we have mean and variance of the sampling distribution of p
x
= E p = E
p
n
1
= E(x) = 1 .n p =
n pn
2 x
and = Var = Var n
p
p
1 1 pq
= . Var(x) = .npq=
2 n2 n
n
= SD p pq
= n
p
When sampling is without replacement, we can use the finite population correction factor, so
sampling distribution of p has its
Mean =p
p
Variance
=
2 pq N n
.
p n N1
and standard deviation = pq . N n
p nN 1
371
Caritas Business College BUSINESS STATISTICS – OAD 221
As the sample size n increases, the central limit theorem applies here as well. The rate at
which the distribution approaches a normal distribution does depend, however, on the shape
of the distribution of the parent population.
if the population is symmetrically distributed, the distribution of p approaches the
normal distribution relatively fast
if the population distributions are very different from a symmetrical distribution, a
relatively large sample size is required to achieve a good normal approximation for
the distribution of p
In order to use the normal approximation for the sampling distribution of p , the sample size
needs to be large. A commonly used rule of thumb says that the normal approximation to the
distribution of p may be used only if both n p and n q are greater than 5.
We now state the central limit theorem when sampling for the population proportion p .
When sampling is done from a population with proportion p, the sampling
distribution of the sample proportion p approaches to a normal distribution
with proportion p and standard deviationpq n as the sample size n
increases.
2
For "Large Enough" n: p ~ N (p, pq n )
The estimated standard deviation of p is also called its standard error. We demonstrate
the use of the theorem in Example 12-2
Example 12-2
A manufacturer of screws has noticed that on an average 0.02 proportion of screws
produced are defective. A random sample of 400 screws is examined for the proportion
372
Caritas Business College BUSINESS STATISTICS – OAD 221
of defective
373
Caritas Business College BUSINESS STATISTICS – OAD 221
screws. Find the probability that the proportion of the defective screws ( p ) in the sample
is between 0.01 and 0.03?
Solution: Given that:
Population proportion p = 0.02
So q = 0.08 (= 1-0.02)
Sample size n = 400
Since the population is infinite and also the sample size is large, the central limit theorem
2
applies. So p ~ N (p, pq n )
2
p ~ N (0.02, (0.02)(0.08) 400 )
p p
We can find the required probability using standard normal variable Z =
pq / n
Z
0.01 0.02 0.03 0.02
P(0.01 < p < 0.03) = P
(0.02)(0.08)
400 (0.02)(0.08)
400
0.01
0.01
=P Z 0.007
= P(-1.43 < Z < 1.43)
= 2 P(0 < Z < 1.43)
= 0.8472
So there is a very high probability that the sample will result in a proportion between 0.01
and 0.03.
12.5 SAMPLING DISTRIBUTION OF THE DIFFERENCE OF SAMPLE MEANS
In order to bring out the sampling distribution of the difference of sample means, let us
assume we have two populations labeled as 1 and 2. So that
μ1 and μ2 denote the two population means.
σ1 and σ2 denote the two population standard deviations
n1 and n2 denote the two sample sizes
374
Caritas Business College BUSINESS STATISTICS – OAD 221
denote the two sample means
X 1 and
X2
Let us consider independent random sampling from the populations so that the sample
sizes need not be same for both populations.
Sinc are random variables so is their
e X 1 and difference X 1 - X 2 . As a random variable,
X2
X1 - has a probability distribution. This probability distribution is the sampling
X2
distribution of X 1 - X 2 .
The sampling distribution X1 - is the probability distribution of all
of X2
possible values the random X1 - may take when independent
variable X2
samples of size n1 and n2 are taken from two specified populations.
Mean and Variance of X 1 - X 2
= EX - X = EX EX
X1 X 2 1 1
= μ1 - μ2
= VarX - X 2 VarX
and = VarX
2
2 1
X1
X
2 1
2 2
= 1 2 ; when sampling is with replacement
n1 n2
1 2 N n 22 N n
= . 1 1 . 2 2 ; when sampling is without
n1 N1 1 n2 N 2 1
replacement
375
Caritas Business College BUSINESS STATISTICS – OAD 221
As the sample sizes n1 and n2 increases, the central limit theorem applies here as well. So we
state the central limit theorem when sampling for the difference of population X1 -X2
means
When sampling is done from two populations with means μ1 and μ2
and standard deviations σ1 and σ2 respectively, the sampling
distribution of the
difference of sample approaches to a normal distribution
means X1 -
X2
376
Caritas Business College BUSINESS STATISTICS – OAD 221
2 2
with mean μ1 - μ2 and standard deviation 1 2 as the sample sizes n1
n1n2
and n2 increases.
For "Large Enough" n1 and n2:
2
X1 - 2 2
X2
~ N (μ1 - μ2, 1 2 )
n1n2
The estimated standard deviation X 1 - X 2 is also called its standard error. We
of
demonstrate the use of the theorem in Example 12-3.
Example 12-3
The makers of Duracell batteries claims that the size AA battery lasts on an average of
45 minutes longer than Duracell’s main competitor, the Energizer. Two independent
random
samples of 100 batteries of each kind are selected. 1 minutes and
Assuming 84
2 67 minutes, find the probability that the difference in the average lives of Duracell and
Energizer batteries based on samples does not exceed 54 minutes.
Solution: Given that:
μ1 - μ2 = 45
σ1 = 84 and σ2 = 67
n1 =100 and n2 = 100
Let X 1 and denote the two sample average lives of Duracell and Energizer batteries
X2
respectively. Since the population is infinite and also the sample sizes are large, the
central limit theorem applies.
2
i.e X1 - 2 2
~ N (μ1 - μ2, 1 2 )
X2 n1n2
842 672 2
~ N (45, 100100 )
X1 -X2
377
Caritas Business College BUSINESS STATISTICS – OAD 221
So we can find the required probability using standard normal variable
X 2
X
Z 1 2 1
2 2
1 2
n1n 2
54 45
So P( X 1 - X 2 < 54) = P(Z< )
84 2 67 2
100100
= P(Z < 0.84)
= 1- 0.20045
= 0.79955
So there is a very high probability that the difference in the average lives of Duracell
and Energizer batteries based on samples does not exceed 54 minutes.
12.6 SAMPLING DISTRIBUTION OF THE DIFFERENCE OF SAMPLE
PROPORTIONS
Let us assume we have two binomial populations labeled as 1 and 2. So that
p1 and p2 denote the two population proportions
n1 and n2 denote the two sample sizes
p1 and denote the two sample proportions
p2
Let us consider independent random sampling from the populations so that the sample
sizes need not be same for both populations.
Sinc are random variables so is their
e p1 and difference p1 - p2 . As a random variable,
p2
p1 - has a probability distribution. This probability distribution is the sampling
p2
distribution of p1 - p2 .
The sampling distribution p1 - is the probability distribution of all
of p2
possible values the random p1 - may take when independent
variable p2
378
Caritas Business College BUSINESS STATISTICS – OAD 221
samples of size n1 and n2 are taken from two specified binomial populations.
379
Caritas Business College BUSINESS STATISTICS – OAD 221
Mean and Variance of
p1 - p2
Ep - p2 = Ep Ep
p
p1 = 1
2 1 2
= p1 - p2
= Varp - p2
and = Varp Varp
2 1 2
p1
p2
1
p1q1 p2 q2
= ; when sampling is with replacement
n2
n1
p1q1 N1 n1 p2 q2 N 2 n2
= . . ; when sampling is
n1 N 1 n2 N 1
1
2
without replacement
As the sample sizes n1 and n2 increases, the central limit theorem applies here as well. So
we state the central limit theorem when sampling for the difference of population
proportions
p1 - p2
When sampling is done from two populations with proportions p1 and p2
respectively, the sampling distribution of the difference of sample
proportions
p1 - p2 approaches to a normal distribution with mean p1 - p2 and standard
deviation
p1q1 p2 q2 as the sample sizes n and n increases.
1 2
n1 n2
2
For "Large Enough" n1 and n2: p1 - p1q1 p2 q2 )
~ N (p1 - p2,
p2 n1 n2
380
Caritas Business College BUSINESS STATISTICS – OAD 221
The estimated standard deviation p1 - p2 is also called its standard error. We demonstrate
of
the use of the theorem in Example 12-4.
Example 12-4
It has been experienced that proportions of defaulters (in tax payments) belonging to
business class and professional class are 0.20 and 0.15 respectively. The results of a
sample survey are:
381
Caritas Business College BUSINESS STATISTICS – OAD 221
Business class Professional class
Sample size: n1 = 400 n2 = 420
Proportion of defaulters:
p1 p2 0.14
0.21
Find the probability of drawing two samples with a difference in the two sample proportions
larger than what is observed.
Solution: Given that:
p1 = 0.20 p2 = 0.15
q1 = 1-0.20 = 0.80 q2 = 1-0.15 = 0.85
n1 = 400 n2 = 420
p1 0.21 p2 0.14
Since the population is infinite and also the sample sizes are large, the central limit
theorem applies. i.e.
2
p1 - p2 ~ N (p1 - p2, p1q1 p2 q2 )
n1 n2
2
p1 - p2
~ N (0.05, (0.20)(0.80) (0.15)(0.85) )
400 420
So we can find the required probability using standard normal variable
p p p p
Z 1 2 1 2
p1 q1 p 2 q 2
n1 n2
P( p1 - p2 > 0.07) = P(Z > 0.07 0.05 )
(0.20 )( 0.80 ) (0.15 )( 0.85 )
400 400
= P(Z > 0.76)
= 0.22363
So there is a low probability of drawing two samples with a difference in the two sample
proportions larger than what is observed.
382
Caritas Business College BUSINESS STATISTICS – OAD 221
12.7 SMALL SAMPLING DISTRIBUTIONS
Up to now we were discussing the large sampling distributions in the sense that the various
sampling distributions can be well approximated by a normal distribution for “Large Enough”
sample sizes. In other words, the Z-statistic is used in statistical inference when sample size
is large. It may, however, be appreciated that the sample size may be prohibited from
being large either due to physical limitations or due to practical difficulties of sampling costs
being too high. Consequently, for our statistical inferences, we may often have to contend
ourselves with a small sample size and limited information. The consequences of the sample
being small; n < 30; are that
the central limit theorem ceases to operate, and
the sample variance S2 fails to serve as an unbiased estimator of 2
Thus, the basic difference which the sample size makes is that while the sampling
distributions based on large samples are approximately normal and sample variance S2 is an
unbiased estimator of 2 , the same does not occur when the sample is small.
It may be appreciated that the small sampling distributions are also known as exact sampling
distributions, as the statistical inferences based on them are not subject to approximation.
However, the assumption of population being normal is the basic qualification underlying the
application of small sampling distributions.
In the category of small sampling distributions, the Binomial and Poisson distributions were
already discussed in lesson 9. Now we will discuss three more important small sampling
distributions – the chi-square, the F and the student t-distribution. The purpose of discussing
these distributions at this stage is limited only to understanding the variables, which define
them and their essential properties. The application of these distributions will be highlighted
in the next two lessons.
383
Caritas Business College BUSINESS STATISTICS – OAD 221
The small sampling distributions are defined in terms of the concept of degrees of freedom.
We will discuss this before concept proceeding further.
Degrees of Freedom (df)
The concept of degrees of freedom (df) is important for many statistical calculations and
probability distributions. We may define df associated with a sample statistic as the
number of observations contained in a set of sample data which can be freely chosen.
It refer to the number of independent variables which vary freely without being
influenced by the restrictions imposed by the sample statistic(s) to be computed.
1n
Let x1 , x2 be n observations comprising a sample whose
mean x xi is a value
.............. ni1
xn
known to us. Obviously, we are free to assign any value to n-1 observation out of n
observations. Once the value are freely assigned to n-1observations, freedom to do the
same for the nth observation is lost and its value is automatically determined as
nth observation = n x - sum of n-1 observations
n1
= n x xi
i1
As the value of nth observation must satisfy the restriction
n
xi nx
i1
We say that one degree of freedom, df is lost and the sum n x of n observations has n-1 df
associated with it.
For example, if the sum of four observations is 10, we are free to assign any value to three
observations only, x1 2, x2 1 and x3 4 . Given these values, the value of fourth
say,
observation is automatically determined as
x4 4 xi (x1 x2 x3 )
i1
384
Caritas Business College BUSINESS STATISTICS – OAD 221
x4 10 (2 1 4)
x4 3
Sampling essentially consists of defining various sample statistics and to make use them in
estimating the corresponding population parameters. In this respect, degrees of freedom
may be defined as the number of n independent observations contained in a sample
less the number of parameters m to be estimated on the basis of that sample
information, i.e. df = n-m.
For example, when the population variance 2 is not known, it is to be estimated by a
particular value of its estimator S2; the sample variance. The number of observations in the
sample being n, df = n-m = n-1 because 2 is the only parameter (i.e. m =1) to
be estimated by the sample variance.
12.8 SAMPLING DISTRIBUTION OF THE VARIANCE
We will now discuss the sampling distribution of the variance. We will first introduce
the concept of the sample variance as an unbiased estimator of population
variance and then present the chi-square distribution, which helps us in working out
probabilities for the sample variance.
12.8.1 THE SAMPLE VARIANCE
By now it is implicitly clear that we use the sample mean to estimate the population
mean and sample proportion to estimate the population proportion, when those parameters
are unknown. Similarly, we use a sample statistic called the sample variance to estimate the
population variance.
As will see in the next lesson on Statistical Estimation a sample statistic is an unbiased
estimator of the population parameter when the expected value of sample statistic is equal to
the corresponding population parameter it estimates.
Thus, if we use the sample variance S2 as an unbiased estimator of population variance2
385
Caritas Business College BUSINESS STATISTICS – OAD 221
Then E(S2) = 2
However, it can be shown empirically that while calculating S2 if we divide the sum of square
x x
of deviations from mean (SSD) i.e. n by n, it will not be an unbiased estimator of 2
2
i
1
n 2
x x n 2 2
and E i1 = 2 1 =
n n
n
2 2
x x
i.e. will underestimate the population variance 2 by the factor .
To
n n
2
x
n
x
compensate for this downward bias we x by n-1, so S2 i1 is
divide i n 2 that n1
x
1
an unbiased estimator of population variance 2 and we have:
n 2
i 1x 2
E =
n1
In other words to get the unbiased estimator of population variance 2, we divide the
2
n x x by the degree of freedom n-1
sum
i1
12.8.2 THE CHI-SQUARE DISTRIBUTION
Let X be a random variable representing N values of a population, which is normally
distributed with mean and variance2, i. e.
X = X 1 , X 2.....X N
We may draw a random sample of size n x1 , x2...xn values from this population.
comprising
As brought out in section 12.2, each of the n sample x1 , x2 can be treated as an
values
.............
xn
386
Caritas Business College BUSINESS STATISTICS – OAD 221
independent normal random variable with mean and variance 2. In other words
x ~ N (μ, 2) where i = 1, 2, ………n
i
387
Caritas Business College BUSINESS STATISTICS – OAD 221
Thus each of these n normally distribution random variable may be standardized so that
xi
Z
~ N (0, 12) where i = 1, 2, ………n
i
A sample statistic U may, then, be defined as
U Z 2 Z 2 ......... Z 2
1 2 n
n
2
U i
i1
2
x
n
U i
i1
Which will take different values in repeated random sampling. Obviously, U is a random
variable. It is called chi-square variable, denoted by χ2. Thus the chi-square random
variable is the sum of several independent, squared standard normal random
variables.
The chi-square distribution is the probability distribution of chi-square variable. So
The chi-square distribution is the probability distribution of the sum
of several independent, squared standard normal random variables.
The chi-square distribution is defined as
1 n
2 1
2 2 for χ2 ≥ 0
f(2)Ce (2) d2
where e is the base of natural logarithm, n denotes the sample size (or the number of
independent normal random variables).C is a constant to be so determined that the total
area under the χ2 distribution is unity. χ2 values are determined in terms of degrees of
freedom, df
=n
Properties of χ2 Distribution
1. A χ2 distribution is completely defined by the number of degrees of freedom, df = n.
So there are many χ2 distributions each with its own df.
388
Caritas Business College BUSINESS STATISTICS – OAD 221
2. χ2 is a sample statistic having no corresponding parameter, which makes
χ2distribution a non-parametric distribution.
3. As a sum of squares the χ2 random variable cannot be negative and is,
therefore, bounded on the left by zero.
Figure 12-6 χ2 Distribution with Different Numbers of df
4. The mean of a χ2 distribution is equal to the degrees of freedom df. The variance
of the distribution is equal to twice the number of degrees of freedom df .
E(χ2) = n Var (χ2 ) = 2n
5. Unless the df is large, a χ2 distribution is skewed to the right. As df increases, the
χ2 distribution looks more and more like a normal. Thus for large df
χ2 ~ N (n, 2
2n )
Figure 12-6 shows several χ2 distributions with different numbers of df.
In general, for n ≥ 30, the probability of χ2 taking a value greater than or less than a
particular value can be approximated by using the normal area tables.
6. If 2 , 2 2, 2 are k independent χ2 random variables, with degrees of
,.........
1 2 3 k
freedo
m n
2
, n , n ,.........n . Then their sum 2 2 2 ...... also possesses
1 2 3 k 1 2 3 k
a χ2 distribution with df = n
n n ......... nk .
1
2 3
12.8.3 The χ2Distribution in terms of Sample Variance S2
We can write
389
Caritas Business College BUSINESS STATISTICS – OAD 221
2
n2 x
n 1
i
2 (xi x) (x )
i1 i1
n
21 (x x)(x )
i1 x)2 (x ) 2
i 2(x
i
1
2 i1n (x x)2
n (x ) 2 (xi
i 1
2 n x)
i (x )
2 2 i1
1
2
(n 1)S 2 x
2
/ n
n n n
since ( x x) 2 ; ( x ) n(x ) and ( x x) 0
2 (n 1)S
i i
i1 i1
i1
Now, we know that the LHS of the above equation is a random variable which has chi-square
distribution, with df = n
We also know that if
2
2 x ~ N (μ, n )
x
Then will have a chi-square distribution with df = 1
n
n 1S 2
Since the two terms on the RHS are independent, will also has a chi-square
2
distribution with df = n-1. One degree of freedom is lost because all the deviations are
measured from x and not from ..
Expected Value and Variance of S2
n 1S 2
In practice, therefore, we work with the distribution of 1
2
of S2 directly.
390
Caritas Business College BUSINESS STATISTICS – OAD 221
and not with the distribution
Since has a chi-square distribution with df = n-1
2
391
Caritas Business College BUSINESS STATISTICS – OAD 221
n 1S 2
So E n1
2
n1
E(S 2 ) n 1
2
E(S 2 ) 2
(n 1)S 2
Also Var 2 2(n 1)
Using the definition of variance, we get
2
n 1S 2 (n 1)S 2
E 2 E 2(n 1)
2
n 2
1S 2
or E (n 2(n 1)
2 1)
n 2
2 (n 1)2 2(n n2 1S 2(n 1)
1 S 4 1)
or E
4 2
n ES
or 2 4
2S 2 2 2(n 1)
1 4
2
4
2
n 1
or E(S 2 2 ) 2 2(n 1)
4
2(n 1) 4
or E(S 2 2 ) 2
(n 1)2
2 4
So Var(S 2 )
n1
It may be noted that the conditions necessary for the central limit theorem to be operative in
the case of sample variance S2 are quite restrictive. For the sampling distribution of S2 to be
approximately normal requires not only that the parent population is normal, but also that the
392
Caritas Business College BUSINESS STATISTICS – OAD 221
sample is at least as large as 100.
Example 12-5
393
Caritas Business College BUSINESS STATISTICS – OAD 221
In an automated process, a machine fills cans of coffee. The variance of the filling process is
known to be 30. In order to keep the process in control, from time to time regular checks of
the variance of the filling process are made. This is done by randomly sampling filled cans,
measuring their amounts and computing the sample variance. A random sample of 101
cans is selected for the purpose. What is the probability that the sample variance is between
21.28 and 38.72?
Solution: We have
Population variance 2 = 30
n = 101
We can find the required probability by using the chi-square distribution
n
χ2 = 1S 2
2
So
= P(21.28 < S2 < 38.72) (101 1)21.28 2 (101 1)38.72
P
394
Caritas Business College BUSINESS STATISTICS – OAD 221
30 30
= P(70.93 < χ2 < 129.06)
= P(χ2 > 70.93) - P(χ2 > 129.06)
≈ 0.990 – 0.025
= 0.965
Since our population is normal and also sample size is quite large, we can also estimate
the required probability using normal distribution.
2 4 2
2 2
We have S ~ ( , n 1 )
So P(21.28 < < 38.72) = 21.28 Z 38.72 2
2
S2 2 4
P
2 4 n1
n 1
395
Caritas Business College BUSINESS STATISTICS – OAD 221
21.28 30 38.72 30
= P2x30x30 Z 2x30x30
101 1 101 1
8.72 8.72
=P Z
4.36 4.36
= P 2 Z 2
= 2P0 Z 2
= 2x0.4772
= 0.9544
Which is approximately the same as calculated above using χ2distribution
12.9 THE F -DISTRIBUTION
Let us assume two normal population with variances
2 an 2 repetitively. For a random
1
d
2
sample of size n1 drawn from the first population, we have the chi-square variable
n 1S 2
2 1 1
1 2
1
which process a χ2 distribution with ν1 = n1 -1 df
Similarly, for a random sample of size n2 drawn from the second population, we have the chi-
square variable
2
n 1S
2 2 2
2 2
2
which process a χ2 distribution with ν2 = n2 -1 df
A new sample statistic defined as
21
F
v1
2
2
v2
is a random variable known as F statistic, named in honor of the English statistician
396
Caritas Business College BUSINESS STATISTICS – OAD 221
Sir Ronald A Fisher.
397
Caritas Business College BUSINESS STATISTICS – OAD 221
Being a random variable it has a probability distribution, which is known as F distribution.
The F distribution is the distribution of the ratio of two chi-square
random variables that are independent of each other, each of which is
divided by its own degrees of freedom.
Properties of F- Distribution
1. The F distribution is defined by two kinds of degrees of freedom – the degrees of
freedom of the numerator always listed as the first item in the parentheses and the
degrees of freedom of the denominator always listed as the second item in the
parentheses. So there are a large number of F distributions for each pair of v1 and v2.
Figure 12-7 shows several F distributions with different v1 and v2.
2. As a ratio of two squared quantities, the F random variable cannot be negative and
is, therefore, bounded on the left by zero.
Figure 12-7 F- Distribution with different v1 and v2
3. The F(v , has no mean for v2 ≤ 2 and no variance for v2 ≤ 4. However, for v2
v )
1 2
>2, the mean and for v2 > 4, the variance is given
as
2v 2 (v 2)
v2 v
2 1 2
E( F )= Var( F )=
(v1 ,v2 ) v2 (v1 ,v2 ) v 2) (v 4)
2
1 2
2 (v2
4. Unless the v2 is large, a F distribution is skewed to the right. As v2 increases, the F
398
Caritas Business College BUSINESS STATISTICS – OAD 221
distribution looks more and more like a normal. In general, for v2 ≥ 30, the probability
399
Caritas Business College BUSINESS STATISTICS – OAD 221
of F taking a value greater than or less than a particular value can be approximated by
using the normal area tables.
5. The F distributions defined F(v , and F(v , are reciprocal of each other.
as v ) as v)
1 2 2 1
i.e.
1
F(v ,v ) =
1 (v2F ,v1 )
12.10 THE t-DISTRIBUTION
Let us assume a normal population with mean μ and variance 2 . If xi represent the n
values of a sample drawn from this population. Then
xi
Z
~ N (0, 12) where i = 1, 2, ………n
i
2 n 2
n
x
xi x
and U i
i1
~ χ2 (n-1 df) where i = 1, 2, ………n
2
i1
A new sample statistic T may, then, be defined as
xi
T
x x
n
i
2
1
i1
n1 2
xi
T
x x
n
i 2
i1
n1
xi
T S
This statistic - the ratio of the standard normal variable Z to the square root of the χ2
variable divided by its degree of freedom - is known as ‘t’ statistic or student ‘t’ statistic,
400
Caritas Business College BUSINESS STATISTICS – OAD 221
named after the pen name of Sir W S Gosset, who discovered the distribution of the
quantity.
401
Caritas Business College BUSINESS STATISTICS – OAD 221
xi
The random follows t-distribution with n-1 degrees of freedom.
variable
S
xi
~ t (n-1 df) where i = 1, 2, ………n
S
12.10.1 The t-distribution in terms of Sampling Distribution of Sample Mean
2
We know X ~ N (μ, n )
X ~ N (0, 12 )
So
n
xi
xi
Putting for , we get
in T
X 2
n
n xi x
1 i1
n1 2
X
n
T
x x
n
i
2
1
i1
n1 2
X
or T
x x
n
i 2
i1
1
n(n 1)
X
or T
1i1
n xi x 2
nn 1
X
or T
402
Caritas Business College BUSINESS STATISTICS – OAD 221
S
n
When defined as above, T again follows t-distribution with n-1 degrees of freedom.
403
Caritas Business College BUSINESS STATISTICS – OAD 221
XS
n ~ t (n-1 df) where i = 1, 2, ………n
Properties of t- Distribution
1. The t-distribution like Z distribution, is unimodal, symmetric about mean 0, and the t-
variable varies from -∝ and∝
2. The t-distribution is defined by the degrees of freedom v = n-1, the df associated
with the distribution are the df associated with the sample standard deviation.
3. The t-distribution has no mean for n = 2 i.e. for v = 1 and no variance for n ≤ 3 i.e. for v
≤ 2. However, for v >1, the mean and for v > 2, the variance is given as
v
E(T) = 0 Var(T) =
v
2
Figure 12-8 t-Distribution with different df
v
4. The of the t-distribution must always be greater than 1, so it is more
variance v2
variable as against Z distribution which has variance 1. This follows from the fact what
while Z values vary from sample to sample owing to the change in the X alone, the
variation in T values are due to changes in both X and S.
5. The variance of t-distribution approaches 1 as the sample size n tends to increase. In
general, for n ≥ 30, the variance of t-distribution is approximately the same as that of Z
distribution. In other words the t-distribution is approximately normal for n ≥ 30.
404
Caritas Business College BUSINESS STATISTICS – OAD 221
12.11 SELF-ASSESSMENT QUESTIONS
1. What is a sampling distribution, and what are the uses of sampling distributions?
2. How does the size of population and the kind of random sampling determine
the shape of the sampling distributions?
3. (a) A sample of size n = 5 is selected from a population. Under what conditions is
the sampling distribution of X normal?
(b) Suppose the population mean is μ = 125 and the population standard deviation
is 20. What are the expected value and the standard deviation of X ?
4. What is the most significant aspect of the central limit theorem? Discuss the
practical utility of central limit theorem in applied statistics.
5. Under what conditions is the central limit theorem most useful in sampling for
making statistical inferences about the population mean?
6. If the population mean is 1,247, the population variance is 10,000, and the sample
size is 100, what is the probability that X will be less than 1,230?
7. When sampling is from a population with standard deviation σ = 55, using a sample of
size n = 150, what is the probability that X will be at least 8 units away from the
population mean μ?
8. The Colosseum, once the most popular monument in Rome, dates from about AD 70.
Since then, earthquakes have caused considerable damage to the huge structure,
and engineers are currently trying to make sure the building will survive future
shocks. The Colosseum can be divided into several thousand small sections.
Suppose that the average section can withstand a quake measuring 3.4 on the
Richter scale with a standard deviation of 1.5. A random sample of 100 sections is
selected and tested for the maximum earthquake force they can withstand. What is
the probability that the
405
Caritas Business College BUSINESS STATISTICS – OAD 221
average section in the sample can withstand an earthquake measuring at least 3.6
on the Richter scale?
9. On June 10, 1997, the average price per share on the Big Board Composite Index in
New York rose 15 cents. Assume the population standard deviation that day was 5
cents. If a random sample of 50 stocks is selected that day, what is the probability
that the average price change in this sample was a rise between 14 and 16 cents?
10. An economist wishes to estimate the average family income in a certain population.
The population standard deviation is known to be Rs 4,000, and the economist uses
a random sample of size n = 225. What is the probability that the sample mean will
fall within Rs 750 of the population mean?
11. When sampling is done from a population with population proportion p = 0.2, using a
sample size n = 15, what is the sampling distribution of p ? Is it reasonable to use a
normal approximation for this sampling distribution? Explain.
12. When sampling is done for the proportion of defective items in a large shipment,
where the population proportion is 0.18 and the sample size is 200, what is the
probability that the sample proportion will be at least 0.20?
13. A study of the investment industry claims that 55% of all mutual funds outperformed
the stock market as a whole last year. An analyst wants to test this claim and obtains
a random sample of 280 mutual funds. The analyst finds that only 128 of the funds
outperformed the market during the year. Determine the probability that another
random sample would lead to a sample proportion as low as or lower than the one
obtained by the analyst, assuming the proportion of all mutual funds that out-
performed the market is indeed 0.55.
14. In recent years, convertible sport coupes have become very popular in Japan.
Toyota is currently shipping Celicas to Los Angeles, where a customizer does a
roof lift and
406
Caritas Business College BUSINESS STATISTICS – OAD 221
ships them back to Japan. Suppose that 25% of all Japanese in a given income and
lifestyle category are interested in buying Celica convertibles. A random sample of
100 Japanese consumers in the category of interest is to be selected. What is the
prob- ability that at least 20% of those in the sample will express an interest in a
Celica con- vertible?
15. What are the limitations of small samples?
16. What do you understand by small sampling distributions? Why are the small
sampling distributions called exact distributions?
17. What do you understand by the concept of degrees of freedom?
18. Define the χ2 statistic. What are important properties of χ2 distribution?
19. Define the F statistic. What are important properties of F distribution?
20. Define the t statistic. What are important properties of t-distribution? How does t
statistic differ from Z statistic?
12.12 SUGGESTED READINGS
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons.,
New Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
407
You might also like
- Principle of Information System: Chapter 10No ratings yetPrinciple of Information System: Chapter 106 pages
- OBE Course Sheet Tentative Assessment Criteria V7 PDFNo ratings yetOBE Course Sheet Tentative Assessment Criteria V7 PDF156 pages
- Data Communication and Computer Networking: ContentNo ratings yetData Communication and Computer Networking: Content42 pages
- CS3492 Database Management Systems Lecture Notes 2100% (1)CS3492 Database Management Systems Lecture Notes 2170 pages
- Bca 2 Sem Fundamentals of Statistics 77415 Dec 20200% (1)Bca 2 Sem Fundamentals of Statistics 77415 Dec 20202 pages
- Web Programming BCA - Unit 1 Study Materials (BHARATHIAR UNIVERSITY)No ratings yetWeb Programming BCA - Unit 1 Study Materials (BHARATHIAR UNIVERSITY)9 pages
- Fact Finding Techniques Fact Finding TechniquesNo ratings yetFact Finding Techniques Fact Finding Techniques29 pages
- IS - Chapter 1, Overview of Information SystemNo ratings yetIS - Chapter 1, Overview of Information System34 pages
- File Systems and Databases: Database Systems: Design, Implementation, and Management, 4th EditionNo ratings yetFile Systems and Databases: Database Systems: Design, Implementation, and Management, 4th Edition50 pages
- UG BBA Syllabus NEP 1st and 2nd Sem 2023No ratings yetUG BBA Syllabus NEP 1st and 2nd Sem 20238 pages
- Prototyping Model - Evolution and Spiral ModelsNo ratings yetPrototyping Model - Evolution and Spiral Models5 pages
- IT2403 Systems Analysis and Design: (Compulsory)No ratings yetIT2403 Systems Analysis and Design: (Compulsory)6 pages
- System Analysis and Design Notes For Bca 2nd Semester VbspuNo ratings yetSystem Analysis and Design Notes For Bca 2nd Semester Vbspu20 pages
- Lab 12 - Java Database Connectivity (JDBC)No ratings yetLab 12 - Java Database Connectivity (JDBC)15 pages
- ITEC 205 Information Management: Information and Decision MakingNo ratings yetITEC 205 Information Management: Information and Decision Making42 pages
- Heterogeneity in Parallel and Distributed SystemsNo ratings yetHeterogeneity in Parallel and Distributed Systems5 pages
- Developing A Program: Stewart Venit Elizabeth DrakeNo ratings yetDeveloping A Program: Stewart Venit Elizabeth Drake39 pages
- Field Project Final Report Submission Guidelines 2025No ratings yetField Project Final Report Submission Guidelines 202521 pages
- Chapter 10: Algorithms 10.1. Deterministic and Non-Deterministic AlgorithmNo ratings yetChapter 10: Algorithms 10.1. Deterministic and Non-Deterministic Algorithm5 pages
- Department of Computer Science & Engineering Question Bank Unit I Introduction and Conceptual Modelling PART-A (2 Marks)0% (1)Department of Computer Science & Engineering Question Bank Unit I Introduction and Conceptual Modelling PART-A (2 Marks)7 pages
- Teacher: Abbas Ali Email:: Subject: StatisticsNo ratings yetTeacher: Abbas Ali Email:: Subject: Statistics10 pages
- Basics of Business Statistics For Executive MBANo ratings yetBasics of Business Statistics For Executive MBA68 pages
- Introduction To Statistics Data CollectionNo ratings yetIntroduction To Statistics Data Collection17 pages
- Energy Consumption Data Mining TechniquesNo ratings yetEnergy Consumption Data Mining Techniques18 pages
- Report Summary On Optimizing Ticketing Operations and Management For Kochi MetroNo ratings yetReport Summary On Optimizing Ticketing Operations and Management For Kochi Metro15 pages
- Exploring The Potential of Multi Source Unsupervised Domain Adaptation in Crop Mapping Using Sentinel 2 ImagesNo ratings yetExploring The Potential of Multi Source Unsupervised Domain Adaptation in Crop Mapping Using Sentinel 2 Images20 pages
- 230 - Rice Price Transmission Between Wholesalers and Retailers in TheNo ratings yet230 - Rice Price Transmission Between Wholesalers and Retailers in The35 pages
- Stock Trading System Using SVM and Box TheoryNo ratings yetStock Trading System Using SVM and Box Theory8 pages
- Literature Review On Time Series Forecasting67% (3)Literature Review On Time Series Forecasting5 pages
- Water Resources Systems Analysis 1566706424100% (4)Water Resources Systems Analysis 1566706424608 pages
- Chapter 3 Business Cycle Measurement: Macroeconomics, 6e (Williamson)No ratings yetChapter 3 Business Cycle Measurement: Macroeconomics, 6e (Williamson)19 pages
- Symbiosis Skills and Professional University KiwaleNo ratings yetSymbiosis Skills and Professional University Kiwale32 pages