Introduction to Statistics for

Engineers

Chapter One

Introduction to Statistics

1
Outline
Introduction to statistics

Importance of statistics

Limitation of statistics

Application areas of statistics

Classification of statistics

Main terms in statistics

Steps In Statistical Investigation

 The Engineering Method And Statistical Thinking

2
Definitions:
It is a science which helps us to collect, analyze and
present data systematically.

It is the process of collecting, processing,

summarizing, presenting, analysing and interpreting of
data in order to study and describe a given problem.

Statistics is the art of learning from data.

Statistics may be regard as (i)the study of populations,

(ii) the study of variation, and (iii) the study of methods
of the reduction of data.
3
3
 Importance of Statistics:
 It simplifies mass of data (condensation);
 Helps to get concrete information about any
problem;
 Helps for reliable and objective decision making;
 It presents facts in a precise & definite form;
 It facilitates comparison(Measures of central
tendency and measures of dispersion);
 It facilitates Predictions (Time series and regression
analysis are the most commonly used methods
towards prediction.)
 It helps in formulation of suitable policies;
4
4
Limitation of statistics:
 Statistics does not deal with individual
items;

 Statistics deals only with quantitatively

expressed items, it does not study
qualitative phenomena;

 Statistical results are not universally true;

 Statistics is liable to be misused.

5
5
 Application areas of statistics
Some of the diverse fields in which Statistical methodology
has extensive applications are:

Engineering:
Improving product design, testing product performance,
determining reliability and maintainability, working out safer
systems of flight control for airports, etc.
Business:
Estimating the volume of retail sales, designing optimum
inventory control system, producing auditing and accounting
procedures, improving working conditions in industrial plants,
assessing the market for new products.
6
Quality Control: Cont’d
Determining techniques for evaluation of quality through
adequate sampling, in process control, consumer survey and
experimental design in product development etc.
* Realizing its importance, large organizations are maintaining
their own Statistical Quality Control Department *.
Economics:
Measuring indicators such as volume of trade, size of labor
force, and standard of living, analyzing consumer behavior,
computation of national income accounts, formulation of
economic laws, etc.
* Particularly, Regression analysis extensively uses in the field
of Economics*.
7
 Health and Medicine: Cont’d
Developing and testing new drugs, delivering improved medical care,
preventing diagnosing, and treating disease, etc. Specifically, inferential
Statistics has a tremendous application in the fields of health and
medicine.
 Biology:
Exploring the interactions of species with their environment, creating
theoretical models of the nervous system, studying genetically evolution,
etc.
 Psychology:
Measuring learning ability, intelligence, and personality characteristics,
creating psychological scales and abnormal behavior, etc.
 Sociology:
Testing theories about social systems, designing and conducting sample
surveys to study social attitudes, exploring cross-cultural differences,
studying the growth of human population, etc.
8
8
 Classification of statistics
There are two main branches of statistics:
1. Descriptive statistics

2. Inferential statistics
1. Descriptive statistics:
 It is the first phase of Statistics;

 involve any kind of data processing designed to the collection,

organization, presentation, and analyzing the important features
of the data with out attempting to infer/conclude any thing
that goes beyond the known data.

 In short, descriptive Statistics describes the nature or

characteristics of the observed data (usually a sample) without
making conclusion or generalization. 9
9
 Cont’d
The following are some examples of descriptive
Statistics:
The daily average temperature range of AA was 25 0c
last week .

The maximum amount of coffee export of Eth. (as

observed from the last 20 years) was in the year 2004.

The average age of athletes participated in London

Marathon was 25 years.

75% of the instructors in AAU are male.

The scores of 50 students in a Mathematics exam are

found to range from 20 to 90. 10
10
2. Inferential statistics (Inductive Statistics)
 It is a second phase of Statistics which deals with techniques of
making a generalization that lie outside the scope of Descriptive
Statistics;

 It is concerned with the process of drawing conclusions

(inferences) about specific characteristics of a population based
on information obtained from samples;

 It is a process of performing hypothesis testing, determining

relationships among variables, and making predictions.

 The area of inferential statistics entirely needs the whole

aims to give reasonable estimates of unknown population
parameters.
11
11
 Cont’d
The following are some examples of inferential Statistics:
 The result obtained from the analysis of the income of 1000 randomly
selected citizens in Ethiopia suggests that the average monthly income of
a citizen is estimated to be 600 Birr.
 Here in the above example we are trying to represent the income of
about entire population of Ethiopia by a sample of 1000 citizens, hence
we are making inference or generalization.
 Based on the trend analysis on the past observations/data, the average
exchange rate for a dollar is expected to be 21 birr in the coming month.
 The national statistical Bereau of Ethiopia declares the out come of its
survey as “The population of Eth. in the year 2020 will likely to be
120,000,000.”
 From the survey obtained on 15 randomly selected towns of Eth. it is
estimated that 0.1% of the whole urban dwellers are victims of AIDS
virus. 12
 Exercise: Descriptive Statistics or Inferential Statistics
1. The manager of quality control declares the out come of its survey
as “the average life span of the imported light bulbs is 3000 hrs.
Inferential
2. Of all the patients taking the drug at a local health center 80% of
them suffer from side effect developed.
Descriptive
4. The national statistical bureau of Eth. declares the out come of its
survey for the last 30 years as “the average annual growth of the
people of Ethiopia is 2.8%. Inferential
5. Based on the survey made for the last 10 years 30,000 tourists are
expected to visit Ethiopia. Descriptive.
6. Based on the survey made for the last 20 years the maximum
number of tourists visited Eth. were in the year 1993. Descriptive.
7. The Ethiopian tourism commission has announced that (as observed
for the last 20 years) the average number of tourists arrived
Ethiopia per year is 3000. Inferential
8. The maximum difference of the salaries of the workers of the
company until the end of last year was birr 5000. Descriptive.
13
 Main terms in statistics
Data: Certainly known facts from which conclusions may be drawn.

Statistical data: Raw material for a statistical investigation which

are obtained when ever measurement or observations are made.

i. Quantitative data: data of a certain group of individuals which is

expressed numerically.

Example: Heights, Weights, Ages and, etc of a certain group of

individuals.

ii. Qualitative data: data of a certain group of individuals that is

not expressed numerically as it is. Example: Colors, Languages,
Nationalities, Religions, health, poverty etc of a certain group of
individuals. 14
 Cont’d
Variable: A variable is a factor or characteristic that can take on
different possible values or outcomes. Example: Income, height,
weight, sex, age, etc of a certain group of individuals are examples
of variables. A variable can be qualitative or quantitative (numeric).
Population: A complete set of observation (data) of the entire group
of individuals under consideration e.g. The number of students in
this class, the population in Addis Ababa etc. A population can be
finite or infinite.
Sample: A set of data drawn from population containing a part
which can reasonably serve as a basis for valid generalization about
the population.Or a sample is a portion of a population selected for
further analysis.
Sample size: The number of items under investigation.
15
 Cont’d

Survey (experiment): it is a device of obtaining the desired

data. Two types of survey:

1. Census Survey: A way of obtaining data referring the entire

population which is said to provide a total coverage of the
population.

2. Sample Survey: A way of obtaining data referring a portion

of the entire population which is said to provide only a partial

coverage of the population.

16
 Steps/stages in Statistical Investigation
1. Collection of Data:
Data collection is the process of gathering information or
data about the variable of interest. Data are inputs for
Statistical investigation. Data may be obtained either
from primary source or secondary source.
2. Organization of Data
Organization of data includes three major steps.
1 . Editing: checking and omitting inconsistencies,
irrelevancies.
2. Classification : task of grouping the collected and edited
data .
3. Tabulation: put the classified data in the form of table.
17
17
 Cont’d
3. Presentation of Data
The purpose of presentation in the statistical analysis is to display
what is contained in the data in the form of Charts, Pictures,
Diagrams and Graphs for an easy and better understanding of the
data.
4. Analyzing of Data
 In a statistical investigation, the process of analyzing data
includes finding the various statistical constants from the
collected mass of data such as measures of central tendencies
(averages) , measures of dispersions and soon.
 It merely involves mathematical operations: different measures of
central tendencies (averages), measures of variations, regression
analysis etc. In its extreme case, analysis requires the knowledge
of advanced mathematics. 18
18
 Cont’d
5. Interpretation of Data
 involve interpreting the statistical constants computed in
analyzing data for the formation of valid conclusions and
inferences.
 It is the most difficult and skill requiring stage.
 It is at this stage that Statistics seems to be very much
viable to be misused.
 Correct interpretation of results will lead to a valid
conclusion of the study and hence can aid in taking correct
decisions.
 Improper (incorrect) interpretation may lead to wrong
conclusions and makes the whole objective of the study
useless. 19
19
20
 THE ENGINEERING METHOD AND
STATISTICAL THINKING
 An engineer is someone who solves problems of
interest to society by the efficient application
of scientific principles.

 Engineers accomplish this by either refining an

existing product or process or by designing a
new product or process that meets customers’
expectations and needs.
21
The steps in the engineering method are as follows:

Develop a clear and brief description of the problem.

1. Identify, the important factors that affect this
problem or that may play a role in its solution.
2. Propose a model for the problem, using scientific or
engineering knowledge of the phenomenon being
studied. State any limitations or assumptions of the
model.
3. Conduct appropriate experiments and collect data to
test or validate the tentative model or conclusions
made in steps 2 and 3.
22
 Cont’d

5. Refine the model on the basis of the observed data.

6. Manipulate the model to assist in developing a
solution to the problem.

7. Conduct an appropriate experiment to confirm that

the proposed solution to the problem is both
effective and efficient.

8. Draw conclusions or make recommendations based

on the problem solution.

23
 Cont’d

 The engineering method features a strong interplay

between the problem, the factors that may
influence its solution, a model of the phenomenon,
and experimentation to verify the adequacy of the
model and the proposed solution to the problem.

 Specifically, statistical techniques can be a powerful

aid in designing new products and systems, improving
existing designs, and designing, developing, and
improving production processes.
24
 Cont’d

 Therefore, Engineers must know how to

efficiently plan experiments, collect data,
analyze and interpret the data, and understand
how the observed data are related to the model
that have been proposed for the problem under
study.

25
 Data collection and representation
The term “Data Collection” refers to all the issues
related to data sources, scope of investigation and sampling
techniques.

The quality of data greatly affects the final output of an

investigation.

With inaccurate and inadequate data, the whole analysis

is likely to be faulty and also the decisions to be taken will
also be misleading. 26
 Cont’d
Classification of Data based on source:
1. Primary: data collected for the purpose of specific study.
It can be obtained by:
Direct personal observation
Direct or indirect oral interviews
Administrating questionnaires

2. Secondary: refers to data collected earlier for some

purpose other than the analysis currently being
undertaken.
It can be obtained from:
 External Secondary data Sources( for eg. gov’t and non
gov’t publications)
 Internal Secondary data Sources: the data generated
within the organization in the process of routine
business activities
27
Advantages and Disadvantages of Primary and
Secondary data
The advantages of primary data over of secondary
data:-

• The primary data gives more reliable, accurate and

adequate information, which is suitable to the
objective and purpose of an investigation.

• Primary source usually shows data in greater detail.

• Primary data is free from errors that may arise

from copying of figures from publications.
28
 Cont’d
The disadvantages of primary data are:

The process of collecting primary data is time

consuming and costly.

Often, primary data gives misleading

information due to lack of integrity of
investigators and non-cooperation of
respondents in providing answers to certain
delicate questions.
29
 Cont’d
The advantage of Secondary data:
 It is readily available and hence convenient and much
quicker to obtain than primary data.
 It reduces time, cost and effort as compared to primary
data.
 Secondary data may be available in subjects (cases)
where it is impossible to collect primary data. Such a
case can be regions where there is war.
The disadvantages of Secondary data are:
 Data obtained may not be sufficiently accurate.
 Data that exactly suit our purpose may not be found.
 Error may be made while copying figures
30
 Cont’d
• The choice between primary data and
secondary data is determined by
following factors
 Nature and scope of the enquiry
 Availability of financial resources
 Availability of time
 Degree of accuracy desired
 Collecting agency
31
 Cont’d
Methods of collecting primary data
I. Personal Enquiry Method (Interview method)
II. Direct Observation (personal observation)
III.Questionnaire method (written questions)

I. Personal Enquiry Method (Interview method):

 In personal enquiry method, a question sheet is
prepared which is called schedule.
 Depending on the nature of the interview, personal
enquiry method is further classified into two types.
1.Direct Personal Interview
2. Indirect Personal Enquiry (Interview)
32
 Cont’d
II. Direct Observation:
 In this approach, an investigator stays at the
place of survey and notes down the observation
himself.

III. Questionnaire Method:

 Under this method, a list of questions related to

the survey is prepared and sent to the various
respondents by post, Web sites, e-mail, etc.

33
 Sampling Techniques
 Inferential statistics is a systematic method of inferring
satisfactory conclusions about the population on the basis
of examining a few representative units termed as sample.
 The process of selecting samples is called sampling
 The number of units in the sample is called Sample size.
 The size of sample for a study is determined on the basis
of the following factors
 The size of the population
 The availability of resources
 The degree of accuracy
 The homogeneity or heterogeneity of the population
 The nature of the study
 The method of sampling technique adopted
 The nature of respondents
34
 Reason for sampling
We study a sample of population instead of
considering the entire population due to one or the
other of the following reasons.
 Shortage of money, time and labor
 Limited data available
 Minimize destruction
 Obtaining more complete detailed and accurate
information about the characteristics of the
population with less time, energy and expenditure
35
 Cont’d
A good sample possesses two characteristics, which are:

i. Representativeness of the Universe and

ii. Adequate in Size

 So the selection of a sample should be done in a

manner that every item in the universe must have an
equal chance of inclusion in the sample.

The sampling methods may broadly be classified as:

i. Random/Probability sampling

ii. Non random/Non-probability sampling

36
 Cont’d
Random sampling method is a method of selection
of a sample such that each item within the
population has equal chance of being selected.
Random sampling method is further divided into the
following
i. Simple
ii. Stratified,
iii. Systematic and
iv. Cluster sampling
i. Simple Random Sampling Method:
This method involves very simple method of drawing
a sample from a given population used when the
population under consideration is homogenous.
37
 Cont’d
ii. Stratified Random Sampling Method:
This method of sampling is used when the population under
study is heterogeneous.

iii. Systematic Random Sampling Method:

This method of sampling is a common method of selecting a
sample when a complete list of the population is available

iv. Cluster Random Sampling:

In cluster random sampling, the total population is divided into
a number of relatively small non overlapping divisions called
clusters and some of these clusters can randomly be selected
for the inclusion of the over all sample.
38
Non-random (Non-probability) Sampling Method
non-random sampling technique is further divided into the following:

 Judgment

 Convenient and

 Quota sampling.
Judgment Sampling: -
In Judgment sampling, personal judgment plays a significant role in
the selection of the sample

Convenient Sampling: - Convenient Sample is a Sample obtained by

selecting population units that are convenient to select for the
investigator with regard to saving time, money and labor and
obtaining required information.
39
 Cont’d

Quota sampling: - In Quota sampling the

population is subdivided into a number of strata’s
(groups) and instead of taking random samples
from each stratum, investigator takes simply
quotas from the different strata.

The sample with in prescribed quota is selected by

personal judgment of the investigator

40
 Data Presentation
Raw data:- It is collected numerical data which has
not been arranged in order of magnitude.

An array is an arranged numerical data in order of

magnitude.

Data presentation methods:

Tabular method

 Graphical method

 Diagrammatic method
41
41
 Cont’d
1. Tabular presentation of data:

 The collected raw data should be put into an

ordered array in either ascending or
descending order so that it can be organized
in to a Frequency Distribution (FD)

 Numerical data arranged in order of magnitude

along with the corresponding frequency is
called frequency distribution (FD).

 FD is of two kinds: namely ungrouped /and

grouped frequency distribution.
42
42
 Cont’d

A. Ungrouped (Discrete) Frequency Distribution

 It is a tabular arrangement of
numerical data in order of magnitude
showing the distinct values with the
corresponding frequencies.

43
Example:
Suppose the following are test score of 16 students in a
class, write un grouped frequency distribution.
“14, 17, 10, 19, 14, 10, 14, 8, 10, 17, 19, 8, 10, 14, 17, 14”

Sol: the ungrouped frequency distribution:

Array: 8,8,10,10,10,10,14,14,14,14,14, 17,17,17,19,19.
Then the ungrouped frequency distribution is then
grouped:-
Test score 8 10 14 17 19
Frequency 2 4 5 3 2

 The difference between the highest and the lowest

value in a given set of observation is called the range
(R) R= L- S, R= 19-8 = 11
44
 B. Grouped (continuous) Frequency
Distribution (GFD)
 It is a tabular arrangement of data in order of
magnitude by classes together with the corresponding
class frequencies.

 In order to estimate the number of classes, the ff

formula is used:

Number of classes = 1+3.322(log N) where N is the

Number of observation.

The Class size = Range (round up)

(class width) 1+3.322(log N)
45
Example:
Grouped/Continuous frequency distribution
where several numbers are grouped into one
class.
e.g. Student age Frequency

18-25 5
26-32 15
33-39 10
46
46
 Components of grouped frequency distribution

1. Lower class limit:

is the smallest number that can actually belong to
the respective classes.
2. Upper class limit:
is the largest number that can actually belong to the
respective classes.
3. Class boundaries:
are numbers used to separate adjoining classes which
should not coincide with the actual observations.
4. Class mark:
is the midpoint of the class.

47
 Cont’d
5. Class width/ Class intervals
is the difference between two consecutive lower
class limits or the two consecutive upper class
limits. (OR)
can be obtained by taking the difference of two
adjoining class marks or two adjoining lower class
boundaries.
Class width = Range/Number of class desired.
Where: Number of classes=1+3.322(log N) where N is
the Number of observation.

6. Unit of measure
is the smallest possible positive difference
between any two measurements in the given data
set that shows the degree of precision.
48
 Cont’d
 Class boundaries:

can be obtained by taking the averages of the

upper class limit of one class and the lower class
limit of the next class.

 Lower class boundaries:

can be obtained by subtracting half a unit of

measure from the lower class limits.

 Upper class boundaries:

can be obtained by adding half the unit of

measure to the upper class limits.
49
Example1 :
Suppose the table below is the frequency distribution
of test score of 50 students.
Then the frequency table has 6 classes (class intervals).
Test score Frequency
11-15 7
16-20 8
21-25 10
26-30 12
31-35 9
36-40 4
What is the Unit of Measure, LCLs, UCLs, LCBs, UCBs, CW, and CM

50
 Cont’d
 The unit of measure is 1

 The lower class limits are:-11, 16, 21, 26, 31, 36

 The upper class limits are:-15, 20, 25, 30, 35, 40

 The class marks are:- 13((11+15)/2), 18, 23, 28, 33, 38

 The lower class boundaries are:- 10.5(11-0.5), 15.5, …,

35.5

 The upper class boundaries are:- 15.5(15+0.5), ….35.5,

40.5

 Class width (size) is 5.

51
52
 Rules to construct Grouped
Frequency Distribution (GFD):
i. Find the unit of measure of the given data;
ii. Find the range;
iii. Determine the number of classes required;
iv. Find class width (size);
v. Determine a lowest class limit and then find the successive
lower and upper class limits forming non over lapping
intervals such that each observation falls into exactly one
of the class intervals;
vi. Find the number of observations falling into each class
intervals that is taken as the frequency of the class (class
interval) which is best done using a tally.
53
Exercise:-
Construct a GFD of the following aptitude test scores
of 40 applicants for accountancy positions in a company
with

a. 6 classes b. 8 classes

96 89 58 61 46 59 75 54
41 56 77 49 58 60 63 82
66 64 69 67 62 55 67 70
78 65 52 76 69 86 44 76
57 68 64 52 53 74 68 39
54
 Types of Grouped Frequency
Distribution

1. Relative frequency distribution (RFD)

2. Cumulative Frequency Distribution (CFD)

3. Relative Cumulative Frequency Distribution

(RCFD)

55
 Cont’d
1. Relative frequency distribution (RFD):

 A table presenting the ratio of the

frequency of each class to the total
frequency of all the classes.

 Relative frequency generally expressed as

a percentage, used to show the percent of
the total number of observation in each
class.

56
For example
Test score F RFD PFD
37.5-47.5 4 4/40=0.1 10%
47.5-57.5 8 8/40=0.2 20%
57.5-67.5 13 13/40=0.325 32.5%
67.5-77.5 10 10/40=0.25 25%
77.5-87.5 3 3/40=0.075 7.5%
87.5-97.5 2 2/40=0.05 5%
57
 Cont’d
2. Cumulative Frequency Distribution (CFD):

It is applicable when we want to know how many

observations lie below or above a certain value/class
boundary.

CFD is of two types, LCFD and MCFD:

 Less than Cumulative Frequency Distribution
(LCFD): shows the collection of cases lying below
the upper class boundaries of each class.
 More than Cumulative Frequency Distribution
(MCFD): shows the collection of cases lying above
the lower class boundaries of each class.
58
58
 Cont’d

LRCF MRCF
Test score CF Test score CF

Less than 37.5 0 more than 37.5 40

Less than47.5 4 more than 47.5 36
Less than 57.5 12 more than 57.5 28
Less than 67.5 25 more than 67.5 15
Less than 77.5 35 more than 77.5 5
Less than 87.5 38 more than 87.5 2
Less than 97.5 40 more than 97.5 0

59
Chapter 1- Introduction to statistics 59
3. Relative Cumulative Frequency Distribution (RCFD)
 It is used to determine the ratio or the percentage of observations
that lie below or above a certain value/class boundary, to the total
frequency of all the classes. These are of two types: The LRCFD and
MRCFD.

 Less than Relative Cumulative Frequency Distribution (LRCFD): A

table presenting the ratio of the cumulative frequency less than
upper class boundary of each class to the total frequency of all
the classes

 More than Relative Cumulative Frequency Distribution (MRCFD): A

table presenting the ratio of the cumulative frequency more than
lower class boundary of each class to the total frequency of all
the classes. 60
 Cont’d
LRCFD
Test score LCF LRCF LPCF
Less than 37.5 0 0/40=0 0%
Less than 47.5 4 4/40=0.1 10%
Less than 57.5 12 12/40=0.3 30%
Less than 67.5 25 25/40=0.625 62.5%
Less than 77.5 35 35/40=0.875 87.5%
Less than 87.5 38 38/40=0.95 95%
Less than 97.5 40 40/40=1 100%

61
 Cont’d
MRCFD
Test score MC F MR C F MP C F
More than 37.5 40 40/40=1 100%
More than 47.5 36 36/40=0.9 90%
More than 57.5 28 28/40=0.7 70%
More than 67.5 15 15/40=0.375 37.5%
More than 77.5 5 5/40=0.125 12.5%
More than 87.5 2 2/40=0.05 5%
More than 97.5 0 0/40=0 0%
62
Graphic Methods of Data presentation
1. Histogram
2. Frequency Polygon (Line graph)

3. Cumulative frequency curve (o-give)

63
63
1. Histogram:
A graphical presentation of grouped
frequency distribution consisting of a series
of adjacent rectangles whose bases are the
class intervals specified in terms of class
boundaries (equal to the class width of the
corresponding classes) shown on the x-axis
and whose heights are proportional to the
corresponding class frequencies shown on
the y-axis.
64
 Cont’d

Histogram: E.g.

Freq.
20

15

10
3-D Column 1
5

0
20 - 3030 - 4040 - 5050 - 60 60 - 70 70 -80

65
Chapter 1- Introduction to statistics 65
 Cont’d
Uses for a Histogram
A Histogram can be used:

 to display large amounts of data values in a

relatively simple chart form.

 to tell relative frequency of occurrence.

 to easily see the distribution of the data.

 to see if there is variation in the data.

 to make future predictions based on the data.

 Steps to draw Histogram
i. Mark the class boundaries on the horizontal
axis (x- axis) and the class frequencies along
the vertical axis ( y- axis) according to a
suitable scale.

ii. With each interval as a base draw a

rectangle whose height equals the frequency
of the corresponding class interval. It
describes the shape of the data.
67
2. Frequency Polygon:
It is a line graph of grouped frequency
distribution in which the class frequency
is plotted against class mark (Midpoint)
that are subsequently connected by a
series of line segments to form line graph
including classes with zero frequencies at
both ends of the distribution to form a
polygon.
68
68
 Cont’d
Frequency Polygon:

20

15

10 Line 1

0
20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 -80 80

69
69
 Steps to draw Frequency polygon
i. Mark the class mid points on the x-axis and
the frequency on the y-axis.
ii. Mark dots which correspond to the
frequency of the marked class mid points.
iii. Join each successive dot by a series of
line segments to form line graph, including
classes with zero frequencies at both ends
of the distribution to form a polygon.
70
Find the midpoints of each class
Create a frequency polygon using the data
Create a frequency polygon using the data
3. O-GIVE curve (Cumulative Frequency Curve /
percentage Cumulative Frequency Curve)
{The number of values less than the upper class boundary for the
current class. This is a running total of the frequencies.}

 It is a line graph presenting the cumulative frequency

distribution.

 O-gives are of two types: The Less than O-give and The
More than O-give.

 The Less than O-give shows the cumulative frequency

less than the upper class boundary of each class; and

 The More than O-give shows the cumulative frequency

more than the lower class boundary of each class. 74
74
 Cont’d

Ogive: E.g.
45
40
35
30
25
Line 1
20
15
10
5
0
20 30 40 50 60 70 80
75
Chapter 1- Introduction to statistics 75
Steps to draw O-gives
i. Mark class boundaries on the x-axis and mark non
overlapping intervals of equal length on the y-axis to
represent the cumulative frequencies.
ii. For each class boundaries marked on the x-axis, plot a
point with height equal to the corresponding cumulative
frequencies.
iii. Connect the marked points by a series of line segments
where the less than O-give is done by plotting the less
than cumulative frequency against the upper class
boundaries
76
Chapter 1- Introduction to statistics 76
Draw the x and y axis

Plot the points

Diagrammatic Presentation Of Data

• Bar charts

• Pie chart

• Pictograph and

• Pareto diagram

81
 Outline

1.3 Measures of Central Tendency

1.4 Measures of Dispersion

1.5 Measure of skewness and kurtosis

82
 1.3 Measures of Central Tendency

 Central value refers to the location of the

centre of the distribution of data.

 Measure of central value:

The mean

The mode

The median

Percentile/Quantiles and

Midrange
83
The Mean:
 It is the most commonly used measures of
central value.
Types of Mean:
1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean
4. Quadratic Mean
5. Trimmed Mean
6. Weighted Mean
7. Combination mean

84
 1. Arithmetic Mean (simply Mean)

 The mean is defined as the arithmetic average

of all the values.

 It is represented by x read as x-bar for a

sample and by µ for a population

n N

∑x i ∑x i
x= i =1
µ= i =1
n N
85
 Cont’d
Advantages
• It is the most commonly used measure of location or
central tendency for continuous variables.
• The arithmetic mean uses all observations in the
data set.
• All observations are given equal weight.
Disadvantage
• The mean is affected by extreme values that may
not be representative of the sample.

86
 2. Geometric mean
• It is the nth root of the product of the
data elements.

• It is used in business to find average

rates of growth.

Geometric mean = ,

for all n >=2.

87
 Cont’d
• Example: suppose you have an IRA (Individual
Retirement Account) which earned annual interest
rates of 5%, 10%, and 25%.
Solution: The proper average would be the
geometric mean or the cube root of (1.05 • 1.10 •
1.25) or about 1.13 meaning 13%.
• Note that the data elements must be positive.
Negative growth is represented by positive values
less than 1. Thus, if one of the accounts lost 5%, the
proper multiplier would be 0.95.
88
 3. Harmonic mean
It is used to calculate average rates.

It is found by dividing the number of data

elements by the sum of the reciprocals of
each data element.

Harmonic mean =

89
• Example: Suppose a boy rode a bicycle three miles. Due
to the topography, for the first mile he rode 2 mph; for
the second mile 3 mph; for the final mile the average
speed was 4 mph. What was the average speed for the
three miles?

Solution:
To show simple analysis using Harmonic mean :

3 miles = 3 . = 36 = 2.77mph
1/2+1/3+1/4 13/12 13

90
 4. Quadratic mean

• It is another name for Root Mean Square or

RMS.

• RMS is typically used for data whose

arithmetic mean is zero.

Chapter 1- Introduction to statistics 91

 Cont’d
Example

• Suppose measurements of 120, -150, and 75

volts were obtained.

Solution: The corresponding quadratic mean is

√((120)2 + (-150)2 + (75)2)/3) or 119 volts
RMS.

• The quadratic mean gives a physical measure

of the average distance from zero.

92
 5. Trimmed mean
It is usually refers to the arithmetic mean
without the top 10% and bottom 10% of
the ordered scores.

Removes extreme scores on both the high

and low ends of the data

93
 6. Weighted mean
 It is the average of differently weighted
scores;

 It takes into account some measure of

weight attached to different scores.

94
 The Mode

The mode is the most frequent or most

typical value.

The mode will not always be the central

value; in fact it may sometimes be an
extreme value. Also, a sample may have more
than one mode Bimodal or Multimodal

95
 Cont’d

Example

‘23, 22, 12, 14, 22, 18, 20, 22, 18, 18’

The mode is 18 and 22 - bimodal

96
 Cont’d
Advantages
Requires no calculations.

Represents the value that occurs most often.

Disadvantage
The mode for continuous measurements is
dependent on the grouping of the intervals.

We may not have mode

97
 The Median

The median is the middle value of a group of an

odd number of observations when the data is
arranged in increasing or decreasing order
magnitude.

If the number of values is even, the median is

the average of the two middle values.

98
 Cont’d
Example

‘23, 22, 12, 14, 22, 18, 20, 22, 18, 18’

Array: 12, 14, 18, 18, 18, 20, 22, 22,22,23

• The median (18+20)/2 = 19

99
 Cont’d
Advantages
• The median always exists and is unique.

• The median is not affected by extreme values.

Disadvantages
• The values must be sorted in order of
magnitude.

• The median uses only one (or two) observations.

100
 Percentiles / Quartiles
 Percentiles are values that divide a distribution
into two groups where the Pth percentile is larger
than P% of the values.
 Some specific percentiles have special names:
First Quartile : Q1 = the 25 percentile
Median : Q2 = the 50 percentile
 A percentile provides information about how the
data are spread over the interval from the
smallest value to the largest value.
101
 Cont’d
Interpretation
• Q1 = a. This means that 25% of the data
values are smaller than a

• Q2 = b. This means that 50 % of the data has

values smaller than b.

• Q3 = c. This means that 75 % of the data has

values smaller than c.

102
 Midrange
The midrange is the average of largest and
smallest observation.

Midrange = (Largest +Smallest)/2

The percentile estimate (P25 + P75)/2 is

sometimes used when there are a large
number of observations.

103
104
 1.4 Measure of Dispersion (of Variability)

• So far we have looked at ways of summarising

data by showing some sort of average (central
tendency).

• But it is often useful to show how much these

figures differ from the average.

• This measure is called dispersion.

 Cont’d
• Measures of dispersion are descriptive statistics
that describe how similar a set of scores are to
each other.

The more similar the scores are to each other,

the lower the measure of dispersion will be.

The less similar the scores are to each other, the

higher the measure of dispersion will be

In general, the more spread out a distribution is,

the larger the measure of dispersion will be.
106
 Cont’d
• A measure of dispersion indicates how the
observations are spread about the central
value.

Measures of dispersion are:

 The range

 The variance

 The standard deviation and

 The coefficient of variation

107
 The range
The range is the difference between the
largest and the smallest value in the
sample.

The range is the easiest of all measures of

dispersion to calculate.

R = Maximum Value - Minimum Value.

108
 Cont’d
Advantage
• The range is easily understood and gives a
quick estimate of dispersion.
• The range is easy to calculate
Disadvantage
• The range is inefficient because it only uses
the extreme value and ignores all other
available data. The larger the sample size,
the more inefficient the range becomes.
109
 The Variance
The variance is the average of the squared
differences between each data value and the mean.
If the data set is a sample, the variance is denoted
by s2.
∑ ( xi − x )
2
s =
2
n −1
If the data set is a population, the variance is
denoted by σ 2
∑ ( xi − µ ) 2
σ =
2
N
 Cont’d

Advantages
• The variance is an efficient estimator

• Variances can be added and averaged

Disadvantage
• The calculation of the variance can be
tedious without the aid of a calculator or
computer.
111
 The Standard Deviation
• The standard deviation is one of the most important measures
of dispersion. It is much more accurate than the range or inter
quartile range.

• It takes into account all values and is not excessively affected

by extreme values.

• The square root of the variance is known as the standard

deviation.

• If the data set is a sample, the standard deviation is denoted s.

s= s 2

• If the data set is a population, the standard deviation is denoted

by σ σ= σ 2

112
 Cont’d
What does it measure?
• It measures the dispersion (or spread) of
figures around the mean.

• A large number for the standard deviation

means there is a wide spread of values around
the mean, whereas a small number for the
standard deviation implies that the values are
grouped close together around the mean.
 Example
• Find the standard deviation of the
minimum temperatures of 10 weather
stations in Ethiopia on a winters day.

The temperatures are:

5, 9, 3, 2, 7, 9, 8, 2, 2, 3 (˚Centigrade)
 Solution
x ẍ (x - ẍ) (x - ẍ)2
5 5 0 0
9 5 4 16
3 5 -2 4
2 5 -3 9
7 5 2 4
9 5 4 16
8 5 3 9
2 5 -3 9
2 5 -3 9
3 5 -2 4
∑x = 50 ∑(x - ẍ)2 = 80
ẍ = ∑x/n = 50/10 = 5 ∑(x - ẍ)2/N = 8
√∑(x - ẍ)2/N =2.8°C
x = temperature --- ẍ = mean temperature --- √ = square root
∑ = total of --- 2 = squared --- n = number of values
 The coefficient of variation
• The coefficient of variation indicates how large
the standard deviation is in relation to the mean.

• If the data set is a sample, the coefficient of

variation is computed as follows:
s
(100)
x
• If the data set is a population, the coefficient of
variation is computed as follows:
σ
(100)
µ
116
 Cont’d

• The higher the Coefficient of Variation

the more widely spread the values are
around the mean.

• The purpose of the Coefficient of

Variation is to let us compare the spread
of values between different data sets.
117
 The Inter-Quartile Range
• It is the difference between the 25th and the 75th
quartiles.

• The inter-quartile range is the range of the middle

half of the values.

• It is a better measurement to use than the range

because it only refers to the middle half of the
results.

• Basically, the extremes are omitted and cannot affect

the answer.
Interquartile range = Q3 – Q1 118
 Cont’d
• To calculate the inter-quartile range we must
first find the quartiles.

• There are three quartiles, called Q1, Q2 & Q3.

We do not need to worry about Q2 (this is just
the median).

• Q1 is simply the middle value of the bottom half

of the data and Q3 is the middle value of the top
half of the data.
 Example
• We calculate the inter quartile range by taking
Q1 away from Q3 (Q3 – Q1).

Remember data must be placed in order

10 – 25 – 45 – 47 – 49 – 51 – 52 – 52 – 54 – 56 – 57 – 58 – 60 – 62 – 66 – 68 – 70 - 90

Because there is an even number of values (18) we

can split them into two groups of 9.

Q1 Q3
IR = Q3 – Q1 , IR = 62 – 49. IR = 13
 Measure of skewness and kurtosis
Skewness (mean deviation)
• Skewness is a measure of the tendency of the deviations
to be larger in one direction than in the other.
• Skewness is the degree of asymmetry or departure
from symmetry of a distribution.
• If the frequency curve of a distribution has a longer tail
to the right of the central value than to the left, the
distribution is said to be skewed to the right or to
have positive skewness.
• If the reverse is true, it is said that the distribution is
skewed to the left or has negative skewness.
121
 Cont’d
The following formula can
( )
∑ X− X
3

be used to determine 3
= N
∑ (X− X )
s 2

skew:
N

 A bell-shaped distribution which has no skewness, i.e.,

mean = median = mode is called a normal distribution.
If mean > Median > mode, the distribution is positively
skewed distribution or it is said to be skewed to the
right. If mean < Median < mode, the distribution is
negatively skewed distribution or it is said to be
skewed to the left.
Chapter 1- Introduction to statistics 122
 Cont’d

• If s3 < 0, then the distribution has a negative skew

• If s3 > 0 then the distribution has a positive skew
• If s3 = 0 then the distribution is symmetrical
• The more different s3 is from 0, the greater the
skew in the distribution
123
 Measure of kurtosis

• Kurtosis characterises the relative peakedness or

flatness of a distribution compared with the normal
distribution. Positive kurtosis indicates a relatively
peaked distribution. Negative kurtosis indicates a
relatively flat distribution.
4
 
 
The measure of kurtosis  X−X 
∑ 
is given by:  ∑ (X − X ) 
2

 
s4 =  N 
N
124
 s2, s3, & s4
Collectively, the variance (s2), skew (s3), and
kurtosis (s4) describe the shape of the
distribution

125
Skewness and Kurtosis
End of Chapter 1