Forecasting

Methods UNIT 14 CORRELATION

Objectives
After completion of this unit, you should be able to :
• understand the meaning of correlation
• compute the correlation coefficient between two variables from sample
observations
• test for the significance of the correlation coefficient
• identify confidence limits for the population correlation coefficient from
the observed sample correlation coefficient
• compute the rank correlation coefficient when rankings rather than actual
values for variables are known
• appreciate some practical applications of correlation
• become aware of the concept of auto-correlation and its application in
time series analysis.
Structure
14.1 Introduction
14.2 The Correlation Coefficient
14.3 Testing for the Significance of the Correlation Coefficient
14.4 Rank Correlation
14.5 Practical Applications of Correlation
14.6 Auto-correlation and Time Series Analysis
14.7 Summary
14.8 Self-assessment Exercises
14.9 Key Words
14.10 Further Readings

14.1 INTRODUCTION
We often encounter situations where data appears as pairs of figures relating
to two variables. A correlation problem considers the joint variation of two
measurements neither of which is restricted by the experimenter. The
regression problem, which is treated in Unit 15, considers the frequency
distributions of one variable (called the dependent variable) when another
(independent variable) is held fixed at each of several levels.
Examples of correlation problems are found in the study of the relationship
between IQ and aggregate percentage marks obtained by a person in SSC
examination, blood pressure and metabolism or the relation between height

268
and weight of individuals. In these examples both variables are observed as Correlation

they naturally occur, since neither variable is fixed at predetermined levels.

Examples of regression problems can be found in the study of the yields of
crops grown with different amount of fertiliser, the length of life of certain
animals exposed to different amounts of radiation, the hardness of plastics
which are heat-treated for different periods of time, and so on. In these
problems the variation in one measurement is studied for particular levels of
the other variable selected by the experimenter. Thus the factors or
independent variables in regression analysis are not assumed to be random
variables, though the dependent variable is modelled as a random variable for
which intervals of given precision and confidence are often worked out. In
correlation analysis, all variables are assumed to be random variables. For
example, we may have figures on advertisement expenditure (X) and Sales
(Y) of a firm for the last ten years, as shown in Table I. When this data is
plotted on a graph as in Figure I we obtain a scatter diagram. A scatter
diagram gives two very useful types of information. First, we can observe
patterns between variables that indicate whether the variables are related.
Secondly, if the variables are related we can get an idea of what kind of
relationship (linear or non-linear) would describe the relationship.
Correlation examines the first
Table 1: Yearwise data on Advertisement Expenditure and Sales

Year Advertisement Sales in thousand Rs.

Expenditure in (Y)
Thousand Rs. (X)

1988 50 700
1987 50 650
1986 50 600
1985 40 500
1984 30 450
1983 20 400
1982 20 300
1981 15 250
1980 10 210
1979 5 200

question of determining whether an association exists between the two

variables, and if it does, to what extent. Regression examines the second
question of establishing an appropriate relation between the variables.

269
Forecasting Figure I: Scatter Diagram
Methods

The scatter diagram may exhibit different kinds of patterns. Some typical
patterns indicating different correlations between two variables are shown in
Figure II.
What we shall study next is a precise and quantitative measure of the degree
of association between two variables and the correlation coefficient.

14.2 THE CORRELATION COEFFICIENT

Definition and Interpretation
The correlation coefficient measures the degree of association between two
variables X and Y. Pearson's formula for correlation coefficient is given as
�
�(����)(����)
�=� �� ��
(14.1)

Where r is the correlation coefficient between X and Y, σx and σy are the

standard deviations of X and Y respectively and n is the number of values of
�
the pair of variable X and Y in the given data. The expression � Σ(X −
�)(Y − �
(X Y) is known as the covariance between X and Y. Here r is also
called the Pearson's product moment correlation coefficient. You should note
that r is a dimensionless number whose numerical value lies between +1 and
-1. Positive values of r indicate positive (or direct) correlation between the
two variables X and Y i.e. as X increases Y will also increase or as X
decreases Y will also decrease. Negative values of r indicate negative (or
inverse) correlation, thereby meaning that an increase in one variable results
in a decrease in the value of the other variable. A zero correlation means that
there is no association between the two variables. Figure II shows a number
of scatter plots with corresponding values for the correlation coefficient r.

270
 Figure II: Different Types of Association Between Variables Correlation

The following form for carrying out computations of the correlation

coefficient is perhaps more convenient
���
�= (14.2)
√�� � ��� �

where
� = � − �� = deviation of a particular X value from the mean ��
� = � − �� = deviation of a particular Y value from the mean ��
Equation (14.2) can be derived from equation (14.1) by substituting for ��
and �� as follows:

� �
σ� = �� Σ(X − X̄)� and �� = �� Σ(X − Ȳ)� (14.3)

Activity A
Suggest five pairs of variables which you expect to be positively correlated.
…………………………………………………………………………………
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Activity B
Suggest five pairs of variables which you expect to be negatively correlated.
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Forecasting A Sample Calculation: Taking as an illustration t he data of advertisement
Methods
expenditure (X) and Sales (Y) of a company for the 10-year period shown in
Table 1, we proceed to determine the correlation coefficient between these
variables:
Computations are conveniently carried out as shown in Table 2.
Table 2: Calculation of Correlation Coefficient

Sl.No X Y � = x − x� � =�−� x� �� xy
.

1. 50 700 21 274 441 75,076 5,7

2. 50 650 21 224 441 50,176 4,7
3. 50 600 21 174 441 30,276 3,6
4. 40 500 11 74 121 5,476 814
5. 30 450 1 24 1 576 24
6. 20 400 -9 -26 81 676 234
7. 20 300 -9 -126 81 15,876 1,134
8. 15 250 -14 -176 196 30,976 2,464
9. 10 210 -19 -216 361 46,656 4,104
10. 5 200 -24 -226 576 51,076 5,424

Total 290 4,260 0 0 2,740 3,06,840 28,310

290
�� =
= 29
10
4260
�=
Y = 426
10
�� 28310
∴�= = = 0.976
√Σ� � �Σ� � √2740 × 306840
This value of r (= 0.976) indicates a high degree of association between the
variables X and Y. For this particular problem, it indicates that an increase in
advertisement expenditure is likely to yield higher sales.
You may have noticed that in carrying out calculations for the correlation
coefficient in Table 2, large values for � � and � � resulted in a great
computational burden. Simplification in computations can be adopted by
calculating the deviations of the observations from an assumed average rather
than the, actual average, and also scaling these deviations conveniently. To
illustrate this short cut procedure, let us compute the correlation coefficient
for the same data. We shall take U to be the deviation of X values from the
assumed mean of 30 divided by 5. Similarly, V represents the deviation of Y
values from the assumed mean of 400 divided by 10.
The computations are shown in Table 3.

272
 Table 3: Short cut Procedure for Calculation of Correlation Coefficient Correlation

S.No X Y U V UV �� ��
1. 50 700 4 30 120 16 900
2. 50 650 4 25 100 16 625
3 50 600 4 20 80 16 400
4. 40 500 2 10 20 4 100
5. 30 450 0 5 0 0 25
6. 20 400 -2 0 0 4 0
7 20 300 -2 -10 20 4 100
8. 15 250 -3 -15 45 9 225
9. 10 210 -4 -19 76 16 361
10. 5 200 -5 -20 100 25 400
Total -2 26 561 110 3,13
����
Σ�� − �
�=
(∑�)� (��)�
�Σ� � − �Σ� � −
� �

(��)(��)
561 − ��
�=
(��)� (��)�
�110 − �3136 −
�� ��

566.2
=
10.47 × 55.39
= 0.976
We thus obtain the same result as before.
Activity C
Use the short cut procedure to obtain the value of correlation coefficient in
the above example using scaling factor 10 and 100 for X and Y respectively.
(That is, the deviation from the assumed mean is to be divided by 10 for X
values and by 100 for Y values.)
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14.3 TESTING FOR THE SIGNIFICANCE OF

THE CORRELATION COEFFICIENT
Once the correlation coefficient has been calculated from sample data one is
normally interested in asking the question: Is there an association between 273
Forecasting the variables? Or with what confidence can we make a statement about the
Methods association between the variables?
Such questions are best answered statistically by using one of the following
two commonly used procedures :
1) Providing confidence limits for the population correlation coefficient
from the sample size n and the sample correlation coefficient r. If this
confidence interval includes the value zero, then we say that r is not
significant, implying thereby that the population correlation coefficient
may be zero and the value of r may be due to sampling variability.
2) Testing the null hypothesis that population correlation coefficient equals
zero vs. the alternative hypothesis that it does not, by using the t-statistic.
The use of both these procedures is now illustrated.
The value of the sample correlation coefficient is used as an estimate of the
true population correlation p. It is desirable to incude a confidence interval
for the true value along with the sample statistics. There are several methods
for obtaining the confidence interval for p. However, the most straight
forward method is to use a chart such as that shown in Figure III.
Figure III: Confidence Bands for the Population Correlation

Scale of r ( Sample correlation cofficient)

Once r has been calculated, the chart can be used to determine the upper and
lower values of the interval for the sample size used. In this chart the range of
unknown values of p is shown in the vertical scale; while the sample r values
are shown on the horizontal axis, with a number of curves for selected sample
sizes. Notice that for every sample size there are two curves. To read the 95%
confidence limits for an observed sample correlation coefficient of 0.8 for a
sample of size 10, we simply look along the horizontal line for a value of 0.8
(the sample correlation coefficient) and construct a vertical line from there till
it intersects the first curve for n =10. This happens for p = 0.2. This is the
lower limit of the confidence interval. Extending the vertical line upwards, it
again intersects the second n =10 line at p = 0.92, which represents the upper
274
confidence limit. Thus the 95% confidence interval for the population Correlation

correlation coefficient becomes

0.2 ≤ � ≤ 0.92
If a confidence interval for p includes the value zero, then r is not considered
significant since that value of r may be due to nothing more than sampling
variability.
This method of using charts to determine the confidence intervals is
convenient, though of course we must use a different chart for different
confidence limits (e.g. 90%, 95%, 99%).
The alternative approach for testing the significance of r is to use the formula
�
�= (14.4)
�(��� � )(���)

Referring to the table of t-distribution for (n-2) degrees of freedom, we can

find the critical value for t at any desired level of significance (5% level of
significance is commonly used). If the calculated value of t (as obtained by
equation 14.3) is less than or equal to the table value, we accept the
hypothesis (H� : the correlation coefficient equals zero), meaning that the
correlation between the variables is not significantly different from zero:
Suppose we obtain a correlation coefficient of 0.2 for a sample of size 10.
0.2
�= ≅ 0.577
�(1 − 0.04)/8
And from the t-distribution with 8 degrees of freedom for a 5% level of
significance, the table value = 2.306. Thus we conclude that this r of 0.2 for n
= 10 is not significantly different from zero.
It should be mentioned here that in case the same value of the correlation
coefficient of 0.2 was obtained on a sample of size 100 then
0.2
t= ≅ 2021
�(1 − 0.04)/98
And the tabled value for a t-distribution with 98 degrees of freedom and a 5%
level of significance = 1.99. Since the calculated t exceeds this figure of 1.99,
we can conclude that this correlation coefficient of 0.2 on a sample of size
100 could be considered significantly different from zero, or alternatively that
there is statistically significant association between the variables.

14.4 RANK CORRELATION

Quite often data is available in the form of some ranking for different
variables. It is common to resort to rankings on a preferential basis in areas
such as food testing, competitive events (e.g. games, fashion shows, or
beauty contests) and attitudinal surveys. The primary purpose of computing a
correlation coefficient in such situations is to determine the extent to which
the two sets of rankings are in agreement. The coefficient that is determined
from these ranks is known as Spearman's rank correlation coefficient, r. 275
Forecasting This is given by the following formula
Methods
� ∑� �
��� ��
�� = 1 − �(�� ��)
(14.5)

Here n is the number of pairs of observations and �� is the difference in ranks

for the ith observation set.
Suppose the ranks obtained by a set of ten students in a Mathematics test
(variable X) and a Physics test (variable Y) are as shown below :

Rank for 1 2 3 4 5 6 7 8 9 10
variable X
Rank for 3 1 4 2 6 9 8 10 5 7
variable Y

To determine the rank correlation, �� we can organise computations as shown

in Table 4 :
Table 4: Determination of Spearman's Rank Correlation

Individual Rank in Rank in d =Y -X ��

Maths(X) Physics(Y)
1 1 3 +2 4
2 2 1 -I 1
3 3 4 +1 1
4 4 2 -2 4
5 5 6 +1 1
6 6 9 +3 9
7 7 8 +1 1
8 8 10 +2 4
9 9 5 -4 16
10 10 7 -3 9
Total 50

Using the formula (14.5) we obtain

6 × 50
�� = 1 − = 1 − 0.303 = 0.697
10(100 − 1)
We can thus say that there is a high degree of correlation between the
performance in Mathematics and Physics.
We can also test the significance of the value obtained. The null hypothesis is
that the two variables are not associated, i.e. r = O. That is, we are interested
to test the null hypothesis, H� that the two variables are not associated in the
population and that the observed value of �� differs from zero only by chance.
The t-statistic that is used to test this is

276
 Correlation
�−2
� = �� �
1 − ���

10 − 2
= 0.697�
1 − (0.697)�

=2.75
Referring to the table of the t-distribution for n-2 = 8 degrees of freedom, the
critical value for t at a 5% level of significance is 2.306. Since the calculated
value of t is higher than the table value, we reject the null hypothesis
concluding that the performances in Mathematics and Physics are closely
associated.
When two or more items have the same rank, a correction has to be applied to
∑d�� . For example, if the ranks of X are 1, 2, 3, 3, 5, ... showing that there are
�
two items with the same 3rd rank, then instead of writing 3, we write 3 � for
each so that the sum of these items is 7 and the mean of the ranks is
unaffected. But in such cases the standard deviation is affected, and therefore,
a correction is required. For this, ∑d�� is increased by (� � − �)/12 for each
tie, where t is the number of items in each tie.
Activity D
Suppose the ranks in Table 4 were tied as follows: Individuals 3 and 4 both
ranked 3rd in Maths and individuals 6, 7 and 8 ranked 8th in Physics.
Assuming that other rankings remain unaltered, compute the value of
Spearman's rank correlation.
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14.5 PRACTICAL APPLICATIONS OF

CORRELATION
The primary purpose of correlation is to establish an association between any
two random variables. The presence of association does not imply causation,
but the existence of causation certainly implies association. Statistical
evidence can only establish the presence or absence of association between
variables. Whether causation exists or not depends merely on reasoning. For
example, there is reason to believe that higher income causes higher
expenditure on superior quality cloth. However, one must be on the guard
against spurious or nonsense correlation that may be observed between
totally unrelated variables purely by chance.

277
Forecasting Correlation analysis is used as a starting point for selecting useful
Methods independent variables for regression analysis. For instance a construction
company could identify factors like
• population
• construction employment
• building permits issued last year which it feels would affect its sales for
the current year.
These and other factors that may be identified could be checked for mutual
correlation by computing the correlation coefficient of each pair of variables
from the given historical data (this kind of analysis is easily done by using an
appropriate routine on a computer). Only variables having a high correlation
with the yearly sales could be singled out for inclusion in a regression model.
Correlation is also used in factor analysis wherein attempts are made to
resolve a large set of measured variables in terms of relatively few new
Categories, known as factors. The results could be useful in the following
three ways :
1) to reveal the underlying or latent factors that determine the relationship
between the observed data,
2) to make evident relationships between data that had been obscured before
such analysis, and
3) to provide a classification scheme when data scored on various rating
scales have to be grouped together.
Another major application of correlation is in forecasting with the help of
time series models. In using past data (which is often a time series of the
variable of interest available at equal time intervals) one has to identify the
trend, seasonality and random pattern in the data before an appropriate
forecasting model can be built. The notion of auto-correlation and plots of
auto-correlation for various time lags help one to identify the nature of the
underlying process. Details of time series analysis are discussed in Unit 20.
However, some fundamental concepts of auto-correlation and its use for time
series analysis-are outlined below.

14.6 AUTO-CORRELATION AND-TIME SERIES

ANALYSIS
The concept of auto-correlation is similar to that of correlation but applies to
values of the same variable at different time lags. Figure IV shows how a
single variable such as income (X) can be used to construct another variable
(X1) whose only difference from the first is that its values are lagging by one
time period. Then, X and X1 can be treated as two variables and their
correlation found. Such a correlation is referred to as auto-correlation and
shows how a variable relates to itself for a specified time lag. Similarly, one
can construct X2 and find its correlation with X. This correlation will indicate
how values of the same variable that are two periods apart relate to each
278 other.
Figure IV: Example of the Same Variable with Different Time Lags Correlation

Time X Original X1One time lag X2 Two time

variable variable lags variable
constructed from constructed from
X X
t=1 13
2 8 8
3 15 15 15
4 4 4 4
5 4 4 4
6 12 12 12
7 11 11 11
8 7 7 7
9 14 14 14
10 12 12 12

One could construct from one variable another time-lagged variable which is
twelve periods removed. If the data consists of monthly figures, a twelve-
month time lag will show how values of 'the same month but of different
years correlate with each other. If the auto-correlation coefficient is positive,
it implies that there is a seasonal pattern of twelve months duration. On the
other hand, a near zero auto-correlation indicates the absence of a seasonal
pattern. Similarly, if there is a trend in the data, values next to each other will
relate, in the sense that if one increases, the other too will tend to increase in
order to maintain the trend. Finally, in case of completely random data, all
auto-correlations will tend to zero (or not significantly different from zero).
The formula for the auto correlation coefficient at time lag k is:
∑��� � �
��� (X � − X)(X ��� − X)
r� =
∑���� (X� − �
X)�
where
�� denotes the auto-correlation coefficient for time lag k k denotes the length
of the time lag n is the number of observations
X, is the value of the variable at time t and
X is the mean of all the data
Using the data of Figure IV the calculations can be illustrated.
13 + 8 + 15 + ⋯ + 12 100
�� = = = 10
10 10
(13 − 10)(8 − 10) + (8 − 10)(15 − 10) + ⋯ + (14 − 10)(12 − 10)
�� =
(13 − 10)� + (8 − 10)� + ⋯ + (14 − 10)� + (12 − 10)�
−27
= = −0.188
144
279
Forecasting For k = 2, the calculation is as follows :
Methods
∑��
��� (X � − 10)(X ��� − 10)
r� =
∑��
��� (X � − 10)
�

(13 − 10)(15 − 10) + (8 − 10)(4 − 10) + ⋯ + (7 − 10)(12 − 10)

=
(13 − 10)� + (8 − 10)� + ⋯ + (14 − 10)� + (12 − 10)�
−29
= = −.201
144
A plot of the auto-correlations for various lags is often made to identify the
nature of the underlying time series. We, however, reserve the detailed
discussion on such plots and their use for time series analysis for Unit 16.

14.7 SUMMARY
In this unit the concept of correlation or the association between two
variables has been discussed. A scatter plot of the variables may suggest that
the two variables are related but the value of the Pearson correlation
coefficient r quantifies this association. The correlation coefficient r may
assume values between -1 and 1. The sign indicates whether the association
is direct (+ve) or inverse (-ve). A numerical value of r equal to unity indicates
perfect association while a value of zero indicates no association.
Tests for significance of the correlation coefficient have been described.
Spearman's rank correlation for data with ranks is outlined. Applications of
correlation in identifying relevant variables for regression, factor analysis and
in forecasting using time series have been highlighted. Finally the concept of
auto-correlation is defined and illustrated for use in time series analysis.

14.8 SELF-ASSESSMENT EXERCISES

1) What do you understand by the term correlation? Explain how the study
of correlation helps in forecasting demand of a product.
2) A company wants to study the relation between R&D expenditure (X)
and annual profit (Y). The following table presents the information for
the last eight years:
Year R&D Expense (X) Annual Profit (Y)
(Rs. in thousands)
1988 9 45
1987 7 42
1986 5 41
1985 10 60
1984 4 30
1983 5 34
1982 3 25
1981 20

280 a) Plot the data on a scatter diagram.

b) Estimate the sample correlation coefficient. Correlation

c) What are the 95% confidence limits for the population correlation
coefficient?
d) Test the significance of the correlation coefficient using a t-test at a
significance level of 5%.
3) The following data pertains to length of service (in years) and. the annual
income for a sample of ten employees of an industry:
Length of service in years (X) Annual income in thousand
rupees (Y)
6 14
8 17
9 15
10 18
11 16
12 22
14 26
16 25
18 30
20 34
Compute the correlation coefficient between X and Y and test its
significance at levels of 0.01 and 0.05.
4) Twelve salesmen are ranked for efficiency and the length of service as
below:
Salesman Efficiency (X) Length of Service (Y)
A 1 2
B 2 1
C 3 5
D 5 3
E 5 9
F 5 7
G 7 7
H 8 6
I 9 4
J 10 11
K 11 10
L 12 11

a) Find the value of Spearman's rank correlation coefficient, ��

b) Test for the Significance of ��
281
Forecasting 5) An alternative definition of the correlation coefficient between a two-
Methods dimensional random variable (X, Y) is
[(� − �(�))(� − �(�))]
�=
��(�)�(�)
where E(.) represents expectation and V(.) the variance of the random
variable. Show that the above expression can be simplified as follows :
�(��) − �(�)�(�)
�=
��(�)�(�)
(Notice here that the numerator is called the covariance of X and Y).
6) In studying the relationship between the index of industrial production
and index of security prices the following data from the Economic Survey
1980-81 (Government of India Publication) was collected.
70-71 71-72 72-73 73-74 74-75 75-76 76-77 77-78 78-79
Index of 101.3 114.8 1196.6 122.1 125.2 122.2 135.3 140.1 150.1
Industrial
(1970-
100)
Index of 100.0 95.1 96.7 116.0 113.2 96.9 102.9 107.4 130.4
Security
Prices
(1970-
71-100)

a) Find the correlation between the two indices.

b) Test the significance of correlation coefficient at 0.01 level of
significance.
7) Compute and plot the first five auto-correlations (i.e. up-to time lag 5
periods) for the time series given below :
t I 2 3 4 5 6 7 8 9 10
dt 13 8 15 4 4 12 II 7 14 12

14.9 KEY WORDS

Auto-correlation: Similar to correlation in that it described the association
or mutual dependence between values of the same variable but at different
time periods. Auto-correlation coefficients provide important information
about the structure of a data set.
Correlation: Degree of association between two variables.
Correlation Coefficient : A number lying between -1 (Perfect negative
correlation) and + 1 (perfect positive correlation) to quantify the association
between two variables.
Covariance: This is the joint variation between the variables X and Y.
Mathematically defined as
282
 ∑(�� − ��)��� − ��� Correlation

�
for n data points.
Scatter Diagram: An ungrouped plot of two variables, on the X and Y axes.
Time Lag: The length between two time periods, generally used in time
series where one may test, for instance, how values of periods 1, 2; 3, 4
correlate with values of periods 4, 5, 6, 7 (time lag 3 periods).
Time-Series: Set of observations at equal time intervals which may form the
basis of future forecasting.

14.10 FURTHER READINGS

Box, G.E.P., and G.M. Jenkins. Time Series Analysis, Forecasting and
Control, Holden-Day: San Francisco.
Chatterjee, S., & Simonoff, J.S. Handbook of regression analysis (Vol.5).
John Wiley & Sons.
Draper, N. and H. Smith. Applied Regression Analysis, John Wiley: New
York.
Edwards, B., The Readable Maths and Statistics Book, George Allen and
Unwin: London.
Makridakis, S. and S. Wheelwright. Interactive Forecasting: Univariate and
Multivariate Methods, Holden-Day: San Francisco.
Peters, W.S. and G.W: Summers. Statistical Analysis for Business Decisions,
Prentice Hall: Englewood-Cliffs.
Srivastava, U.K., G.V. Shenoy and S.C. Sharma. Quantitative Techniques for
Managerial Decision Making,Wiley Eastern: New Delhi.
Stevenson, W.J. Business Statistics-Concepts and Applications, Harper and
Row: New York.

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