Page 176 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 8: Correlation Analysis

Notes
sum of positive products and hence X i X Y i Y will be negative for all the n observations.

Further, if there is no correlation, the sum of positive products of deviations will be equal to the
sum of negative products of deviations such that X i X Y i Y will be equal to zero.

On the basis of the above, we can consider X i X Y i Y as an absolute measure of
correlation. This measure, like other absolute measures of dispersion, skewness, etc., will depend
upon (i) the number of observations and (ii) the units of measurements of the variables.
In order to avoid its dependence on the number of observations, we take its average, i.e.,
1
X i X Y i Y . This term is called covariance in statistics and is denoted as Cov(X,Y).
n
To eliminate the effect of units of measurement of the variables, the covariance term is divided
by the product of the standard deviation of X and the standard deviation of Y. The resulting
expression is known as the Karl Pearson's coefficient of linear correlation or the product moment
correlation coefficient or simply the coefficient of correlation, between X and Y.

Cov X ,Y
r XY .... (1)
X Y
1
X i X Y i Y
n
or r XY 1 2 1 2 .... (2)
X i X Y i Y
n n
1
Cancelling from the numerator and the denominator, we get
n
X i X Y i Y
r XY 2 2 .... (3)
X i X Y i Y

Consider X i X Y i Y X i X Y i Y X i X

X Y X Y i (second term is zero)
i i

X Y nX Y Y i nY
i i
2 2 2
Similarly we can write X i X X i nX
2 2 2
and Y i Y Y i nY
Substituting these values in equation (3), we have

X Y nXY
i i
r XY
X 2 nX 2 Y 2 nY 2 .... (4)
i i

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