Page 300 - DMTH404_STATISTICS
P. 300
Statistics
Notes
On the basis of the above, we can consider X X Y Y as an absolute measure of
i
i
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 X Y Y . This term is called covariance in statistics and is denoted as Cov(X,Y).
n i i
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)
Y
X
1
X X Y Y
i
i
or r n .... (2)
XY
1 2 1 2
X X Y Y
n i n i
1
Cancelling from the numerator and the denominator, we get
n
X X Y Y
r i i .... (3)
XY 2 2
X X Y Y
i
i
Consider X X Y Y X X Y Y X X
i
i
i
i
i
X Y X Y i (second term is zero)
i
i
X Y nXY Y nY
i
i
i
2 2 2
Similarly we can write X X X nX
i
i
2 2 2
and Y Y Y nY
i
i
Substituting these values in equation (3), we have
X Y nXY
i
i
r XY .... (4)
2
2
2
2
X nX Y nY
i
i
X i Y
X Y i
n
i
i
r XY n n
X 2 Y 2
2
2
X i Y i
n
n
i
i
n n
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