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Unit 23: Regression Analysis

Further when r = 0, equation (12) becomes Y =Y and equation (24) becomes X = X . These Notes
Ci Ci
are the equations of lines parallel to X-axis and Y-axis respectively. These lines also intersect at
d
the point X ,Yi and are mutually perpendicular at this point, as shown in figure 23.4.
Figure 23.4

23.1.3 Correlation Coefficient and the two Regression Coefficients

s Y s X
r
r
Since b = × and d = × , we have
s X s Y
s Y s X 2
.
b d = r r × = r or r = b.d . This shows that correlation coefficient is the geometric
s X s Y
mean of the two regression coefficients.
Remarks:

The following points should be kept in mind about the coefficient of correlation and the regression
coefficients :

Cov X Y) Cov (X Y, ) Cov (X Y, )
(
,
(i) Since r = , b = 2 and d = 2 , therefore the sign of r, b and
s s Y s X s Y
X
d will always be same and this will depend upon the sign of Cov (X, Y).
2
2
(ii) Since bd = r and 0 £ r £ 1, therefore either both b and d are less than unity or if one of them
is greater than unity, the other must be less than unity such that 0 £ b.d £ 1 is always true.
Example 1:
Obtain the two regression equations and find correlation coefficient between X and Y from the
following data :

X : 10 9 7 8 11
Y : 6 3 2 4 5

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