Show that the coefficient of correlation is the geometric mean of the two regression
coefficients.
Answers
Given : coefficient of correlation is the geometric mean of the two regression coefficients.
To find : Prove
Solution:
two regression coefficients a and b of the two regression lines can be stated as follows:
a = Sxy / (Sx)²
b = Sxy / (Sy)²
r = coefficient of correlation
r = Sxy / (Sx . Sy)
coefficient of correlation is the geometric mean of the two regression
coefficients.
if r² = ab
LHS = r² = ( Sxy / (Sx . Sy) )² = (Sxy)²/ ( Sx² . Sy²)
= ( Sxy . Sxy )/ ( Sx² . Sy²)
= ( Sxy / Sx² ) . ( Sxy / Sy² )
= a b
= RHS
QED
Hence proved that
coefficient of correlation is the geometric mean of the two regression
coefficients.
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