= r xy v v is known as the regression coefficient of Y on X , x = X g and y = ( Y – Y ) Note: Instead of finding out the values of correlation coefficient σ x , σ y , etc, we can find the value of regression coefficient by calculating ∑ xy and ∑ y . Case : Deviations taken from Assumed Mean When actual means of X and Y variables are in fractions the calculations can be simplified by taking the deviations from the assumed means. The regression equations and their coefficients are written as follows (i) Regression Equation of Y on X: Y Y = b yx b yx = N dx N dxdy dy - ] ^ g (ii) Regression Equation of X on Y: = b xy Y Y b xy = N dy dy N dxdy dy - ] ^ g where dx = X –A, dy = Y – B , A and B are assumed means or arbitrar y values are taken from X and Y respectively. Properties of Regression Coefficients (i) Correlation Coefficient is the geometric mean between the regression coefficients r = !
xy yx (ii) If one of the regression coefficients is greater than unity, the other must be less than unity. (iii) Both the regression coefficients are of same sign. Example . Calculate the regression coefficient and obtain the lines of regression for the following data Y Y X Y X Y ∑ X = ∑ Y = ∑ X = ∑ Y = ∑ XY = Table .
N = , Y N Y = Regression coefficient of X on Y: b xy = N Y Y N XY Y - ] ]g = = ` b xy = . (i) Regression equation of X on Y