Assumed Means Deviations in F2 Test : When actual means of X and Y variables are in fractions the calculations can be simplified by taking the deviations from assumed means. When deviations are taken from assumed means the entire procedure of finding regression equations remains the same - the only difference is that instead of taking deviations from actual means, we take the deviations from assumed means. The two regression equations are:
X - X = r σx / σy ( y-y)
The value of σx/ σy will now be obtained as follows
r σx/ σy = N Σ dx dy - (Σ dx x Σ dy)/N Σ d2y - (Σ dy)2
dx = (X - A) and dy = (Y - A)
Similarly the regression equation of y on x is
y - y = r σy/ σx (X - X)
r σy / σx = N Σ dx dy - Σ dx Σ dy / N Σ d x2 - (Σ dx)2
It should be noted that in both the cases the numerator is the same the only difference is in the denominator when the regression coefficients are calculated from correlation table their values are obtained as follows.
r σx/ σy = N Σ f dx dy - Σ f dx Σ dy /N Σ fd2y - (Σ f dy)2 x ix / iy
Ix= class interval of X variable and
Iy = class interval of y variable
Similarly r σy/ σx = N Σ fd2x - (Σ f dx)2 X iy/ix
As is clear from above the formulae for calculating regression coefficients in a correlation table are the same - the only difference is that in a correlation table we are given frequencies also and hence we have multiplied every value by f.
Illustration: From the data of illustration calculate regression equation by taking deviations of X series from 5 and of y series form 7.
| X |
(X - 5) dx
|
d2x |
Y |
(Y-7)dy |
d2y |
dx dy |
| 6 |
+1 |
1 |
9 |
+2 |
4 |
+2 |
| 2 |
-3 |
9 |
11 |
+4 |
16 |
-12 |
| 10 |
+5 |
25 |
25 |
-2 |
4 |
-10 |
| 4 |
-1 |
1 |
8 |
1 |
-1 |
-1 |
| 8 |
+3 |
9 |
7 |
0 |
0 |
0 |
| ΣX=30 |
Σ dx=+5 |
Σ d2x=45 |
Σy =40 |
Σ dy = 5 |
Σ d2y = 25 |
Σ dxdy= - 21 |
Regression equation of X on Y; X-X = r σx /σy (Y-Y)
X = ΣX/N = 30/5 6; y = Σ y/N = 40/5 = 8
r σx/σy = N Σ dxdy - Σ dx Σ dy/N Σ d2y - (Σdy)2
= [5 (- 21) - (5)(5)]/(5)(25) - (5)2 = -105 - 25/(- 125 - 25) = - 130 / 100 = - 1.3
X - 6 = - 1.3 (Y - 8)
X - 6 = - 1.3 Y + 10.4 or X = 16.4 - 1.3 y
Regression equation of Y on X ; Y - Y = r σy/ σx = N Σ dx dy - Σ dx Σ dy/N Σ d2x - (Σ dx)2
= 5 ( - 21) - (5) (5) / (5) (45) - (5)2 = - 105 - 25 / 200 = - 0.65
Y - 8 = - 0.65 (X - 6)
Y - 8 = - 0.65 X + 3.9 or Y = 11.9 - 0.65 X >
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