Σ(X-x bar)2 = SX2-1/n(ΣX)2
Σ(Y-y bar)2 = ΣY2-1/n(ΣY)2 }--(2)
Σ(X-x bar)(Y-y bar)=ΣXY-1/n(ΣX)(ΣY)
The last equation in (2) is a new form, but its affinity with the other two is evident. We can now extend the array in (1) by adding first a row of “correction factors”. Subtracting each of these from the number above then gives the required expressions on the left of (2). Thus we have the additional rows
1/n(ΣX)2, 1/n (ΣY)2, 1/n (ΣX)(ΣY) }--(3)
Σ(X-x bar)2, Σ(Y-y bar)2, Σ(X-x bar)(Y-y bar)
where the quantities in the first line of (3) are subtracted from the corresponding quantities in the last line of (1) to give second line of (3).
Dividing through the second line of (3) by n—1 gives the two estimated variances and estimated covariance: sX2, sX2, C---(4)
Now let us suppose that the equation of the true regression line is given by the expression
Y = α + βX ---(5)
Where, for any given X,Y is equal to the corresponding true Y-mean µx. In (5) the symbol β is, in fact, the true regression coefficient of Y and X. It is estimated from the sample by b, which is given by either of the two expressions: |
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