If Sx = Sy = 12, Sx² = Sy² = 45, Sxy = 41 and n = 5, then what is the value of…

Concept

The regression coefficient of y on x measures how y changes per unit change in x. From raw sums it is given by the identity byx = [ n·Σxy − (Σx)(Σy) ] / [ n·Σx2 − (Σx)2 ]. The denominator uses the spread of the independent variable x (its sum of squares), while the numerator is the cross-product term corrected for the means.

Application

Given n = 5, Σx = Σy = 12, Σx2 = Σy2 = 45 and Σxy = 41, substitute step by step:

1. Numerator = n·Σxy − (Σx)(Σy) = 5·41 − 12·12 = 205 − 144 = 61.

2. Denominator = n·Σx2 − (Σx)2 = 5·45 − 122 = 225 − 144 = 81.

3. Therefore byx = 61 / 81.

Cross-check / Contrast

• Because here Σx2 = Σy2 and Σx = Σy, the denominator is the same whichever variable you treat as independent, so bxy shares the same denominator 81 and would put 81 underneath — never on top.

• Swapping numerator and denominator gives the reciprocal 81/61, which is the form of bxy only if the cross-product were divided by the wrong spread — not the value asked for here.

• Re-evaluating 205 − 144 confirms the numerator is exactly 61, not 71; an off-by-ten slip there is the usual source of the other numerators.