Correlation and Regression
Correlation and regression describe whether two business variables move together and how one can be predicted from the other. The exam tests scatter diagrams, Karl Pearson’s coefficient, rank correlation, the two regression lines and the interpretation of slope and r².
Core rules
- Karl Pearson’s r = [nΣxy − ΣxΣy] / √{[nΣx² − (Σx)²][nΣy² − (Σy)²]}. It lies between −1 and 1 and is unit-free.
- r > 0 means a positive linear association; r < 0 negative; r = 0 means no linear association, not necessarily no relationship.
- Regression line of y on x: y − ȳ = b_yx(x − x̄), where b_yx = [nΣxy − ΣxΣy]/[nΣx² − (Σx)²].
- The line of x on y is different; the two lines intersect at (x̄, ȳ). Also r² = b_yx × b_xy when both are computed from the same data.
- Rank correlation uses ranks instead of values and is useful for ordinal business ratings.
Worked example
For n = 5, suppose Σx = 15, Σy = 25, Σxy = 88, Σx² = 55 and Σy² = 145. Then x̄ = 3 and ȳ = 5. The numerator is 5(88) − (15)(25) = 440 − 375 = 65. The first bracket is 5(55) − 225 = 50 and the second is 5(145) − 625 = 100. So r = 65/√(50 × 100) = 65/70.71 = 0.919, a strong positive linear association. The slope of y on x is b_yx = 65/50 = 1.3, so the line is y − 5 = 1.3(x − 3), i.e. y = 1.3x + 1.1. At x = 4 the predicted y is 6.3.
Traps
- Reading r = 0 as proof of no relationship at all.
- Mixing the regression of y on x with x on y; they have different slopes.
- Using a regression line far outside the observed x-range (extrapolation).
- Treating a high correlation as proof of causation.