Correlation and Regression
Correlation measures the strength and direction of association between two variables; regression expresses that association as an equation for prediction. The examiner tests Karl Pearson's coefficient, Spearman's rank correlation with tie correction, the two regression lines and their coefficients, and the interpretation of r.
Core rules
- Karl Pearson r = Σxy/(n σx σy), using deviations from the respective means; r lies between −1 and +1.
- Spearman R = 1 − 6Σd²/[n(n² − 1)]; for repeated ranks use d² corrected by (t³ − t)/12 per tie group.
- Regression of y on x: y − ȳ = byx(x − x̄), with byx = r σy/σx.
- Regression of x on y: x − x̄ = bxy(y − ȳ), with bxy = r σx/σy.
- byx × bxy = r², and both lines pass through (x̄, ȳ).
- r is independent of change of origin and scale, but not of interchange of variables.
Worked example
Given σx = 3, σy = 5 and r = 0.8: byx = 0.8 × (5/3) = 1.333 and bxy = 0.8 × (3/5) = 0.48. Check: 1.333 × 0.48 = 0.64 = r², as required. If x̄ = 10 and ȳ = 20, the line of y on x is y − 20 = 1.333(x − 10), so y = 1.333x + 6.67.
Traps
Quoting byx when the question asks for the regression of x on y; claiming r = 0 means no relationship when it only rules out a linear one; and treating a high correlation as proof of causation.