Correlation and Regression
Correlation measures the strength and direction of a linear association; regression expresses that association as an equation for prediction. DHSE questions typically supply a small paired data set or give regression coefficients and ask for r, the means, or a predicted value.
Correlation
- Karl Pearson coefficient r = Σ(x − x̄)(y − ȳ) / (n σx σy); the computing form uses Σdxdy, Σdx², Σdy² with dx = x − A.
- r lies in [−1, 1], is unaffected by change of origin or scale, and is symmetric in x and y.
- Spearman rank correlation rₛ = 1 − 6Σd²/[n(n² − 1)], d being the difference of ranks; equal values receive the mean rank.
Regression
- The line of y on x minimises squared deviations in the y direction: y − ȳ = b_yx(x − x̄), with b_yx = r σy/σx.
- The line of x on y is x − x̄ = b_xy(y − ȳ), with b_xy = r σx/σy.
- Both lines pass through (x̄, ȳ); r = ±√(b_yx × b_xy), taking the sign of the coefficients; |r| ≤ 1 forces the two coefficients to share a sign.
Worked example
If b_yx = 0.8 and b_xy = 0.45, then r = √(0.8 × 0.45) = √0.36 = 0.6, positive. Given σx = 2, σy = b_yx σx / r = 0.8 × 2 / 0.6 = 2.67.
Traps that cost marks
Using the y-on-x line to predict x; reading r = 0 as “no relationship” when it only rules out a linear one; forgetting the mean-rank correction for tied ranks; assuming b_yx × b_xy equals r rather than r²; mixing the scale factors in the two coefficient formulas.