Karl Pearson's coefficient of correlation

Karl Pearson's coefficient of correlation, r, measures the degree of linear relationship between two variables. Using x = X – X̄ and y = Y – Ȳ (the deviations from the means):

r = Σxy ÷ √(Σx² × Σy²)

Or, directly from the values: r = [ΣXY – (ΣX × ΣY)/N] ÷ √{[ΣX² – (ΣX)²/N] × [ΣY² – (ΣY)²/N]}.

The steps: find the deviations, multiply each pair (xy), square each deviation (x², y²), add each column, and do one division. For example, if Σxy = 30, Σx² = 40 and Σy² = 40, then r = 30 ÷ √(40 × 40) = 30 ÷ 40 = 0.75, a fairly high positive correlation.

With large numbers, the step deviation method makes the work easier: use U = (X – A) ÷ h and V = (Y – C) ÷ k, and calculate r from U and V instead. The answer is the same.

Properties of r:

• It has no unit: it is a pure number.
• It always lies between –1 and +1. A value outside this range means a mistake in the working.
• It is unaffected by a change of origin or scale: adding or subtracting a constant, or multiplying or dividing by one, does not change r. That is why the step deviation method works.
• r = 0 means there is no linear relationship (though there may be a curved one).
• Close to +1 means strong positive; close to –1, strong negative.

Use r only for linear relationships, so check the scatter diagram first.