Correlation

Objective. Interpret product-moment correlation without claiming causation.

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Learn: the key idea

Correlation measures the strength and direction of a linear association between two variables using the product-moment correlation coefficient r, which lies between −1 and 1. A value near 1 shows a strong positive linear trend, near −1 a strong negative trend, and near 0 little linear link. r is calculated from Sxy divided by the square root of Sxx times Syy, using standardised deviations from the means. Crucially, a high |r| only shows association, not that one variable causes the other — a lurking third variable may drive both. For example, ice-cream sales and drowning deaths correlate strongly because both rise with hot weather, not because one causes the other.

Worked example

A dataset gives Sxy = 48, Sxx = 64, Syy = 49. Calculate the product-moment correlation coefficient and comment on the correlation.

1 $r = Sxy \div \sqrt(Sxx \times Syy)$
2 $\sqrt(64 \times 49) = \sqrt3136 = 56$
3 $r = 48 \div 56 = 6/7 \approx 0.857$
4 $This is close to 1, showing strong positive linear correlation$

Answer: r = 6/7 ≈ 0.857 (strong positive correlation)

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Common mistake. Students claim r = 0.9 proves causation, writing "X causes Y". Correlation only measures linear association; always check for confounding variables and never assert causation from r alone.

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