PMCC: what the product moment correlation coefficient means and how to test it

The product moment correlation coefficient (PMCC), written r, measures how close the points on a scatter diagram lie to a straight line. It runs from −1 (perfect negative linear correlation) through 0 (no linear correlation) to +1 (perfect positive linear correlation). In A-level Maths you interpret a given value of r and use it in a hypothesis test for correlation.

What the PMCC tells you

• The sign gives the direction: positive means y tends to rise as x rises; negative means y tends to fall.
• The size gives the strength: values near ±1 mean the points hug a straight line; values near 0 mean no linear pattern.
• It only measures linear correlation. Points on a clear curve can have an r near 0.

AQA's A-level Maths specification asks you to interpret scatter diagrams and correlation informally (L2), and to understand that correlation does not imply causation. It also warns about scatter diagrams that contain distinct sections of the population, where groups mixed together can give a misleading overall pattern.

The PMCC formula

Here Sxx = Σ(x − x̄)², Syy = Σ(y − ȳ)² and Sxy = Σ(x − x̄)(y − ȳ). In practice your calculator works r out from the data. AQA's specification says calculating correlation coefficients is excluded: you are given r and asked to interpret it. Seeing the formula once still shows what r is measuring.

A small example

For the points (1, 2), (2, 4), (3, 5), (4, 4), (5, 5): x̄ = 3 and ȳ = 4, giving Sxx = 10, Syy = 6 and Sxy = 6. So r = 6 ÷ √(10 × 6) ≈ 0.775, a fairly strong positive correlation.

Interpreting a value of r

These words are a rough guide, not fixed cut-offs. Whether a correlation is significant depends on the sample size, which is what the hypothesis test checks.

The hypothesis test for correlation

AQA content O1 extends hypothesis testing to correlation coefficients: you interpret a given r using a given p-value or critical value. The test asks whether the sample shows enough evidence of correlation in the whole population, whose correlation coefficient is written ρ (rho).

1. State the hypotheses. H₀: ρ = 0 (no correlation in the population). H₁: ρ ≠ 0 for a two-tailed test, or ρ > 0 or ρ < 0 for a one-tailed test.
2. Note the significance level, such as 5%, and the sample size n.
3. Compare: if r is beyond the critical value (or the p-value is less than the significance level), reject H₀.
4. Conclude in context: there is (or is not) evidence of correlation between the two variables.

Worked example

A sample of 12 students gives r = 0.62 between hours of revision and test score. Test at the 5% level whether there is positive correlation. The one-tailed critical value for n = 12 at 5% is 0.4973.

H₀: ρ = 0, H₁: ρ > 0. Since 0.62 > 0.4973, reject H₀. The sample gives significant evidence (5% level) that hours of revision and test scores are positively correlated. It doesn't show that revising more causes higher scores.

These critical values come from the t distribution with n − 2 degrees of freedom; exam questions give you the value you need.

PMCC or Spearman's rank?

The PMCC uses the actual values and measures linear correlation. Spearman's rank uses ranks, so it suits ordinal data and relationships that rise or fall consistently but aren't straight lines. See our Spearman's rank guide.

Common mistakes

• Saying correlation proves cause. It shows association only.
• Ignoring the sign when choosing a one-tailed test: H₁: ρ > 0 needs a positive r.
• Writing hypotheses about r. Hypotheses are about the population ρ, not the sample r.
• Treating a curve as no relationship. r near 0 means no linear correlation, not no relationship at all.

Practise with a short timed paper

Correlation and hypothesis testing are in A-level Maths statistics. The daily A-level Maths: Statistics paper is a short timed paper of 8 questions, with a new one every day.

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