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3.7a · Correlation & Regression · Sub-skill
Measuring the strength and direction of a linear relationship between two variables using the correlation coefficient, r.
Build it up, step by step
Click each step below to reveal it — work through them in order the first time round.
Correlation measures how closely two variables are linearly related — as one increases, does the other tend to increase (positive correlation), decrease (negative correlation), or show no clear pattern (no/zero correlation)? Scatter graphs are the visual way to spot correlation.
The product moment correlation coefficient (PMCC), $r$, gives a numerical value for the strength and direction of a linear relationship. It always lies between −1 and +1. r = +1 is perfect positive correlation, r = −1 is perfect negative correlation, and r = 0 means no linear correlation.
Values close to ±1 indicate strong correlation; values close to 0 indicate weak or no correlation. There's no single fixed cut-off for ‘strong’ vs ‘weak’ — context matters, but roughly: |r| > 0.7 is often considered strong. You'll usually calculate r using your calculator's statistical mode from raw data.
A strong correlation between two variables does not prove that one causes the other. There could be a third factor influencing both (a confounding variable), or the relationship could be coincidental. Exam questions frequently test whether you can avoid claiming causation just because r is high.
A scatter graph of hours studied vs exam score gives a PMCC of r = 0.82. Interpret this value.
r = 0.82 is close to +1, indicating a strong positive linear correlation between hours studied and exam score — as hours studied increases, exam score tends to increase too, and the relationship is fairly consistent/strong (though not perfect, since r isn't exactly 1).
Test yourself
A researcher calculates the PMCC between ice cream sales and the number of drowning incidents at beaches over a year, finding r = 0.78.
(a) Describe the strength and direction of this correlation.
(b) The researcher claims ‘buying ice cream causes drowning.’ Explain why this conclusion is not justified by the correlation alone. [4 marks]
(a) r = 0.78 indicates a fairly strong positive correlation between ice cream sales and drowning incidents.
(b) Correlation does not imply causation — there is likely a confounding (third) variable at play, such as hot weather, which independently increases both ice cream sales and swimming/drowning incidents. The correlation doesn't prove buying ice cream directly causes drowning.
Practice
Five short questions on pmcc. Work through them, then reveal the mark scheme to check.