Mathématiques

What Is a Correlation Coefficient?

r runs from −1 to 1. What r = 0.85, r = 0.1 and r = −0.9 each look like on a scatter graph, and the three questions the number cannot answer for you.

Correlation coefficient

A correlation coefficient is a number between −1 and 1 saying how closely the points on a scatter graph lie to a straight line, and in which direction.

Aussi appelé: Pearson's r, r value, Product moment correlation coefficient
Où les élèves le rencontrent: Grade 9 or 10 statistics and Digital SAT data-analysis questions; GCSE and IGCSE papers ask you to describe correlation in words rather than to calculate r.

La réponse en bref
A correlation coefficient, written r, measures how closely points on a scatter graph lie to a straight line. It runs from −1 to 1: r = 0.85 is a tight rising band, r = −0.9 a tight falling one, and r = 0.1 a shapeless cloud.

Un exemple

r = 0.85 | r = 0.1 | r = −0.9

Picture three scatter graphs. At r = 0.85 the points form a clear band rising to the right, with a handful sitting off it — knowing x gives you a good idea of y. At r = 0.1 the points are a cloud with no direction, and knowing x tells you almost nothing. At r = −0.9 the band is just as tight as the first but falls to the right: as one quantity goes up, the other reliably comes down.

The sign and the size do different jobs

Read r in two parts. The sign says which way the relationship runs — positive means the points rise to the right, negative means they fall. The size, ignoring the sign, says how tightly the points hug the line.

Those two are independent, and forgetting that is the usual mistake. An r of −0.9 describes a stronger relationship than an r of +0.6, even though one is negative and the other positive. Compare strengths by comparing distances from zero.

r = 1 — every point exactly on a rising straight line
r = 0.85 — a clear rising band, useful for prediction
r = 0.1 — effectively a cloud, no usable straight-line pattern
r = 0 — no linear relationship at all
r = −0.9 — a tight band falling to the right
r = −1 — every point exactly on a falling straight line

Three things r does not tell you

First, it only measures straightness. Points lying perfectly on a smooth U-shaped curve can produce an r close to zero, because there is no single straight line that fits them. A near-zero r means no linear relationship, not no relationship — and the only way to know which is to look at the graph.

Second, it says nothing about steepness. The gradient of the line and the tightness of the scatter around it are separate numbers. A very strong r on a nearly flat line means you can predict y accurately and that y barely changes.

Third, it says nothing about cause. And in small data sets, a single unusual point can drag r a long way on its own, which is another reason to plot before you trust.

Describing correlation without calculating it

At GCSE and IGCSE you are not asked to compute r. You are asked to look at a scatter graph and describe what you see, and the marks go to two things: a direction and a context.

"Strong positive correlation" is half an answer. "There is a strong positive correlation, so plants given more water tended to grow taller" is the whole one. Naming the variables in the sentence is what separates the two, and it also stops you from claiming more than the graph shows.

Questions fréquentes

What counts as a strong correlation?

There is no universal cut-off, and the honest answer is that it depends on the field: an r of 0.5 would be remarkable in some kinds of research and disappointing in a physics experiment. In school work, values above about 0.8 are usually described as strong and values near 0 as none.

Can a correlation coefficient be greater than 1?

No. The definition confines it to the interval from −1 to 1 inclusive, and a value outside that range means an arithmetic slip. The two endpoints correspond to every point sitting exactly on a straight line, falling for −1 and rising for 1, which is as strong as a linear relationship can get.

What does r = 0 mean?

That there is no linear relationship in the data — knowing x gives no straight-line prediction of y. It does not mean the two are unrelated. A perfect V or U shape can return an r near zero while the connection between the variables is exact, which is why you look at the scatter graph first.

What is r squared?

Squaring r gives the proportion of the variation in y accounted for by the straight-line relationship with x. An r of 0.85 gives r squared of about 0.72, so roughly 72 per cent of the variation in y is explained by the line and the rest by everything else.

Do I need to calculate r for GCSE maths?

No. GCSE and IGCSE papers ask you to describe correlation from a scatter graph, draw a line of best fit and use it to estimate. The formal product moment correlation coefficient belongs to A level statistics. Knowing what the values look like still helps you describe a graph accurately.

Dernière mise à jour 21 août 2026