What r actually measures
Pearson's correlation coefficient, r, measures the strength and direction of the linear association between two quantitative variables. It runs from −1 to +1: −1 is a perfect downward straight line, +1 a perfect upward one, 0 no linear association at all. Because r is built from standardized deviations it is unitless — the correlation between height and weight is the same in centimetres and kilograms as in inches and pounds.
Square it and you get r², the proportion of variance in one variable that the linear relationship accounts for. An r of 0.8 gives r² = 0.64, so about 64% of the variability is explained by the linear fit and 36% is not — a useful reality check, because 0.8 sounds like near-certainty and 64% does not.
Two limits matter more than the formula. First, r only sees straight lines: for the made-up dataset x = −3, −2, −1, 0, 1, 2, 3 with y = x², a deterministic parabola, r is exactly 0 even though y is completely determined by x. Second, r is not robust to outliers. The invented points (1,6), (2,5), (3,4), (4,6), (5,5), (6,4) have r ≈ −0.48; add one distant point at (20,25) and r flips to about +0.95. A single observation reversed the sign and the story.
It is also worth separating r from the regression slope. They share a sign and come from the same quantities, but the slope carries units — 'y changes by this much per unit of x' — while r is a unitless measure of tightness. A steep slope can accompany a weak correlation, and a gentle slope a very strong one.
- r measures linear association only, is unitless, and always lies between −1 and +1.
- r² is the proportion of variance explained by the linear relationship: r = 0.8 means r² = 0.64.
- A perfect curved relationship can give r = 0, and a single outlier can flip r's sign — plot first.
Why a correlation is not a cause
When X and Y are correlated, at least five explanations are on the table. X may cause Y. Y may cause X — reverse causation, common whenever both are measured at the same time. A third variable Z may cause both, making X and Y move together while neither touches the other. The link may be an artefact of how cases were selected. Or it may be chance, especially when many pairs were screened and only the striking one reported.
The third-variable case has its own vocabulary. A confounder is associated with the exposure and also a cause of the outcome; a lurking variable is one you never measured, which is worse, because you cannot adjust for what is not in your dataset. Ice cream sales and drowning deaths rise together across the year, but the driver is warm weather, which sends people to both the freezer and the water. Nothing in the correlation tells you that.
Direction is ambiguous in exactly the questions people care about. Students who report more study hours often score higher, but confident students may study more because the material already feels tractable — and a single cross-sectional correlation cannot separate those two stories.
- Candidate explanations: X causes Y, Y causes X, a third variable causes both, selection artefact, or chance.
- A confounder is measured and adjustable; a lurking variable was never recorded and cannot be adjusted for.
- Correlation is symmetric — r(x, y) = r(y, x) — so on its own it cannot point a direction.
Simpson's paradox: when the aggregate lies
Simpson's paradox is the sharpest demonstration that a summary statistic can point the wrong way: an association that holds in every subgroup reverses once the subgroups are pooled, usually because the groups differ in both the outcome rate and how cases were distributed across them.
Here is a made-up illustration. Two tutoring programmes are compared on pass rates across an introductory and an advanced course. In the introductory course, programme A passes 95 of 100 students (95%) and B passes 270 of 300 (90%). In the advanced course, A passes 150 of 300 (50%) and B passes 45 of 100 (45%). A wins both. Pool them and A passes 245 of 400 (61.25%) while B passes 315 of 400 (78.75%) — B now looks clearly better.
Nothing in the arithmetic is wrong; the pooled comparison answers a different question. Programme A's students were concentrated in the harder course, where everyone does worse, so the aggregate mostly reflects course difficulty rather than programme quality. Aggregation is a modelling choice: before trusting an overall association, ask which groups were combined and whether they differ in ways that also drive the outcome.
- Simpson's paradox: an association present in every subgroup can reverse when subgroups are pooled.
- It arises when subgroups differ in both baseline outcome rate and case mix.
- Always check whether an aggregate comparison hides an uneven split across groups.
What does support a causal claim
The strongest evidence is a randomized controlled experiment. Randomly assigning the treatment breaks the link between it and everything else about a subject, so on average the groups differ only in what you manipulated. Randomization, not sample size, is what upgrades an association into a causal estimate — a huge observational dataset with a confounder in it is a precisely measured wrong answer.
When randomization is impossible or unethical, causal claims are built more slowly from converging evidence: a plausible mechanism, correct temporal order, a dose-response pattern, consistency across populations and designs, and results that survive adjustment for measurable confounders. For everyday reading, three questions do most of the work: was the treatment assigned or merely observed, which third variables went unmeasured, and were groups pooled that should have been kept apart?
- Randomized assignment is what licenses a causal interpretation; sample size alone never does.
- Observational causal claims need mechanism, temporal order, dose-response and consistency.
- Ask: assigned or observed? which confounders were measured? were groups wrongly pooled?
Reporting correlations honestly
Report r with the scatterplot beside it, never alone. The plot shows curvature, clusters, and outliers that r compresses into one number, and it is the fastest way to catch a correlation that exists only because of one extreme point. Include the sample size too: r from a handful of points is extremely unstable.
Use language that matches the design. 'Associated with', 'linked to', and 'predicts' are honest for observational data; 'causes' and 'leads to' are not. Prediction does not require causation — a correlate can forecast an outcome perfectly well while being useless to intervene on, and confusing the two is how organisations end up acting on a variable that moves nothing.
Finally, statistical significance for a correlation only rules out chance; it says nothing about confounding or direction. StatRise's Pearson's Correlation calculator shows each step of computing r on your own numbers, and the Regression Lab simulation lets you add noise and outliers to watch r and R-squared react.
- Always publish the scatterplot and the sample size alongside r.
- Match the verb to the design: 'associated with' for observational data, not 'causes'.
- Significance rules out chance only — it does not rule out confounding or reverse causation.