Correlation, Causation & Confounding Simulator

Two variables can move together for three very different reasons: one moves the other, the movement is a coincidence, or a third variable is quietly driving both. This simulator builds the third case — and the genuine case — in front of you: pick a story, set the third variable's strength, and then hold it fixed to see what is left of the correlation.

How it works

The data is generated from a hidden third variable Z. In the manufactured mode, X and Y are each built from Z plus their own noise — neither touches the other — so any correlation between them is Z's work alone. In the direct mode, X genuinely feeds Y, and Z merely makes people differ in X. Every run is seeded (the mulberry32 generator), so the sample you see is reproducible — the same runs the site's tests verify.

The honest check is the partial correlation, computed exactly from the three pairwise correlations: r_xy.z = (r_xy − r_xz·r_yz) ÷ √((1 − r_xz²)(1 − r_yz²)). It asks: among people who are equal on Z, do X and Y still move together? The coloured view answers the same question geometrically — split the sample into thirds by Z and fit each third separately. Two honest limits, stated plainly: a partial correlation near zero shows the association was carried by Z, but a surviving partial does not by itself prove causation — some other unmeasured variable, or reverse causation, could still be the story. All math runs client-side and is verified against independently computed reference values — see Software & sources.