Simpson's Paradox

When a trend in the aggregate disappears — or reverses — once you account for a grouping variable

A dataset shows a clear negative overall correlation between study hours and exam score — until you account for academic major. Each major shows a positive within-group correlation, reversing the overall trend. Drag the slider to reveal the hidden group structure.

Group by Academic Major 0%

Overall view: one aggregate group

Overall r: —

The lesson: An association that holds in each subgroup can reverse (or vanish) in the aggregate. The mechanism here: harder majors require more study hours yet produce lower average scores (confound). Within each major, more studying does improve scores — but the major-level confound overwhelms this within-group effect in the aggregate.

But do not read this as "always split by subgroup." The within-group estimate is the right one here because major is a common cause of both study hours and score — a confounder. Had the grouping variable instead been a mediator (something study hours cause, which in turn affects the score) or a collider (something both variables cause), splitting on it would have introduced the bias rather than removed it, and the aggregate estimate would have been the correct one. The data alone cannot tell you which case you are in; that takes an argument about what causes what.

Named after Edward H. Simpson (1951), though the phenomenon was described earlier by Pearson (1899) and Yule (1903) — and the name itself was coined by Blyth (1972), not by Simpson.

Citation

Persson, B. N. (2026). Simpson's Paradox [Interactive visualization]. https://bjorn-persson.github.io/visualizations/simpsons-paradox/

@misc{Persson2026simpsonsparadox,
  author = {Björn N. Persson},
  year = {2026},
  title = {Simpson's Paradox},
  note = {Interactive visualization},
  url = {https://bjorn-persson.github.io/visualizations/simpsons-paradox/}}