Simpson's Paradox: When a Trend Reverses After You Combine the Groups
October 29, 2026
Simpson's paradox occurs when a trend that appears in several groups of data reverses when the groups are combined. A well-known real example comes from a 1986 study comparing two treatments for kidney stones. For small stones, Treatment A succeeded in 81 of 87 cases (93.1%) and Treatment B in 234 of 270 (86.7%). For large stones, A succeeded in 192 of 263 cases (73.0%) and B in 55 of 80 (68.8%). Treatment A looks better in both groups.
But combining the groups, A succeeded in 273 of 350 cases (78.0%) and B in 289 of 350 (82.6%). Overall, B looks better. How can A be better for small stones, better for large stones, and worse overall?
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The paradox arises because the groups are of very different sizes and difficulty, and the treatments were assigned unevenly. Treatment A was used mostly on large stones (263 of 350 cases), which are harder to treat and have lower success rates for both treatments. Treatment B was used mostly for small stones (270 of 350), which are easier. B's overall rate is inflated by the easy cases, while A's overall rate is dragged down by the hard cases. The overall rate is a weighted average of the group rates, and the weights differ between the treatments.
| Small stones | Large stones | Overall | |
|---|---|---|---|
| Treatment A | 81/87 = 93.1% | 192/263 = 73.0% | 273/350 = 78.0% |
| Treatment B | 234/270 = 86.7% | 55/80 = 68.8% | 289/350 = 82.6% |
| Better treatment | A | A | B (misleading) |
The Lurking Variable
The stone size is a confounding variable: it influences both which treatment a patient received (doctors tended to use the more invasive Treatment A on severe cases) and the outcome. Aggregating over it mixes the effect of treatment with the effect of case severity. The correct comparison, if the goal is to find which treatment works better for a patient with a given stone size, is within each group. The question that decides whether to pool or split is about causation: what is causing what? If stone size affects which treatment is chosen, split by stone size. If instead the grouping variable is a consequence of the treatment (like a side effect), aggregation might be appropriate.
Other Famous Cases
In 1973, UC Berkeley appeared to have bias against women in graduate admissions: overall, men were admitted at a higher rate (44% vs. 35%). But within most individual departments, women were admitted at equal or higher rates; women had applied in larger numbers to more competitive departments with low admission rates. Baseball provides examples where one player has a higher batting average than another in each of two seasons but a lower average overall, because they had different numbers of at-bats in each season. The paradox occurs whenever group sizes differ and there is a hidden variable.
How to Avoid Being Fooled
Look at disaggregated data. Ask what other variables could affect the outcome and how subjects were assigned to groups. Randomized experiments help by breaking the link between the group variable and the treatment. Use statistical methods such as stratification and regression adjustment to control for confounders. Be cautious about headline comparisons from raw averages in news, business dashboards, and policy debates. A good habit is to check whether a trend holds in every relevant subgroup before accepting an aggregate conclusion.
Frequently Asked Questions
What is Simpson's paradox?
A reversal of a trend when data from several groups are combined.
Why does it happen?
Group sizes differ and a confounding variable affects both group membership and outcome.
Which result is correct, the aggregate or the subgroup?
It depends on the causal structure. When a confounder drives assignment, subgroup comparisons are more reliable.
Can randomization prevent it?
Randomization balances confounders across groups, which prevents this kind of distortion.
What was the Berkeley example?
Overall admission rates favored men, but within most departments women were admitted at equal or higher rates.