Written by Dackohn; last reviewed November 2026.
Simpson’s paradox occurs when a trend that appears in every subgroup reverses when the groups are combined. In a real kidney stone study, treatment A did better for both small and large stones, yet B looked better overall, because A was mostly used on the harder cases. The cause is a third variable that is unevenly spread across the groups.
The kidney stone data
This comes from a 1986 study by Charig and colleagues in the British Medical Journal, comparing two surgical treatments:
| Stone size | Treatment A | Treatment B |
|---|---|---|
| Small stones | 93.1% (81 of 87) | 86.7% (234 of 270) |
| Large stones | 73.0% (192 of 263) | 68.8% (55 of 80) |
| All patients | 78.0% (273 of 350) | 82.6% (289 of 350) |
A wins in each row, yet B wins in total. No arithmetic has gone wrong.
Why it happens
Look at who received what. A was given to 263 patients with large stones and only 87 with small ones. B had the opposite mix: 270 small and 80 large. Large stones are harder to treat whichever treatment is used. So A’s overall figure is dragged down by its harder cases, and B’s is lifted by its easier ones. Stone size is a confounder: it affects both which treatment a patient receives and the outcome.
An overall rate is a weighted average of subgroup rates, and the two treatments have different weights. That is the whole mechanism.
Which number should you trust?
The data alone cannot tell you. You need to know how they arose.
- Split the data when the third variable influenced who got which group, as stone size did here. Treatment A is the better choice.
- Keep the data combined when the third variable is a result of the treatment. Splitting a drug trial by blood pressure measured after treatment would remove part of the effect you want to measure.
Where else it shows up
- University admissions: in a well-known 1973 Berkeley analysis, men were admitted at a higher overall rate, but most departments did not favour men. Women had applied more often to highly competitive departments.
- Averages over time: a company’s overall pay can fall while every group’s pay rises, if many new staff join at lower-paid levels.
How to protect yourself
- Randomise where possible. Random assignment balances confounders on average.
- Check group composition before comparing overall rates.
- Think about causes first, then decide which variables to split by. See correlation vs causation.
Try it yourself
Simpson's Paradox Calculator →Enter your own subgroup counts and check for a reversal. Chi-Square Calculator →Test the association within a single table.Frequently asked questions
What is a simple example of Simpson's paradox?
In the kidney stone study, treatment A had the higher success rate for both small and large stones, but treatment B had the higher rate overall because it was mostly used for small stones.
Is Simpson's paradox a real paradox?
Not in the logical sense. It is a surprising but correct result of combining groups with different weights.
How do I know whether to combine or split the data?
Think about causes. Split by variables that affected who got which group. Do not split by variables the treatment itself changed.
Related reading: Correlation vs causation · Regression to the mean · Bayes’ theorem explained
Sources & further reading: NIST/SEMATECH e-Handbook of Statistical Methods · OpenStax Introductory Statistics. How we compute & validate: methodology. Written & reviewed by Dackohn; last reviewed August 2026.