What Simpson’s paradox is
Simpson’s paradox happens when a trend that appears within every subgroup reverses when the groups are combined. It is not a calculation error. It arises when a third variable affects both which group someone is in and the outcome. Read Simpson’s paradox explained for the full reasoning.
The example: kidney stone treatments
In a 1986 study by Charig and colleagues, treatment A succeeded more often for small stones (93.1% vs 86.7%) and for large stones (73.0% vs 68.8%). Yet overall, treatment B looks better (82.6% vs 78.0%). The reason is that A was given mostly to large stones, the harder cases: 263 of its 350 patients, against 80 of 350 for B.
Which number should you believe?
Statistics alone cannot answer this. You need to know how the data were generated. If the stratum variable influenced who got which treatment, as stone size did here, compare within strata. If the stratum variable was caused by the treatment, as with a side effect measured afterwards, the combined figure may be the right one.
To test a single 2 × 2 table, use the chi-square calculator, or Fisher’s exact test for small counts.
Frequently asked questions
What causes Simpson's paradox?
A confounding variable that is unevenly distributed across the groups and also affects the outcome. Combining the data mixes the effect of the group with the effect of the confounder.
Is Simpson's paradox common in real data?
Reversals are not rare in observational data with unbalanced groups. Well-known examples include medical treatments and university admissions.
Does randomisation prevent it?
Largely, yes. Randomisation balances confounders across groups on average, so a reversal caused by confounding becomes unlikely in large randomised trials.
Should I always split the data into subgroups?
No. Split by variables that existed before the treatment and could affect it. Splitting by something the treatment itself caused can create misleading results.