A treatment can help every group individually, yet appear to hurt when you combine the groups. This isn't a statistical trick—it's a fundamental feature of how aggregation can reverse conclusions.
Adjust the parameters and watch the paradox emerge. The treatment helps both groups, but hurts overall.
Scenario: Medical Treatment Trial
Two hospitals test a new treatment. Hospital A treats mostly mild cases. Hospital B treats mostly severe cases. The treatment helps patients at both hospitals. But when you combine the data...
Hospital A (Mild Cases)
Hospital B (Severe Cases)
Hospital A (Mild Cases)
Hospital B (Severe Cases)
Combined (All Patients)
🤯 The Paradox
Why this happens
The key is unequal group sizes combined with different base rates.
Hospital A (mild cases, high recovery) mostly uses the control. Hospital B (severe cases, low recovery) mostly uses the treatment. When you combine them, the treatment group is dominated by severe cases, making it look worse overall.
The treatment genuinely helps at both hospitals! But the aggregated data hides this because it confounds treatment effect with case severity.
Real-world examples
UC Berkeley Admissions (1973)
Overall, men were admitted at higher rates than women. But in nearly every department, women had equal or higher admission rates. Women applied more to competitive departments.
Baseball Batting Averages
Player A can have a higher batting average than Player B in both halves of the season, yet a lower average for the full season. It depends on how many at-bats they had in each half.
Kidney Stone Treatment
Treatment A had better success for small stones AND large stones, but Treatment B looked better overall—because A was used more on large (harder) stones.