In 1973, Berkeley appeared to admit men at a higher rate than women, hinting at bias. But when admissions were broken down department by department, most departments actually favoured women slightly. The reversal was real, not a mistake. Women had applied in larger numbers to the most competitive departments, where everyone’s odds were low. The aggregate hid the structure.
This is Simpson’s paradox: a relationship that holds within every subgroup can vanish or flip when the subgroups are pooled. A treatment can help both mild and severe patients yet look worse overall, simply because it was given more often to the sicker ones. The lurking variable — department, severity, the way cases were sorted — quietly steers the total.
The pattern is a warning about aggregation: a single number summarising a mixed population can point in a direction that is true of no one inside it. The fix is not better arithmetic — it is asking what was combined, and why.