What is Simpson's paradox?

A trend that holds in every subgroup of the data can disappear or reverse when the subgroups are pooled. In Berkeley's 1973 admissions, the university admitted 44% of men but only 35% of women, yet most departments admitted women at equal or higher rates — because women applied disproportionately to the most competitive departments.

Simpson's paradox

The same data, two opposite stories.

Berkeley, 1973. The university admitted 44% of men but 35% of women. Split it by department and the gap flips — women win almost every one. Toggle split / merge and watch it happen.

View

Dataset

Men

—

— / —

Women

—

— / —

Aggregate gap—

← favours Bfavours A →

How it works

The toggle is the whole trick. Merged pools every applicant into two bars — and the men's bar is taller. Hit Split and the same people fan out into their departments, where the women's bar wins almost every column. Nothing about the data changed; only how we grouped it.

What just happened

Simpson's paradox is when a trend that appears in every subgroup of a dataset disappears or reverses when the subgroups are pooled together.

It happens because the subgroups have different base rates. In Berkeley's 1973 admissions, women disproportionately applied to highly competitive departments (English, education) while men applied to less competitive ones (engineering, chemistry). Each department was fair — often admitting women at higher rates — but the aggregate looked discriminatory.

Why aggregation lies

The lesson: aggregate averages weight by group size, not by group quality. When subgroups have wildly different sizes, the pooled average can sit below every subgroup it summarizes.

Women's overall rate is dragged down not because any department rejected them more, but because so many of them queued for the departments that reject everyone more.

How to spot it in real life

Always ask "compared to what?" Before accepting an aggregate finding, look at the disaggregated subgroups. If trends conflict, the system is more interesting than the aggregate suggests.

"Mortality higher in Hospital A": maybe Hospital A treats sicker patients.
"Drug less effective in this trial": maybe the trial enrolled different demographics.
"This city has lower wages": maybe it has a younger workforce.

Confounders are everywhere. Simpson's paradox is the visible tip.

Citations

Bickel, P.J., Hammel, E.A. & O'Connell, J.W. (1975). "Sex Bias in Graduate Admissions: Data from Berkeley." Science 187(4175): 398–404. Source of the 44% / 35% aggregate and per-department figures.
Charig, C.R. et al. (1986). "Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy." British Medical Journal 292: 879–882. Kidney stone data.
Cohen, J. & Cohen, P. (1996). Batting-average example (David Justice vs Derek Jeter, 1995–96).

Built by Joshua Chua · Fable rebuild · vanilla JS + Canvas