What is Simpson's paradox?

It is a reversal in which a trend present in combined data shrinks, vanishes or flips once the data are split by a third variable. The third variable has to be unevenly distributed between the groups being compared. Neither the combined figure nor the split figure is a miscalculation; they answer different questions, and which one to trust depends on how the data came to exist.

Did Berkeley turn out to have no bias at all?

Not quite. Bickel, Hammel and O'Connell found that once applications were grouped by department, the data showed a small but statistically significant bias in favour of women, not against them. In the six largest departments women's admission rate was higher than men's in A, B, D and F and lower in C and E. The authors' point was that the pooled gap came from which departments people applied to, so the evidence did not support a charge of bias by the admitting departments.

How do I check whether a headline comparison hides a Simpson's reversal?

List the variables that could decide which side of the comparison a person lands on, such as department, job type, age or shift pattern. Recompute the gap inside each level of those variables. If the sign changes, or the gap shrinks sharply, the combined number was mostly measuring group mix. If it holds in every subgroup, the headline is sturdier.

Berkeley admitted 44% of men and 35% of women. Inside four of the six biggest departments, women did better

Simpson's paradox explained with the 1973 Berkeley graduate admissions data, where a pooled gap against women sat beside department-level rates that favoured them, plus a toy example of how a wake-time survey could point the wrong way.

By the DontSnooze Team · November 1, 2026 · 2 min read

Simpson’s paradox is a reversal: a trend in combined data weakens, vanishes or flips once the data are split into groups. It appears when a hidden variable is spread unevenly across the two sides being compared.

The standard specimen is Berkeley, fall 1973. Of 8,442 men who applied to graduate study, 44% were admitted. Of 4,321 women, 35% were (Bickel, Hammel and O’Connell, Science, 1975). That looks like bias against women. Split by department it stops looking that way: in four of the six largest departments, a higher share of women than men were admitted. Department A took 82% of its 108 women and 62% of its 825 men. A larger share of women had applied to departments that reject most applicants, so the pooled rate reflected where people applied. The authors went further than “no bias”: with applications grouped by department, they found a small but statistically significant tilt in favour of women.

The same shape in a wake-time survey

The numbers below are made up to show the shape. No real sleep dataset is being described.

Suppose 200 people report whether they feel rested, 100 early wakers and 100 late. Among office workers, 80% of 90 early wakers report feeling rested, against 90% of 10 late wakers. Among shift workers, 40% of 10 early wakers do, against 50% of 90 late wakers. Early wakers lose in both groups. Pooled, though, 76% of early wakers report feeling rested and only 54% of late wakers do, because the early side is mostly office workers, who feel rested more often at any wake time.

Shift pattern, age or chronotype could each play that hidden role. The problem is a cousin of the selection effect in the survivorship-bias post on Abraham Wald, and it is one reason the 5 AM data is so hard to read. At Berkeley the sorting variable was the department each applicant chose. In a wake-time survey it would be whatever job or life pattern put a person on the early side, and the survey form never records it unless someone asks.

Posted in simpsons paradox statistics berkeley admissions survey design early rising

The same shape in a wake-time survey

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