Corporate life · Simpson's paradox
Simpson's Paradox at Work: When Every Department Says the Opposite of the Total
The short version: an overall rate can point the opposite way from every group inside it when the groups differ a lot and people are spread unevenly across them. In our example, women are promoted more often than men in both departments, but overall men are promoted at 43% and women at 22.5%, because most women work in the department where promotions are rare. The best-known real case: in 1973, the University of California, Berkeley, admitted about 44% of male and 35% of female graduate applicants, yet most departments admitted women at the same rate or higher. The totals and the groups answer different questions.
HR has published the promotion figures. Across the company, 43% of men who applied for promotion got one, against 22.5% of women. A director has called it a scandal and asked each department head to explain.
Both department heads came back with the same answer: in their department, women were promoted more often than men. The director has asked for the figures to be checked. They have been checked. All of them are correct.
The figures
| Men | Women | |
|---|---|---|
| Department A (sales): promoted | 50 of 100 (50%) | 12 of 20 (60%) |
| Department B (engineering): promoted | 2 of 20 (10%) | 15 of 100 (15%) |
| Whole company: promoted | 52 of 120 (43%) | 27 of 120 (22.5%) |
In each department, women do better. In total, men do much better. Nothing is miscounted. The difference comes from where people are: most of the men applied in department A, where half of applicants get promoted; most of the women applied in department B, where very few do. The total mixes the two departments in different proportions for each group.
This is Simpson's paradox: a pattern in every group reverses when the groups are combined, because the groups differ in size and in their baseline rate.
The famous case
In autumn 1973, the University of California, Berkeley, received graduate applications from 8,442 men and 4,321 women. It admitted about 44% of the men and 35% of the women. That looked like clear bias, and statisticians were asked to look closer.
Department by department, the gap mostly disappeared: in most departments, women were admitted at the same rate as men or higher. Women had applied more often to departments that admitted few applicants of either sex, and men to departments that admitted many. The overall gap was real, but it came from where people applied, not from how departments treated them.
So which number is right?
Both, for different questions. "Are departments treating equal applicants unequally?" is answered by the department figures. "Why do women in this company end up promoted less often?" is answered by the total, and the answer points somewhere else: why are promotions so rare in department B, and why do women work there? That might be a real problem worth fixing; it's just not the problem the director was looking for.
Statistics for noobs: always split, then ask why
Whenever you compare two groups' rates, ask whether they're spread differently across some third thing: department, region, age, product, difficulty. If they are, look at the comparison inside each slice as well as in total. If the answers differ, the third thing is a confounding variable, and the story is about it.
Where it costs you money
A fund can beat its rival in every single year and still have a lower total return, if it had more money invested in the bad years. A strategy can win in every market regime and lose overall. Before choosing on a total, check the comparison inside each period.
The moral
The director accepted the department figures and declared the matter closed. Kevin asked why department B promotes only 15% of applicants and why it's mostly women applying there. The director said that was a different question. Kevin agreed, and pointed out that it's the one the total was asking.
What this example can't tell you
- The company is a model. The numbers were chosen to show the effect clearly; real data are rarely this tidy.
- Splitting can mislead too. If the slices themselves are the result of unfairness, for example people being steered into departments, the within-department figures hide it.
- Berkeley is one study. It shows how the paradox works in admissions data, not what happens in any particular company.
The director, the department heads and the promotion figures are an illustration. The Berkeley figures come from the 1975 study in Science. Educational content, not HR, legal or financial advice.