A pooled acceptance rate can differ between two groups although each department accepts both at the same rate, because the groups apply to different departments. Standardising the rates removes the effect of the mix. The wrong model is that a gap in the pooled rate is a gap in treatment.
Core ideas
The lesson reuses the arithmetic of “Correlation, Confounding and Simpson's Paradox” for a question about fairness. Three ideas carry it.
An acceptance rate is the number admitted over the number who applied. Compared within a department, it asks whether like applicants were treated alike. Compared across departments pooled together, it asks something else.
The pooled rate for a group is an average of its department rates weighted by where its members applied. Two groups with the same rate in every department can have different pooled rates if they applied in different proportions.
Equal treatment says applicants with the same qualification in the same department should have the same chance. Equal outcomes says the overall rates for the groups should match. The two can disagree about the same data.
The material
A table in which both departments treat the two groups alike and the overall rates do not match.
A university has an open department and a selective one. In the open department three applicants in five are admitted, and in the selective department one in five. Men and women are admitted at exactly those rates in both. The only difference between the groups is where they applied: of every hundred men, eighty applied to the open department and twenty to the selective; of every hundred women, twenty applied to the open department and eighty to the selective.
Within the open department both groups are admitted at 3/5, and within the selective department both at 1/5. Pooled, the men's rate is 52/100, which is 13/25, and the women's is 28/100, which is 7/25. A glance at the pooled rates suggests the men are admitted at nearly twice the rate. A glance at the departments shows no group treated differently anywhere.
The pattern is the one Bickel, Hammel and O'Connell found in the graduate admissions at Berkeley in 1973, where the pooled rates suggested a bias against women that the departments, taken one at a time, did not show. The counts here are made small enough to follow by hand.
Both statements are true of the same counts. What differs is the question each answers. The within-department comparison asks whether, for the choice the applicant has already made, the outcome depends on their group. The pooled comparison asks whether the two groups end up in the same overall position.
A standardised rate weights each department's rate by the same shares for both groups, here each department's share of all applicants, so that the mix of applications cannot move the comparison.
Half of all applicants, 100 of 200, applied to each department. The standardised rate for either group is (1/2)·(3/5) + (1/2)·(1/5), which is 2/5. The two groups are equal at 2/5, and the gap in the pooled rates has vanished, which shows that the gap came from the mix.
Equal treatment holds in this table and equal outcomes fails. Each is a respectable notion of fairness, and a policy that secures one need not secure the other. A rule that equalised the pooled rates would have to admit applicants to the selective department at different rates for men and women, which breaks equal treatment in the department.
What the lab cannot decide is whether the mix is itself the unfairness. If women apply to the selective department because the open one is closed to them in practice, or because of what they were taught was possible, the equal rates within departments conceal a disadvantage upstream. If they apply there from preference, they do not. The same counts fit both stories, and which is true is not in the table.
Make women's rates higher in each department, 3/4 against 7/10 in the open one and 1/8 against 1/10 in the selective one, with the same application mix. The pooled rates are then 58/100 for men and 25/100 for women: the women do better in each department and worse overall. The lab marks this as a reversal, and the standardised rates put the women ahead.
Interactive
the same counts, compared group by group and all together
Method
Five moves, and the third is the one that separates the two notions of fairness.
Keep the fractions as they are. A rate rounded early hides the equalities this lesson depends on.
Compare the two groups within a department only. This is the equal-treatment comparison.
Add admitted over added applied for each group. This is the equal-outcomes comparison, and it may disagree with the last step.
Weight each department by its share of all applicants, the same for both groups, and compare again. If the gap vanishes, it came from the application mix.
Whether the mix is itself unfair is a question about the world that lies outside the counts. State which answer your conclusion depends on.
Men and women admitted at 3/5 in the open department and 1/5 in the selective one.
Within each department the groups are treated alike, and standardised to the same mix they are equal at 2/5. The pooled gap, 13/25 against 7/25, is explained by who applied where.
Given admitted and applied counts for two groups in two departments, compute the four rates and the two pooled rates, standardise them, report whether the pooled comparison reverses, and name the claim about the mix on which the fairness verdict depends.
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