17.6. Simpson’s Paradox 713
CS 2 men admitted out of 5 candidates 40%
50 women admitted out of 100 candidates 50%
EE 70 men admitted out of 100 candidates 70%
4 women admitted out of 5 candidates 80%
Overall 72 men admitted, 105 candidates 69 %
54 women admitted, 105 candidates 51 %Table 17.1 A scenario in which men are overall more likely than women to be
admitted to a school, despite being less likely to be admitted into any given pro-
gram.
Assume that all candidates are either men or women, and that no candidate be-
longs to both departments. That is, the eventsFEE,FCS,MEE, andMCSare all
disjoint.
In these terms, the plaintiff is making the following argument:
PrAjMEE[MCS>PrAjFEE[FCS:
In plain English, across the university, the total probability that a woman candidate
is admitted is less than the probability for a man.
The university retorts thatin any given department, a woman candidate has
chancesequal to or greaterthan those of a male candidate; more formally, that
PrAjMEEPrAjFEEand
PrAjMCSPrAjFCS:
It is easy to believe that these two positions are contradictory. But Table 17.1
shows a set of admission statistics for which the assertions of both the plaintiff
and the university hold. In this case, a higher percentage of female applicants
were admitted to each department, but overall a higher percentage of males were
accepted! So the apparently contradictory claims can in fact both be true. How can
we make sense of this seemingly paradoxical situation?
Initially, we and the plaintiffs both assumed that the overall admissions statis-
tics for the university could only be explained by discrimination. However, the
department-by-department breakdown shows that the source of the discrepancy is
that the CS department lets in about 20% fewer candidates overall, but attracts a
far larger number of woman applicants than the more permissive EE department^3.
(^3) At the actual university in the lawsuit, the “exclusive” departments more popular among women
were those that did not require a mathematical foundation, such as English and education. Women’s
disproportionate choice of these careers reflects gender bias, but one which predates the university’s
involvement.