530 18. PROBABILITY AND STATISTICSFIGURE 3. Comparison of the chi-square distribution and the fre-
quencies for the sum of the squares of ten independent standard
normal random variables when the experiment is performed 1000
times.One of the many pitfalls of statistical inference was pointed out by Pearson's
colleague George Udny Yule (1871- 1951) in 1903. Following up on Pearson's 1899
paper "On the spurious correlation produced by forming a mixture of heterogeneous
but uncorrelated material," Yule produced a set of two 2x2 tables, each of which
had no correlation, but produced a correlation when combined (see David and
Edwards, 2001, p. 137). Yule's result was, for some reason, not given his name;
but because it was publicized by Edward Hugh Simpson in 1951, 20 it came to be
known as Simpson's paradox. 21
Simpson's paradox is a counterintuitive oddity, not a contradiction. It arises
frequently in practice. An example of it occurred in the admissions data from
the graduate school of the University of California at Berkeley in 1973. These
data raised some warning flags. Of the 12,763 applicants, 5232 were admitted,
giving an admission rate of 41%. However, investigation revealed that 44% of
the male applicants had been admitted and only 35% of the female applicants.
There were 8442 male applicants, 3738 of whom were accepted, and 4321 female
applicants, 1494 of whom were accepted. Simple chi-square testing showed that the
hypothesis that these numbers represent a random deviation from a sex-independent
acceptance rate of 41% was not plausible. There was unquestionably bias. The
question was: Did this bias amount to discrimination? If so, who was doing the
discriminating?
For more information on this case study and a very surprising conclusion, see
"Sex bias in graduate admissions: data from Berkeley," Science, 187, 7 February
1975, 398-404. In that paper, the authors analyzed the very evident bias in admis-
sions to look for evidence of discrimination. Since admission decisions are made
by the individual departments, it seemed logical to determine which departments
had a noticeably higher admission rate for men than for women. Surprisingly, the
authors found only four such departments (out of 101), and the imbalance resulting
from those four departments was more than offset by six other departments that
had a higher admission rate for women. It appears that the source of the bias was
20 See "The interpretation of interaction in contingency tables," Journal of the Royal Statistical
Society, Series Â, 13, 238-241.
21 The name Simpson's paradox goes back at least to the article of C. R. Blyth, "On Simpson's
paradox and the sure-thing principle," in the Journal of the American Statistical Association, 67
(1972), 364-366.