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Simpson's paradox

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Simpson's paradox, or the Yule-Simpson effect, is a phenomenon noted in statistics, which states:

The ratio of success of several sub-populations can be reversed in the ratio of success of the population as a whole.

This effect has been noticed to occur in Bernoulli Trials, especially in social science, medicine and baseball batting averages.

It is not this statement by Homer Simpson:

To alcohol! The cause of, and solution to, all of life's problems.


The easiest way to see this is with examples:

Example 1 - Berkeley’s sex bias case[edit]

Berkeley University was one of the first universities to be sued for sexual discrimination. In the fall of 1973, 12,763 people applied for admission through the Graduate Admission Office. 8,442 of the applicants were males, and 4,321 were females. 44% of the males were successful, but only 35% of the females were admitted.[1]

Looking at the gender breakdown for the 6 most popular majors:[note 1]

A B C D E F Overall
Male 62% 63% 37% 33% 28% 6% 52%
Female 82% 68% 34% 35% 24% 7% 42%

We can see Simpson's paradox in that in 4 of the cases a higher ratio of females were admitted and the other two the difference is not great. However, this ratio does not accurately represent the overall ratio between females admitted and males admitted—in fact, a higher ratio of males were admitted overall, a reversal of most of the ratios found in the individual majors.

The cause of the paradox can be found by the number of applicants.

A B C D E F Overall
Male 825 560 325 417 191 373 2691
Female 108 25 395 375 393 341 1835

Of the 1,835 female applicants 1,504, or 82%, applied for courses in which ~33% (or less) applicants were admitted. Of the 2,691 male applicants 1,385, or 51.5%, applied for courses in which ~ 66% applicants were admitted. Berkeley won the case.

See also[edit]

External links[edit]


  1. For reasons of confidentiality, Berkeley did not release the departments' names for the paper.


  1. Bickel, Hammel and O’Connell, Sex Bias in Graduate Admissions: Data from Berkley Science, Vol. 187, No.4175, pp.398-404.