Math 143 C/E
Probability and Statistics
Spring, 2001

Simpson's Paradox: UC-Berkeley Graduate Admissions, 1973

The University of California-Berkeley was charged with having discriminated against women in the graduate admissions process for the fall quarter of 1973. The 2-way table below gives tallies for acceptances and denials for both male and female applicants in the six largest graduate programs at the institution at that time.
Men
-------		 Women
-------		 Total
-------
Admitted 1195   559 1754
Denied 1486 1276 2762
Total 2681 1835 4516
Perform a chi-square test on this data to see if there is evidence for an association between gender and acceptance into one of these top six graduate programs.

Answer: The admittance rate is different based on gender P < 0.0005.

The next table calls into question the results of the previous exercise. Here is a breakdown by program. (Note: there are now three variables being displayed: gender, status (accepted or denied) and program — that makes this a type of 3-way table.)

Men
Accepted
-------		 Men
Denied
------		 Acceptance
Rate
--------		 ||
||
||		 Women
Accepted
-------		 Women
Denied
------		 Acceptance
Rate
--------
Program A 511 314 61.94% ||   89   19 82.41%
Program B 352 208 62.86% ||   17     8 68.00%
Program C 120 205 36.92% || 202 391 34.06%
Program D 137 270 33.66% || 132 243 35.20%
Program E   53 138 27.75% ||   95 298 24.17%
Program F   22 351   5.90% ||   24 317   7.04%

While it is not always the case that an association revealed by the chi-square test is due to some lurking variable, when it is (as was the case here with the lurking variable “Program”), the chi-square test cannot be expected to account for it.

Source: “Workshop Statistics: Discovery with Data and Minitab”, by Allan J. Rossman and Beth L. Chance
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