Simpson’s Paradox

Simpson’s paradox for quantitative data: a positive trend ( blue line and red line  ) appears for two separate groups, whereas a negative trend (dotted line) appears when the groups are combined.

Simpson’s paradox, or the Yule–Simpson effect, is a phenomenon in probability and statistics, in which a trend appears in several different groups of data but disappears or reverses when these groups are combined. It is sometimes given the descriptive title reversal paradox or amalgamation paradox.


For these particular sets of graphs, if you look in aggregate the consumer marketing emails had a higher click-through rate, however when you split into the different customer spend buckets, the business emails do better. For some reason, most of the business emails went to the low value customers with lower click rates, and so while in aggregate the business emails did worse, when you account for spend they did better.



One of the best-known examples of Simpson’s paradox is a study of gender bias among graduate school admissions to University of California, Berkeley. The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance


Applicants Admitted
Men 8442 44%
Women 4321 35%

But when examining the individual departments, it appeared that six out of 85 departments were significantly biased against men, whereas only four were significantly biased against women. In fact, the pooled and corrected data showed a “small but statistically significant bias in favor of women.”[16] The data from the six largest departments are listed below.

Department Men Women
Applicants Admitted Applicants Admitted
A 825 62% 108 82%
B 560 63%  25 68%
C 325 37% 593 34%
D 417 33% 375 35%
E 191 28% 393 24%
F 373  6% 341 7%

The research paper by Bickel et al. concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as in the English Department), whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants (such as in engineering and chemistry).



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