Solution to Gender Pay Conundrum

This Biz Quiz came courtesy of Economics Professor Robert Thornton, who holds the Charles William MacFarlane Professorship.

Suppose that an economist is investigating the allegation that wage discrimination exists against female employees of a certain firm with two locational branches. Investigating the charge, the economist finds that it is indeed the case that the female employees of the firm are receiving lower average pay than males in each of the two branches. However, in comparing the overall levels of average male and female pay for both locations combined, the economist finds that females are actually receiving higher average pay than males. How can this be?



This surprising result is a manifestation of what is known as “Simpson’s Paradox” (1951).

The paradox occurs when a conclusion is reversed depending on whether one looks at aggregated data or disaggregated data (Anderson, Sweeney, and Williams). For the example used here—a firm with two locational branches, both of which are paying female employees lower pay than males—see the table below. It can be seen that, although average hourly female pay is $0.50 below that of males at both location 1 and at location 2, when the overall (weighted) average is computed (the sum of gender pay at each location multiplied by its proportional weight), average female pay exceeds average male pay by $1.00.

The conditions for Simpson’s Paradox to occur are rather technical (See Thornton and Innes, 1985). But it can be seen here that female pay at the higher paying location must be greater than male pay at the lower paying location. Furthermore, the proportion of females working at the lower paying location must be sufficiently small.


Is Simpson’s Paradox rare? Yes, but when it does occur it often is a headline-grabber. For example:

  • A number of years ago at the University of California Berkeley, admissions data showed apparent bias against female applicants. For each of the academic departments investigated, a smaller proportion of female applicants than male applicants was admitted. However, when the data for all departments were combined, a greater proportion of female applicants than male applicants was admitted;
  • The publishers of a certain magazine were pleased to note an increase in the overall subscription renewal rate among current subscribers for a particular year. Yet when the renewal rate data were broken down by categories of subscribers, the publishers found a decrease in renewal rates for each category.


Editor's Note: Thank you to all who submitted answers to the Lehigh Biz Quiz. Our random winner, chosen from those who submitted correct answers, is Arthur L Poole III '76 MBA '82. Congratulations!


Anderson D., Sweeney, D., and Williams, T. Statistics for Business and Economics, 12th ed. South-Western Cengage Learning, 2012.
Simpson, E.H. “The Interpretation of Interaction in Contingency Tables,” Journal of the Royal Statistical Society, Series B, Vol. 13, 1951.
Thornton, R. And Innes, J. “On Simpson’s Paradox in Economic Statistics,” Oxford Bulletin of Economics and Statistics, Vol. 47, No. 4, November 1985.

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