New York Times Misses the Median vs. Arithmetic Mean Distinction:

The New York Times reports:

In study after study and in country after country, men report more, often many more, sexual partners than women.

One survey, recently reported by the federal government, concluded that men had a median of seven female sex partners. Women had a median of four male sex partners. Another study, by British researchers, stated that men had 12.7 heterosexual partners in their lifetimes and women had 6.5.

But there is just one problem, mathematicians say. It is logically impossible for heterosexual men to have more partners on average than heterosexual women. Those survey results cannot be correct.

Well, heterosexual men generally can't have more partners on average than heterosexual women, where "average" means "sum up the partner counts and divide by the number of people." I say generally because there are possible boundary cases (e.g., if you asked all live men and women in the U.S. how many opposite-sex sexual partners they've had, and they answered completely accurately, the numbers wouldn't match because some of their partners are now dead, or are now outside the country, or because the partners are prostitutes and the study doesn't ask prostitutes or ask them enough), but it's generally so. As the article points out, "invoking women who are outside the survey population cannot begin to explain a difference of 75 percent in the number of partners, as occurred in the study saying men had seven partners and women four." (Perhaps prostitution might explain the difference, but the numbers that I've seen suggests that it doesn't suffice.)

But the medians may well be very different. Just as a sample, imagine a population with men A, B, C, D, E, and women P, Q, R, S, T, in which the sex partners map out this way:

 A B C D E P Y Q Y R Y S Y Y Y Y Y T Y Y Y Y Y

The median number of sex partners for the women is 1 (since the women's partner counts are 1, 1, 1, 5, 5); the median number of sex partners for the men is 3 (since the men's partner counts are 2, 2, 3, 3, 3). The arithmetic means for both are 2.6, since there are 13 male-female pairings; but the medians differ substantially.

Now this having been said, the bottom line is likely still correct; as I understand it, there is substantial reason to believe that men overreport their sexual partner counts and women underreport them. And it may well be that the 6.5 and 12.7 numbers are arithmetic means (the article doesn't say), which would be substantial evidence for this theory. But the "men had a median of seven female sex partners" / "[w]omen had a median of four male sex partners" data is extremely weak evidence, if evidence at all; and it surely can't be confounded with the "average" in the sense of arithmetic mean, which the article does.

Thanks to reader John Crawford for the pointer.