Number of Sexual Partners: Median v. Mean

Via Brad DeLong comes a challenge to certain survey evidence from Gina Kolata:

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. It is about time for mathematicians to set the record straight, said David Gale, an emeritus professor of mathematics at the University of California, Berkeley. “Surveys and studies to the contrary notwithstanding, the conclusion that men have substantially more sex partners than women is not and cannot be true for purely logical reasons,” Dr. Gale said. He even provided a proof, writing in an e-mail message:

“By way of dramatization, we change the context slightly and will prove what will be called the High School Prom Theorem. We suppose that on the day after the prom, each girl is asked to give the number of boys she danced with. These numbers are then added up giving a number G. The same information is then obtained from the boys, giving a number B.

Theorem: G=B

Proof: Both G and B are equal to C, the number of couples who danced together at the prom. Q.E.D.”

We’ll return to Brad who spotted a major error but let’s finish with this:

One is that men are going outside the population to find partners, to prostitutes, for example, who are not part of the survey, or are having sex when they travel to other countries. Another, of course, is that men exaggerate the number of partners they have and women underestimate.

Is this an estimation problem or is it as Brad notes:

Our own David Gale from the tenth floor is made to look ridiculous by Gina Kolata – you see, she didn’t tell him that the survey didn’t ask about means – about averages = but about medians. Which means that she doesn’t know the difference between means and medians.

Let’s speculate a bit. Suppose these sample medians do capture the population medians. That means about the half the men have 13 partners or more. I doubt many of them have more than 15 or 20. About half the men have 12 partners or less and some might have had only one partner in life. Or even none. So B’s mean is likely less than B’s median. Now ladies – excuse me for this – but Ms. Kolata did raise the issue of prostitutes – as in a few women have lots and lots of partners, which would suggest the G’s mean exceeds its median.

Now for you loyal husbands who could care less about this, hug your wife and tell you how terrific she is. Even if both of you are “below average” by one silly measure.