Theorem: When It Comes To Sex, The Typical Guy = The Typical Girl

Reading The New York Times yesterday, I was mystified by a story on math and sex. It's commonly reported that men have more sex partners than women. Some mathematicians have claimed this is impossible, on the grounds that men and women must have equal numbers of sex partners, since they have sex with each other (bracketing, I guess, the various complexities about homosexuality and "threesomes").

I was mystified because of all the people they asked, no one pointed out that there's a simple confusion here between "median" and "average." The mathematician's proof shows that the average number of partners are the same. But this leaves open the possibility that most men have more sex partners than most women.

As we all know, averages can be the same while distributions are very different. So, for example, if a few women have sex with lots of men, while many women have sex with only one man, then there is an obvious sense in which men have "more" partners: lots of men are having sex with more than one woman, while few women are having sex with more than one man. But the average number of sex partners will be the same.

For those who like this sort of thing, I drew up an example. Remember, the median is the number at which half the sample is above and half is below.

Suppose {1, 2, 3, 4, 5} and {A, B, C, D, E} have sex in the following combinations:

{1A, 1B, 1C, 1D, 1E, 2A, 3B, 4C, 5D}.

The median number of partners for numbers is Med {5, 1, 1, 1, 1} = 1
The median number of partners for letters is Med {2, 2, 2, 2, 1} = 2

There is a sense in which the numbers here have fewer sex partners than the letters: most numbers have only one partners, while most letters have 2. But the average number of partners for each is the same: 1.8.

I don't know if this is what it's like for women and men, but it seems possible. It's weird that none of the experts cited in the article mentioned this. The Times article even shifts between reporting results for "medians" when discussing the received view on sex difference, then moves to averages when discussing the impossibility of such difference.

What The Times should have been reporting on in this story is why the received view is based on medians and not averages. If a few women are having lots of sex, are those women less significant when it comes to making judgments about "how many"? Why so? After all, usually when we say "the typical person," we're talking about the average person. And as the mathematicians show, there's a sense in which the typical woman is having the same number of sex partners as the typical guy.

At least, this is so leaving aside complexities such as men going to prostitutes, who, the CDC researcher mentioned, are "not part of the survey," or going outside the country. These outlier factors may explain differences in averages, if there are any. But there is no need to invoke them to explain differences in medians.

UPDATE: The Times published a follow-up, here. It seems there are differences in reported averages as well as reported medians; indeed, the raw data of what is reported is, in a sense, internally inconsistent.