The New York Times slips up on sexual math.

A mathematician's guide to the news.
Aug. 13 2007 6:09 PM

Mean Girls

The New York Times slips up on sexual math.

Illustration by Mark Alan Stamaty. Click image to expand.

In Sunday's New York Times, science writer Gina Kolata took on studies suggesting that men tend to have more sexual partners than women do. This CDC study, for one, shows that American men between the ages of 20 and 59 report accumulating a median of seven female bedmates, while for women the corresponding figure is just four. The problem, Kolata writes, is that these numbers present a mathematical contradiction. "It is logically impossible for heterosexual men to have more partners on average than heterosexual women," she explains. "Those survey results cannot be correct." Kolata even quotes a theorem to this effect, backed up by mathematician David Gale of Berkeley: The average number of partners has to be the same for men and women.

It's not every day I get to read a mathematical theorem in the New York Times, so I hate to complain. But Kolata isn't quite right here. The problem is hiding in the distinction between the median (the number reported by the CDC study) and the mean (the number Gale was talking about). The mean is what people usually call the "average." To calculate the mean number of sexual partners among a group of men, you add together each man's sexual partners, then divide by the total number of men. The median, on the other hand, is the number you'd get if you line all the men up in order of their number of partners, then ask the man in the middle to state his count.

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Consider a village with 200 people, evenly divided by sex. Ninety of the women are virgins, but none of the men is. Each man has slept with just one of the sexually active women; each woman who's had sex, then, has had 10 partners. In this case, the median woman has zero sexual partners, but the median man has one. So we see a big difference in medians between the male and female populations, just as in the CDC data.

The means are a different story. Each male villager has one partner, for a total of 100; dividing by the total male population of 100 gives a mean of one. Among the women, the 10 nonvirgins have 10 sexual partners each, totaling 100 again; divide by the female population, and you'll find that the mean number of sexual partners per woman is also one. This equality is no coincidence. In a closed group like our village, the total number of opposite-sex partnerships has to be the same, whether you count these partnerships from the male or female point of view. If the questionnaire gives a different result, the questioned are lying.

In practice, means and medians are often roughly comparable. The exceptions come from situations where a small slice of the population has a lot of sex—or whatever other item is being measured. An old joke is illustrative here: 10 statisticians in a bar. Ted Turner walks in. The statisticians start to whoop and holler. "What's going on?" asks Turner. One statistician explains, "On average, we just got a whole lot richer!"

The joke here is that average, to a statistician, means mean—but average, to, well, an average person, means something more like typical. Ted Turner's presence in the bar raises the mean income of the drinkers quite a lot, but the median hardly at all. And when we ask questions about sexual behavior, it's usually typical men and women we want to know about—not averages that can be dragged upward by a few hypothetical Ted Turners of sex.

Not that Kolata's conclusion is inaccurate. As she points out, "Another study, by British researchers, stated that men had 12.7 heterosexual partners in their lifetimes and women had 6.5." These numbers, though Kolata doesn't say so, are means, not medians. In this case, it's indeed mathematically impossible that the numbers are correct. The medians in the British sample? Seven and four, same as in the American study—so you can stop worrying about a transatlantic promiscuity gap. Note that the means are indeed a lot higher than the medians, suggesting that a certain amount of sexual Ted Turnerism is taking place. Since it's as prevalent among men as women, however, it doesn't create the mysterious gender discrepancy.

So what does? One possibility, as Kolata points out, is that people are drawing sexual partners from outside the sample. One hetero Lothario in the next village over could single-handedly increase the median number of sexual partners for village women without (directly) affecting the sexual fortunes of the men. In the CDC study, these outsiders might be prostitutes, or people outside the study's 20 to 59 age range. (Though one imagines there's as much sexual contact between male twentysomethings and female teens as between middle-aged women and sexagenarian men, so this might be a wash.) Kolata observes, too, that people simply might not be telling researchers the truth about their sexual lives. The importance of inaccurate self-reporting is emphasized in the two most thorough papers on the topic I could find, a 1996 article by Wadsworth et al. in the Journal of the Royal Statistical Society and Michael Wiederman's 1997 article from the Journal of Sex Research. Wiederman considers all the explanations I could think of (and plenty more) for the impossible discrepancy of means, and concludes that the culprit is inaccurate self-reporting. One tip-off: The discrepancy shrinks somewhat if you ask people only about the number of partners they've had in the last five years, and even more if you restrict the questioning to the last year, which strongly suggests that unreliable memory is playing a part.

In the end, then, Kolata is right. Studies that report these numbers should emphasize that the reported difference between men and women is an anomaly that can't be taken at face value. But in making this subtle mathematical point, she chose to gloss over a much simpler one—that the mean and median are not the same. The mean is easier to analyze mathematically. But if you want to know how you measure up to the typical American's sex life, it's the median you're after.

Jordan Ellenberg is a professor of mathematics at the University of Wisconsin and the author of How Not to Be Wrong. He blogs at Quomodocumque.

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