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.
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!"