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Mean GirlsThe New York Times slips up on sexual math.

By Jordan Ellenberg
Posted Monday, Aug. 13, 2007, at 6:09 PM ET
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.

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.

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Jordan Ellenberg is an associate professor of mathematics at the University of Wisconsin. His first novel is The Grasshopper King.
Illustration by Mark Alan Stamaty.
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Remarks from the Fray:

You've equivocated the median and the mode. Yes, the median is the value such that half of the data points are below and half are above. Thus, the median might not exist as an actual data point if there are an even number of data points. In that case, you take the middle point between the two in the middle. That is, if I have $10 and you have $100, the median is $55, an amount neither of us have.

But you brought up the concept of "typical" and that isn't the median. That's the mode. The mode is the most common value in a data set. If there are multiple values that appear just as often, then the group has multiple modes. If we have six people, one with $10, two with $20, and three with $100, then the average is $58.33, the median is $60, but the mode is $100.

Each value is important. But since we know that the average is heavily affected by outliers, we don't rely on that single number to tell us anything. That's where things like standard deviation and variance come in. It lets us know just how varied around the average things are.

The problem with the median is that it doesn't tell you just how much spread there is. If there are three people, one with $49, one with $50, and one with $51, the median is $50. But if those three people have one with $0, one with $50, and one with $100, the median is still $50 and we don't have any way of seeing how varied the population is.

The problem with the mode is that it is simply the most common result. This is the seeming paradox that most people don't have the most common outcome. If we have 101 people, the first 100 of which have that much money ($1 for person 1, $2 for person 2, etc.) and person 101 has $50, then the mode is $50, even though 98% of people don't have the mode.

That's why good studies report mean, median, and mode as well as standard deviation.

--Rrhain

(To reply, click here.)

This huge and impossible discrepancy between men's and women's responses calls into question the whole enterprise of self-reported sexual surveys. It turns out that, typically, the reported number of partners is the only available check on the internal consistency of these surveys and, as Kolata only recently discovered, they all fail spectacularly.

How would you judge a so-called scientific methodology that consistently and obviously fails its only consistency test? Both Kolata and Ellenberg treat this as a quirk in search of an explanation, but fail to explore the deeper implication. If men and women are systematically misreporting on something as straightforward as their number of partners, why should we believe that they are telling the truth on all the other questions?

--lloyd667

(To reply, click here.)

It turns out that a quick review of the actual data reported in the study demonstrates that it would take a pretty impressive deviation at the top end of the number of partners for women to have the average number of partners come out even. The reported data breaks down this way: 0-1 partner: men 16.6%, women 25.0% 2-6 partners: men 33.8%, women, 44.3% 7-14 partners: men 20.7%, women 21.3% 15 or more partners: men 28.9%, women 9.4% The trend is pretty clear - women are overrepresented in the groups with the fewest partners and men are overrepresented in the group with most partners. Nearly half the men reported 7 or more partners, while nearly 70% of the women reported 6 or fewer. Given these facts, to get the average number of partners for women to equal the average number of partners for men, you have to start making some pretty amazing assumptions.

Statistics tells us that, as the sample size gets larger in a randomly-distributed sample the chance that the mean and the median deviate by a meaningful amount gets lower and lower. This survey covered more than 25,000 people over a period of 4 years. With a sample that big, the likelihood that the mean and the median differ enough for it to matter is pretty darned low.

--randy-khan

(To reply, click here.)

Apparently nobody here remembers American Pie 2 and the "Rule of Three." In that movie, the Rule of Three was a simple rule that said that for men, divide the number of women he says he's slept with by three to get the real answer; for women, multiply it by three to get the real answer. It seems apparent to me that that's what's causing the discrepancy in these studies: men may overstate their number of sexual partners, while women may understate theirs.

--tdd

(To reply, click here.)

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