Lies, Damned Lies, And Sexual Statistics

Lies, Damned Lies, and Sexual Statistics Romann M. Weber September 21, 2008 You may have seen one of a number of news reports in the past few years that reported on surveys of the sexual behavior of Americans. In particular, you may have seen lines such as this one: “Overall, women report an average of six sex partners in their lifetimes, men, 20” [1]. Now, if you’re a woman, maybe you read this and thought to yourself, “Boy, men sure are dogs. No wonder they can’t seem to commit to a relationship. They’re always off trying to find something to rub up against!” And if you’re a man, perhaps you thought that you have really underperformed relative to your fellow males. But did you stop to question the validity of the numbers? Maybe you leafed through the full survey results.1 There are colorful graphs and some statistical terminology. The conclusions must be legitimate, right? Not even close. Perhaps it did occur to you that it seems strange for the average number of sexual partners for men to be so much larger than that for women. Shouldn’t they be closer together? (Yes.) Should they be the same? (Not necessarily.) Why are the statistics so different? Well, people lie, especially about sex. To motivate our discussion a bit, let’s have a look at the chart below.

1 http://abcnews.go.com/images/Politics/959a1AmericanSexSurvey.pdf

1

This is a matrix, of sorts, representing a hypothetical population of 20 men and 21 women. The sexual relationships among those in the group are indicated by whether there is a numeral one in the cell that lies at the intersection of a row representing a male and a column representing a woman. For instance, we see that male number 11 and female number 12 have gotten it on with each other, whereas male 14 and female 14 have not known each other in the biblical sense. Notice also that male number 12 is a virgin. Male number 10 is the stud in the group. We can see a lot of ones in his row, and we can see from the sum in the far-right column that male number 10 has slept with more than half of the women in the population (over 57 percent, in fact). You probably know how to compute an arithmetic average, which is just the sum of all the numbers on a list divided by the total number of items on that list. To find the average number of partners for the men, we add the numbers in the right-most column, which gives us the total number of sexual relationships for all of the men.2 The average number of partners among the men is 3.0. We use the same idea to compute the average for the women, this time adding up the numbers in the bottom row and dividing by the total number of women. The average number of partners among women is about 2.86. Notice that the sum of the relationships for the men is the same as the sum of the relationships for women, namely 60. This is necessarily so. After all, we can count up all of the ones on our table either row by row or column by column. Either way, we should get the same answer. It also makes intuitive sense, since for a man to have a sexual relationship with a woman, it is required that a woman have a sexual relationship with a man! This is going to be the case in any complete population under consideration: The total number of relationships will be the same for both the men and the women. The average can be a little different, however, because it is not necessarily the case that there are exactly as many men as there are women. I haven’t looked it up, but for the purposes of the rest of this document, let us say that there are 51 percent women and 49 percent men in the American population. I think that is actually pretty close to being true. We can represent the proportion of men in the population by the letter p (where p = 0.49). Necessarily, then, the proportion of women in the population is given by 1 − p. If the total population is the sum of the men and women, given by N = M + W , then M = pN and W = (1 − p)N . As we showed earlier, the sum of sexual relationships is the same for both the men and the women. Let us represent this number by S. The average number of partners for men is therefore given by µM = S/M = S/pN , and the average number of partners for women is given by µW = S/W = S/(1 − p)N . We will assume that the average is higher for men (which it should be if there are fewer men in the population). We can represent the difference of the 2 Here I refer to “sexual relationships” rather than “partners” in order to avoid confusion, since some women will be counted more than once if they slept with different men.

2

averages by S S − pN (1 − p)N S(1 − 2p) = . pN (1 − p)

∆µ =

But recall from above that µM = S/pN , so we can write   1 − 2p ∆µ = µM . 1−p We see, then, that the difference between the averages is a function of the average for the men and the proportion of the population made up by men. If the 49 percent figure I mentioned above is correct, then it should be the case that µF ≈ 0.96µM . At any rate, the difference between the averages cannot be greater than four percent of the average for the men, a pretty trivial amount. The difference shrinks further the closer the men and women get to a 50-50 split in the population. In our model, if the men are telling the truth, the average number of sexual partners for women should be 19.2, not six. But the men are probably lying. Many probably overstated their sexual histories, bumping the average up. The women are probably also lying by understating their sexual histories to avoid coming off as tramps.3 We are left feeling disappointed in this survey, which was apparently never vetted by a competent statistician. It appears that, for now, the truth about Americans’ sex lives will stay between the sheets.

References [1] “ABC News Primetime Live Poll: The American Sex Survey,” October 21, 2004.

3 The other possibility is that these numbers are correct but that participants were drawn from wildly different populations: relatively normal women and bar-hopping studs. This is a possibility, but it is remarkably improbable.

3