overheard in providence

Yesterday I rambled on about Facebook in lieu of discussing a New York Times article explaining why women and men must have the exact same average number of heterosexual partners, despite what survey’s reveal. Unfortunately the article fails to acknowledge the difference between a medians and averages, nor does it offer any real explanation for the erroneous survey outcomes.

As I mentioned yesterday, comments on Ezra Klein’s blog fill in some of these details. My old friend Matthew Yglesias also has the right idea, and Eugene Volokh spells out the mean/median discrepancy in more detail. Despite all of this discussion, a number of important considerations have gone unmentioned, hence this more comprehensive post.

First the facts. The New York Times cites a federal survey stating men have a median number of 7 heterosexual partners, while women have only 4. It also cites a British survey in which men have 12.7 partners and women have 6.5. There is simply no way 12.7 is a median from a large survey (even 6.5 is very unlikely) so we should assume these numbers are means (Jordan Ellenberg of Slate backs me up here.)

The Times also explains that the average number of heterosexual partners men and women have should be equal. Why is this? Suppose a population has M men and W women, and within that population N male-female pairs have had sex. Men have an average of N/M partners, while women have an average of N/W. In the real world, M and W are very close (the Times didn’t mention this assumption), so the averages are nearly identical.

As the article points out, this proof of equality doesn’t account for one gender having a lot of sex outside of the population being surveyed (for example, US men might have lots of sex overseas). Still, are most men having half of their sex overseas? Of course not. So what gives?

One possible culprit, as many have already pointed out, is “the whore effect”. In the female population, there will be a very small percentage of women (prostitutes) who have had a very large number of sexual partners. These outliers increase the female population’s mean, but not its median. Think about it, you survey 100 women, 5 report 0, 5 report 1, 10 report 2, 20 report 3, 30 report 4, the rest 5 (or more). The median is 4, and if any one of these women were replaced with a prostitute, the median is still 4. The average, on the other hand, can increase by a great deal. If the prostitute had slept with 300 people, the average would increase by 3.

So if a population contains female prostitutes, but not male prostitutes, having heterosexual sex, they can bring up the average of the female population without changing the mean. Still, what about the British survey which gave means? It turns out prostitutes influence the mean as well. They do so in two ways.

First (and less significant), a prostitute being surveyed likely does not know exactly how many people they have slept with. They probably stopped keeping count. As a result, they would generally give a conservative estimate, artificially lowering the female mean. Second (and this one is more interesting), any researcher taking a survey is likely to throw away a small percentage (say 1 percent) of outliers. They do this with good reason.

Suppose you are surveying a thousand women, and they all give you numbers like 5, 10, 20, and then all of a sudden one says “1000″. This number is unlikely to be exactly correct, but it significantly effects the sample mean. Even worse, it hugely effects the variance, which you’ll probably use to calculate a confidence interval (the mean plus or minus some range). So if for some reason 1 in 1000 respondents decides to mess with you, they can greatly impact the quality of your results. Rather than screen for liars, it makes sense to throw out the biggest outliers. Unfortunately, this masks the existence of legitimate outliers, in this case prostitutes.

So the whore effect is real, even when estimating the mean, but how big is it? Happily (or sadly?) it cannot be that large. Suppose 1 in 1000 women are prostitutes (a rather high estimate) which on average have slept with 1000 men each (again, a high estimate). The exclusion of these prostitutes would lower the female average by 1. Similarly, if men discounted the sex they had with prostitutes, the male median would unlikely change by more than 1. To see why, assume 1 in 10 men have slept with prostitutes, and figure at least half of these have slept with 6 or more non-prostitutes. Including female prostitutes, 50 percent of men slept with 7 or more women, so we can assume 55 percent have slept with at least 6. When prostitutes are excluded, at most 5 percent of male totals dip below 6, so the median remains at least 6.

So we are forced to conclude that the impact of female prostitutes is small, and that men or women (or both) are not being truthful when surveyed. Still, I doubt the liars are solely to blame. As commenter Kathy G writes on Ezra’s post, both parties in a sexual pairing may not agree about what happened. For example, if a survey doesn’t explicitly call sex consensual intercourse, all bets are off. There could easily be a tendency to skew one’s totals by defining sex more or less liberally.

At least traditionally, I’m sure men are more likely to receive oral sex than women, so ambiguous sexual encounters would probably tend to bolster the male totals. Also there can be ambiguity over consent, which would once again lead to an increase numbers for males. Finally there’s the issue of gratification. If an encounter leaves one party less satisfied than the other perhaps they’d be less inclined to remember it as sex. Sadly, it’s hard to estimate the impact of each of these effects.

So there you have it, a concise 10 paragraphs of supplimental commentary. The conclusion: there may be some whores out there, but men and women still need to get their stories straight. The funny thing is, had this article been better written, I would have found it a lot less interesting. Sadly, all the discussion it prompted among bloggers may discourage writers from trying to incorporate mathematical observations in future articles.

Finally, I should mention that in the course of writing this I came across an actual paper on the subject (although I haven’t read it).