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Medians?

Jeff noticed this news article by Gina Kolata:

EVERYONE knows men are promiscuous by nature. It’s part of the genetic strategy that evolved to help men spread their genes far and wide. The strategy is different for a woman, who has to go through so much just to have a baby and then nurture it. She is genetically programmed to want just one man who will stick with her and help raise their children.

Surveys bear this out. In study after study and in country after country, men report more, often many more, sexual partners than women.

One survey, recently reported by the federal government, concluded that men had a median of seven female sex partners. Women had a median of four male sex partners. Another study, by British researchers, stated that men had 12.7 heterosexual partners in their lifetimes and women had 6.5.

But there is just one problem, mathematicians say. It is logically impossible for heterosexual men to have more partners on average than heterosexual women. Those survey results cannot be correct.

Jeff’s response: MEDIANS??!!

Indeed, there’s no reason the two distributions should have the same median. I gotta say, it’s disappointing that the reporter talked to mathematicians rather than statisticians. (Next time, I’d recommend asking David Dunson for a quote on this sort of thing.) I’m also surprised that they considered that respondents might be lying but not that they might be using different definitions of sex partner. Finally, it’s amusing that the Brits report more sex partners than Americans, contrary to stereotypes.

24 Comments

  1. Thank you! I've been watching all day to see if anyone else brought up this problem, and it's upsetting to see how many people are just saying "ooh, an expert said it, so it must be true!" I discuss the fact that Gale is using means vs medians in my own post today:
    http://differenceblog.livejournal.com/78608.html

  2. Bob O'H says:

    That's curious. For men, there are two alternative mating strategies: monogamy, and invest in your few offspring, or polygamy where you have plenty of offspring, but don't invest so much in each one, so they are less well off. For women, only the latter strategy makes sense. One might therefore expect a larger variance in number of sexual partners for males, but this suggests that the opposite might be true (because the mean number of sexual partners should be about the same assuming an even sex ratio, so a lowering of the median must be balanced by a few larger numbers).

    Of course, the "men are liars" explanation may also hold, but I'm surprised the effect is that large.

    Bob

  3. Vince says:

    Isn't it sad that people don't know the difference between the median and the average?

    The quoted mathematical argument doesn't exactly prove that the two distributions have the same average. It seems that there is a bipartite graph describing the heterosexual partner network of the population. The "High School Prom Theorem" is just a bipartite variant of the "Handshaking Lemma," i.e. the total degree of each partition is equal to the total number of edges. That doesn't explain averages unless the partitions have equal numbers of vertices. Still, the degree distributions of the partitions could be very different.

    They really should've asked a statistician. sigh.

  4. John S. says:

    Here is an interesting blog post about this:

    http://justoneminute.typepad.com/main/2007/08/the

  5. Barry says:

    (a) I wonder who the 'mathematicians' were. I've made such stupid errors before, so I'm not preaching from a position of superiority, but that is a foolish erro.

    (b) Bob O'H, there are a number of mating strategies for each sex. For men, there are these additional strategies:
    1) Maintain 1 or a few primary mates, and screw around on the side.
    2) Don't have an official mate, because you can't afford one/the more powerful males got them – so try some covert cheating with women claimed by other men.

    For women:
    1) Get (or get taken by) one male as an official mate, but screw around on the side. For example, marry the sleazy old fat guy who's got money, status and power, and, uh, 'cultivate' that hot young gardener.

    2) Keep a rotating set of mates, as they are in the area. This would work better with males who are both mobile and transitory (as in an expectation that they'll leave and not come back, or that they'll get killed soon). The idea here is that one doesn't have a strong relationship with any of them, but can make it up with numbers, so to speak.

    3) Marry a lower status man (e.g., not much money/status/power), and have covert relationships with one or more men who are more desirable, or who have access to money/status/power.

    Needless to say, these can all be very dangerous, but that's frequently a major factor in the mating game.

  6. digitusmedius says:

    And, of course, there's no independent way to confirm that the partners are heterosexual. Being one, I'd suspect that men would assume that their female partner was heterosexual when, in fact she may not be. Would women be less likely to make this automatic assumption about the men they sleep with? Anyone? And, being very confused in general about statistics (remember, it's the third category of lies) can someone enlighten me as to why this being about MEDIANS changes everything? Or, did I just fall for the joke.

  7. Austin says:

    "the Brits report more sex partners than Americans, contrary to stereotypes."

    What stereoptype is that baby!

  8. digitusmedius says:

    P.S. And women could be lying, too, for that matter.

  9. Matt Weiner says:

    According to Jordan Ellenberg the British do not report higher numbers of partners than the Americans; rather the difference is between the British mean and the American median:

    As [Kolata] 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.

    http://www.slate.com/id/2172186/pagenum/2/

  10. Mikey says:

    Suppose you have ten men, and ten women. All the men sleep with the first woman, and none of the other nine women sleep with anyone. Without lying, all ten men say they have slept with one or more women. 90% of the women report that they have never slept with anyone. The percentages don't have to match.

  11. Andrew says:

    Matt

    Thanks for the info.

    P.S. When I was in grad school, a similar study came out that reported something like a median or average of 7 lifetime sex partners. We weren't sure if they corrected for the fact that people's lives are ongoing, so 7 lifetime partners so far isn't the same as 7 in your whole life. In any case, my favorite remark was from one of the other Ph.D. students in the program who said of the study: "I don't believe these numbers. I've already had 7 sex partners this semester!"

  12. Phil says:

    Mikey, In your example each man has 1 partner. So the mean number of partners = 1 for men. On the distaff side, 9 of the women have 0 partners while 1 of the women has 10 partners. So the mean number of partners = (9*0 + 1*10)/10 = 1 for women, same as for men. The means agree, as they must. The medians do not have to agree, and in your example they don't.

  13. digitusmedius says:

    Could the discrepancy in the study be accounted for by Matt's example above? (If I understand his point correctly there could be small number of very promiscuous women in the study group that increased the median for men but might have a negligible effect on the median for women).

  14. digitusmedius says:

    correction: Mikey posted the example to which I referred.

  15. Andrew says:

    Digitus,

    I suspect the discrepancy arises from the combination of many factors that have been discussed, including skewed distributions, imperfect sampling, different definitions of the survey response, recall problems, and even lying.

  16. Peter says:

    I've looked at this, and men and women do report different mean number of partners (not just different medians). I've seen studies of this from a lot of countries.

    Clearly there are lots of problems.

    For one thing, even if you define sex precisely in your questionnaire, people will give odd answers….I've seen ethnographic work citing all sorts of odd things (one of my favorites was that sex didn't count if you were trying to get pregnant).

    Then there's the vast undersampling of commercial sex workers: A female CSW can have a *lot* more partners than a man.

    Then there's plain forgetting, especially at higher numbers.

    Then there's lying, or some artful combination of lying, different definitions, and mis-remembering, especially with lifetime partners.

  17. Corey Yanofsky says:

    From http://www.salon.com/mwt/broadsheet/2007/08/13/se

    Since so many folks wrote into our letters thread taking issue with the fact that professor Gale commented that the average number of sexual partners needs to be the same while the actual survey reports the median number of partners, I shot him an e-mail, giving him a chance to respond. Here's what he said:

    I've gotten several messages making the same point. If you look at Gina's article you will see that I never attacked the statement about medians. I tried to carefully avoid saying anything directly about the median statement in the article because, as you realize, it could be correct even with accurate data. What I did was to get a copy of the CDC report and used the data in its tables. It groups people into four groups and gives the percentage of men and women in each group:

    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.

    From these figures you can estimate the total partners claimed by each sex. I got between 40 percent and 75 percent more male than female partners depending on how you guess the average on each interval. Thus, the raw data is inconsistent (so it doesn't matter whether you take averages or medians or any other statistic). I hope this clarifies.

  18. truth machine says:

    Um, 12.7 and 6.5 are obviously means. OTOH, Gale doesn't even bother to mention that his "proof" is only valid if the number of boys equals the number of girls (and all the partners were of the opposite sex, and they all have the same notion of "dance partner" — perhaps someone gyrating in the middle of the room considers everyone within a 15 foot radius to be a "partner"). But his statement that the numbers can't be trusted is obvious without doing any proofs or calculations, when the claims being made are of numbers of sexual partners, rather than numbers of sexual partners reported. Gale is absolutely right that the papers are furthering a stereotype, although it's just one of many examples of media mistreatment of empirical studies.

  19. truth machine says:

    One might therefore expect a larger variance in number of sexual partners for males, but this suggests that the opposite might be true

    That Y-Chromosomal Adam is considerably more recent than Mitochondrial Eve establishes that the variance has been historically greater for males.

  20. Peter says:

    The distribution of the 15+ is going to be vastly different for men and women.

  21. John says:

    Gina Kolata has a follow-up article in the Times: http://www.nytimes.com/2007/08/19/weekinreview/19….

  22. Andrew says:

    John,

    To repeat my original statement, it's disappointing that the reporter talked to mathematicians rather than statisticians.

  23. new yorker says:

    I'm not a mathematician, but I think I have a model for how there might be a difference between the means even with honest reporting, equal sizes of male and female populations, exact definition of sexual act, etc.etc.

    As someone pointed out above, lifetime partners to date is not the same as partners over a liftetime. Now assume that the you work with a study population controlled for age. Let's say, for simplicity's sake, that 2% of the population, per sex, reported being currently x years old for each 15

  24. new yorker says:

    Ouch. That was cropped in a way I didn't intend. So, without the example numbers. Assume that the world's male and female populations are equal in total number and age distribution. Assume that sexual habits are perfectly historically static. Assume an unambiguous definition of intercourse and perfectly honest responses. And assume that the study population is simply the world population (getting rid of the prostitute and sex-tourism biases). Then men's and women's average numbers of total lifetime heterosexual partners should be perfectly equal. Still, if women go through their phases of relative promiscuity significantly later in life, their average number of lifetime-to-date heterosexual partners will be lower, no?