## It Does, In Fact, Take Two to Tango

This week I want to discuss two interesting subjects: sex and graph theory.  At the same time.

Consider these rather intentionally vague questions as motivation: do men tend to have more sexual partners than women?  Or the other way around?  In either case, how can we appropriately measure any differences?

It appears I am coming to the subject late, but as we will see, some important observations still seem to have been missed.  My interest was first piqued by the following minor note hidden in the most recent issue of the College Mathematics Journal:

“Megan McArdle attempts to illustrate her point that survey respondents are unreliable (“Misleading Indicator,” November Atlantic) by telling us that it is mathematically impossible for men to report an average number of female sexual partners that is much higher than the average number of male partners reported by women.  I agree that survey respondents are unreliable, but so is McArdle’s math.  It may be unlikely that the number of partners reported by honest males would be higher, but it is not mathematically impossible.” – Fred Graf, Concord, NH

This seems to be a continuation of debate that began at least several years ago, stemming from a couple of studies with some interesting observations.  The following is from a 2007 New York Times article about the studies:

“One [CDC] 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.”

There are two questions worth asking here: (1) do either of these results make sense, and (2) what conclusions may be drawn from them?

Much of the past discussion has focused on the following argument that the results do not make sense: suppose that there are m men and w women.  Model the heterosexual partnerships between them with a bipartite graph, with a vertex for each of the m men in one partite set, a vertex for each of the w women in the other partite set, and an edge between vertices representing the two corresponding people having had sex with each other in their lifetimes.  Let t be the total number of edges in this graph (i.e., the total number of sexual partnerships).  Then the average number of female partners per male is t/m, and the average number of male partners per female is t/w.

If $m = w$, that is, if there are an equal number of men and women, then these averages are the same.  More importantly, even if $m \neq w$, then the averages differ only by the ratio of the two population sizes, not by any measure of promiscuity of one or the other.

Given this, how can we explain the nearly 2:1 ratio suggested by the British study?  And what can be said about the difference of medians (7 for men, 4 for women) indicated by the CDC study?  (There was some unfortunate confusion about the distinction between mean and median that generated more heat than light; you can read more at this apparently now-abandoned blog.)

Regarding the British study, there are really only three possibilities:

1. There are nearly twice as many women as men in the surveyed population.  I am reasonably confident that this is not the case.
2. Survey respondents reported partnerships with persons outside the population being sampled.  Prostitution (men reporting sex with prostitutes that were not themselves surveyed) and survivor bias (reporting partners who are now dead and thus were obviously not surveyed) are two specific explanations along these lines.  But I think neither is sufficient to explain the extent of discrepancy.  For prostitution to be the primary cause of the difference, for example, every man in the survey would have to report an average of about 6 encounters with prostitutes!  (Or half of the men would need to report about 12 encounters, etc.)
3. Respondents were simply dishonest.  I think this is by far the most plausible explanation, as suggested by McArdle, as well as by David Gale, the mathematician at Berkeley who contributed to the debate.  Men tend to inflate their sexual histories, and women tend to minimize theirs.

Finally, regarding the CDC survey, reporting the median vs. the mean does have the potential to provide more insight into what is happening… but I hesitate to trust these results at all, for two reasons.  I wonder how many people bothered to actually read the paper in question, available here.  In the main text of the paper, the medians of 7 and 4 for men and women, respectively, are indeed mentioned.  However, close inspection of the more detailed tables at the end of the paper reveals some interesting things.

First, the median numbers of partners are given as… 6.8 and 3.7, respectively.  Did the authors round these values to 7 and 4?  More importantly, I admit to being confused about how the median of a set of integers can be anything other than another integer or an integer plus 0.5.

Also, a footnote below the tables indicates that “median values exclude [respondents] with no… sexual partners.”  It is not clear to me why this was done, particularly considering that one of the categories into which responses were grouped was “0-1 partners.”

In summary, I think it is certainly interesting to consider how men’s and women’s sexual experiences differ, “in distribution,” so to speak; how men and women view their sexual histories; and how they share and/or distort them with each other.  But I don’t think either of the studies discussed here succeed in shedding much light on those issues.

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### 2 Responses to It Does, In Fact, Take Two to Tango

1. nomasir says:

Nice use of “in distribution”

On that note, though slightly tangentially, it would be nice to see the distribution of respondents, and perhaps attempt to draw inference from that. The mean is clearly not a sufficient statistic.

• The CDC paper does provide some information about the distribution, but it is rather confusing. The table mentioned in the post essentially provides a histogram of responses with bins of 0-1, 2-6, 7-14, and 15+ partners. It is not clear whether these four bins were the only possible “multiple choice” responses in the survey, or whether these were arbitrary groupings of numeric responses after the data was collected, although I would guess the latter.

For example, the bins for the above four categories for women (reporting number of male partners) were (0.250, 0.444, 0.212, 0.094)… with a median number of partners of 3.7, which must mean someone reported a fractional number of partners, which sounds like a cool trick. The corresponding bins for the guys were (0.167, 0.338, 0.207, 0.289), with a median of 6.8 partners.