Quod Erat Demonstrandum, poorly
Gina Kolata, in yesterday's New York Times, discusses a couple of mathematicians who have a logical problem with the number of sex partners reported by men and women. David Gale and Ronald Graham question how it is possible for men and women to have on average different numbers of sexual partners. Specifically, Gale questions the results of Fryar et al's (CDC, 2007) report: "Drug Use and Sexual Behaviors Reported by Adults: United States, 1999-2002", released in June, which showed that men reported a median lifetime sexual partners of 7 for men and 4 for women.
By way of proof, Gale offers the following:By way of dramatization, we change the context slightly and will prove what will be called the High School Prom Theorem. We suppose that on the day after the prom, each girl is asked to give the number of boys she danced with. These numbers are then added up giving a number G. The same information is then obtained from the boys, giving a number B.
Theorem: G=B
Proof: Both G and B are equal to C, the number of couples who danced together at the prom. Q.E.D.
Now, I readily recognize that "Math is Hard!" and that most people don't have an intuitive grasp of statistics. But it strikes me that Gale and Fryar are discussing two very different issues. By totalling all of the sexual partners (or dance partners) of the men and women, Gale is looking for the mean, not the median (which was reported in the CDC report). As David Lane's website illustrates, the skew of a distribution affects the relationship between the mean and the median. The means could be equal and the medians different if the distribution of lifetime sexual partners for women were positively skewed, and the distribution for men were negatively skewed. I don't feel that addressing this issue is too much to ask of a mathematician.
One other possible confound that the NYT addresses is that prostitutes do not fall within the sample, but that men who visits prostitutes do. An issue that isn't addressed is that the CDC statistics do not distinguish between same- and opposite-sex partners. The NHANES questionnaire appears to ask about both male and female partners, but no data on same-sex encounters are reported. If men are counting their same-sex partners as "sex" and women are not, even identical behavior patterns could appear different.
By way of proof, Gale offers the following:By way of dramatization, we change the context slightly and will prove what will be called the High School Prom Theorem. We suppose that on the day after the prom, each girl is asked to give the number of boys she danced with. These numbers are then added up giving a number G. The same information is then obtained from the boys, giving a number B.
Theorem: G=B
Proof: Both G and B are equal to C, the number of couples who danced together at the prom. Q.E.D.
Now, I readily recognize that "Math is Hard!" and that most people don't have an intuitive grasp of statistics. But it strikes me that Gale and Fryar are discussing two very different issues. By totalling all of the sexual partners (or dance partners) of the men and women, Gale is looking for the mean, not the median (which was reported in the CDC report). As David Lane's website illustrates, the skew of a distribution affects the relationship between the mean and the median. The means could be equal and the medians different if the distribution of lifetime sexual partners for women were positively skewed, and the distribution for men were negatively skewed. I don't feel that addressing this issue is too much to ask of a mathematician.
One other possible confound that the NYT addresses is that prostitutes do not fall within the sample, but that men who visits prostitutes do. An issue that isn't addressed is that the CDC statistics do not distinguish between same- and opposite-sex partners. The NHANES questionnaire appears to ask about both male and female partners, but no data on same-sex encounters are reported. If men are counting their same-sex partners as "sex" and women are not, even identical behavior patterns could appear different.