The NY Times Week In Review front-pages an article so mathematically misdirected that it is laugh-out-loud funny for anyone with an understanding of basic statistics, a group which apparently includes no Times editors. Fortunately, the article includes mad sex pairings, a high school prom, and a mortified professor, so you know it is hot. Here we go:
The Myth, the Math, the Sex
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.
So far, so good, but here comes the curve ball:
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.
It is about time for mathematicians to set the record straight, said David Gale, an emeritus professor of mathematics at the University of California, Berkeley.
“Surveys and studies to the contrary notwithstanding, the conclusion that men have substantially more sex partners than women is not and cannot be true for purely logical reasons,” Dr. Gale said.
Oh My Goodness - is this professor emeritus really unfamiliar with the difference between "mean" and "median"? I assume not, and yet... the Times lead clearly referred to partners of the "median" man and woman.
Well, let's see his reasoning:
He even provided a proof, writing in an e-mail message:
“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.
Proof: Both G and B are equal to C, the number of couples who danced together at the prom. Q.E.D.”
For heaven's sake - I will grant that *if* the reporting by men and women is accurate then the means must be equal. However, one reason for using medians rather than means is to mute the behavior of the, hmm, possibly misbehaving tails. Of the distribution. Since this is apparently not obvious to Times editors, let me illustrate by returning to the prom.
Suppose 100 young men and woman attend the prom. Ninety of the young ladies choose to dance only with their dates; ten dancing femmes set out to dance with every guy in attendance, and succeed (Guys are soooo easy. Who knew?).
So how does the reporting go at the end of the evening? Ninety young men report eleven dance partners - the ten Dancing Femmes plus their date. Ten guys have a mere ten partners, since their original date was one of the Dancing Femmes. From the guys we get total partners equal to (90*11 + 10*10), which equals 1,090 partners. *Averaged* over the one hundred guys, that is 10.9 partners per guy, with a *median* number of partners equal to eleven.
And how about the ladies? Ninety of them report just one partner, their date. Ten of them report one hundred partners. Total partners from the ladies' ledger is (90*1 + 10*100), which also equals 1,090. That produces a mean of 10.9 partners per lady, but the median number of partners is only 1.
The median is different from the mean because the median effectively ignored the behavior of the small group of mad dancers. How about that? Or should I say, Q.E.D?
Groan. I have no doubt that mocking emails are piling up in David Gale's inbox; I further have no doubt that Prof. Gale is wishing he had been a bit more clear on what the Federal study was saying about medians and what he was saying about means. As to whether the reporter inadvertently misrepresented the study to him (Mean, median, hey, I'm on deadline!), or Gale misunderstood the question, who knows? But clearly neither the Times reporter nor her editors understood what they were presenting.
But let me just add this - Gale blew it, and the ends did not justify the means.
For those who care, the article overcomes this ghastly start to make some plausible points about cultural pre-dispositions to misreporting by both sexes.
MORE: Singular Values makes a similar point.
TROUBLING: A reader revolt:
Well, on behalf of the males here, I appreciate your efforts, TM. I really do.
But, let's face it, we're lying.
I have nothing to say on the record.
MEANS AND MEDIANS AT BERKELEY: Brad DeLong illustrated the difference at David Brooks' expense in July.