[math-fun] Sex, Lies and Government Conducted Surveys
On September 15 the government released a report "Sexual Behavior and Selected Health Measures" reporting on a survey "conducted by the Centers for Disease Control and Prevention's [sic] (CDC) National Center for Health Statistics (NCHS)" http://www.cdc.gov/nchs/data/ad/ad362.pdf The objective of the survey is to get information on public health concerns, fertility, sexually transmitted disease, etc. However I bring it up here because it contains the most glaring examples I have yet seen of published inconsistency. It's the well worn story that males have many more "sexual partners" than females. Interested people should look at the report. For example, Tables 10 and 11 of the survey show that the median number of partners "in lifetime" for males over forty is 8 while that for females is 3.8. To immediately recognize the inconsistency imagine the same survey with but with the words sexual partners replaced by spouses. For more of the same look at Figure 6 on page 6. More inconsistency: when men and women are asked for the number of partners over the past 12 months, Tables 1 and 2, the numbers come out the same, within epsilon. . As I said, this is an old story. See the following from New York Review of Books ten years ago by Richard Lewontin. http://www.nybooks.com/articles/article-preview?article_id=1923 The authors are aware of the problem and suggest explanations on pages 12 and 13 some of which are, well, amusing. "Some researchers have suggested that some of this difference is due to a small percentage of men who report very large numbers of partners," Another "explanation" is that women my have partners over age 44 who were not polled in the survey (this would of course skew it in the other direction). Is this an example of innumeracy in high places? Lewontin writes in http://www.nybooks.com/articles/1871 "The willingness of both social and natural scientists to recognize contradictions in their findings, and then to ignore them when it is convenient, is a serious disease of inquiry." David The authors
I'm sure that you are correct, but there are a couple of complicating factors. 1. The statistics are very highly skewed for both men & women -- some have very few (0 or 1), while others have thousands (e.g., prostitutes). While those with thousands are very few, they really screw up the averages. As these people are very rare, most people who don't know of them wouldn't believe in their existence, and therefore wouldn't believe in studies that include them. 2. Male homosexuals (on average) have a far higher number of partners than female homosexuals. (I've had this confirmed to me by homosexuals of both genders.) The Canadian flight attendant (male) who is reputed to have been "Ground Zero" in the AIDS epidemic, apparently had thousands of partners all over the world.
Date: Mon, 26 Sep 2005 03:55:11 -0700 From: David Gale <gale@math.berkeley.edu> For example, Tables 10 and 11 of the survey show that the median number of partners "in lifetime" for males over forty is 8 while that for females is 3.8. To immediately recognize the inconsistency imagine the same survey with but with the words sexual partners replaced by spouses. It's entirely possible that this data contains reporting error, of course, but I don't think that it is inconsistent; it seems entirely possible to me that these statistics are both correct. Suppose there were 20 men and 20 women in a population. Suppose that 15 of the women have no sexual partners, and the other 5 women each have sex with all 20 men. In that situation, the median number of partners for the men is 5, and the median number of partners for the women is 0. Since men and women will of course have the same mean number of partners, the fact that the median for women is much smaller than it is for men suggests that the distribution of number-of-partners for women is more skewed than it is for men. This is confirmed by other studies I've seen that reported the full distribution, rather than just the mean or median. A few female prostitutes who have sex with a very large number of men, raising the female mean a lot with virtually no effect on the female median, is certainly part of the cause of such a skew. For more of the same look at Figure 6 on page 6. More inconsistency: when men and women are asked for the number of partners over the past 12 months, Tables 1 and 2, the numbers come out the same, within epsilon. Again, I see no inconsistency. I can certainly construct data where this would be true, but the long-term median number of partners for men is greater than for women. Is this an example of innumeracy in high places? I think that at least some of the innumeracy is here on the math-fun list, where no-one seems to have noticed that in expecting the median for men and women to be the same, you are attributing to the median a property that is held by the mean, but need not be held by the median. Andy Latto andy.latto@pobox.com
Wed, 28 Sep 2005 17:22:20 +0000 (GMT) Andy Latto <andy.latto@pobox.com> Since men and women will of course have the same mean number of partners, Everyone seems to accept this, even up to the "of course", but why? It's possible that I am missing something obvious, and I'm the one who is innumerate --- but isn't this (mean number of partners equal for both men and women) only true if you make some very specific assumptions? It seems to me that even under the assumption of perfectly accurate reporting, this need not be true. First of all, even if you are assuming only heterosexual pairings, the number of women in the pool may be larger than the number of men. Consider 20 women, 10 men, every man and woman pair up, then the mean number of partners for women is 10 and for men it is 20. Second of all, if you include homosexual pairings, then all bets are off, no? Consider a pool of 20 monogamous heterosexual women, 1 heterosexual man, and 19 homosexual men, where all the men are aggressively polygamous. For the purpose of a mathematical example, and not to cater to stereotypical prejudices, let's assume that the men want to, and willingly do, pair with every possible partner (the 20 women for the lone heterosexual man, and the other 18 homosexual men for the 19 homosexual men). The 20 monogamous women, each partner with the one heterosexual man (presumably unaware that he's not monogamous). The homosexual men each pair up with every possible homosexual partner. Then we'd have a mean of 1 partner for the women, but a mean of about 18.1 partners for the men. [Please note, I am *not* saying that there are no obvious inconsistencies in the report (which I haven't read), I am just saying that it is possible for men and women to have different mean number of partners because (a) there may be more women than men, and (b) not all "sexual partnerships" consist of one man and one woman. Similarly, to jump on an earlier example meant to highlight this seeming inconsistency, it is certainly possible that the mean number of pieces of email sent by each person is smaller than the mean number of pieces of email received by each person (multiple recipients). The ratio could be reversed, too, by the phenomena of mis-addressed email (if enough people were clumsy error-prone typists).]. Also: Date: Mon, 26 Sep 2005 03:55:11 -0700 From: David Gale <gale@math.berkeley.edu> For example, Tables 10 and 11 of the survey show that the median number of partners "in lifetime" for males over forty is 8 while that for females is 3.8. To immediately recognize the inconsistency imagine the same survey with but with the words sexual partners replaced by spouses. It's entirely possible that this data contains reporting error, of course, but I don't think that it is inconsistent; it seems entirely possible to me that these statistics are both correct. I am not arguing with your main point, but I am not sure how the *median* can be 3.8, frankly. Suppose there were 20 men and 20 women in a population. Suppose that 15 of the women have no sexual partners, and the other 5 women each have sex with all 20 men. In that situation, the median number of partners for the men is 5, and the median number of partners for the women is 0. the fact that the median for women is much smaller than it is for men suggests that the distribution of number-of-partners for women is more skewed than it is for men. This is confirmed by other studies I've seen that reported the full distribution, rather than just the mean or median. A few female prostitutes who have sex with a very large number of men, raising the female mean a lot with virtually no effect on the female median, is certainly part of the cause of such a skew. .... Is this an example of innumeracy in high places? I think that at least some of the innumeracy is here on the math-fun list, where no-one seems to have noticed that in expecting the median for men and women to be the same, you are attributing to the median a property that is held by the mean, but need not be held by the median. Andy Latto andy.latto@pobox.com
Andy Latto is of course right that the MEDIANS of the male and female distributions can be different. I came up with the same example as he did where the female population consists of v virgins and p prostitutes and v>p. Then the female median is 0 and the mail is p. So then I looked at the distributions given in top line of tables 10 and 11 and used them to estimate the total number of partners of males and females. It was an estimate because when they say 32 % 0f men reported between 3 and 6 partners I used 4.5 etc. Doing it this way I found men have about 50% more partners than women. Mike Greenwald asks, how a median can be 3.8. Good question! Maybe I'm revealing my own innumeracy but then please tell me what it means for an integer valued random variable to have a fractional median. I suppose if exactly half the values were greater than n then any number between n-1 and n could be a median, n+1/2 would be a natural candidate. Tables 10 and 11 have fractional medians everywhere. One funny datum is that the median number of life partners for women up to age 40 is 3.9 while for those up to 44 it goes down to 3.8. I suppose this could be sampling error, (unless it shows that women are getting more promiscuous over time). Anyhow, someone please help me out on those fractional medians David At 10:22 AM 9/28/2005, you wrote:
Date: Mon, 26 Sep 2005 03:55:11 -0700 From: David Gale <gale@math.berkeley.edu> For example, Tables 10 and 11 of the survey show that the median number of partners "in lifetime" for males over forty is 8 while that for females is 3.8. To immediately recognize the inconsistency imagine the same survey with but with the words sexual partners replaced by spouses.
It's entirely possible that this data contains reporting error, of course, but I don't think that it is inconsistent; it seems entirely possible to me that these statistics are both correct.
Suppose there were 20 men and 20 women in a population. Suppose that 15 of the women have no sexual partners, and the other 5 women each have sex with all 20 men. In that situation, the median number of partners for the men is 5, and the median number of partners for the women is 0.
Since men and women will of course have the same mean number of partners, the fact that the median for women is much smaller than it is for men suggests that the distribution of number-of-partners for women is more skewed than it is for men. This is confirmed by other studies I've seen that reported the full distribution, rather than just the mean or median. A few female prostitutes who have sex with a very large number of men, raising the female mean a lot with virtually no effect on the female median, is certainly part of the cause of such a skew.
For more of the same look at Figure 6 on page 6. More inconsistency: when men and women are asked for the number of partners over the past 12 months, Tables 1 and 2, the numbers come out the same, within epsilon.
Again, I see no inconsistency. I can certainly construct data where this would be true, but the long-term median number of partners for men is greater than for women.
Is this an example of innumeracy in high places?
I think that at least some of the innumeracy is here on the math-fun list, where no-one seems to have noticed that in expecting the median for men and women to be the same, you are attributing to the median a property that is held by the mean, but need not be held by the median.
Andy Latto andy.latto@pobox.com
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