### 2.18.2005

## Issues of intrinsic aptitude hamper Larry Summers's critics

Here's (part of) what Larry Summers actually said:

Naturally, in this New York Times piece the authors get from "there's a bigger spread in the mathematical abilities of men than there is in women" to "women aren't as good as men".

Granted, Larry Summers's evidence for his proposition consist of one study of twelfth-graders and a bunch of back-of-the-envelope calculations, but it's clear that he's put some amount of thought into his theory. Just looking at a table of values for the cumulative distribution function N, N(-2) = 0.0228 and N(-1.6) = 0.0548. So a 20% difference in the standard deviations for men and women would lead to approximately a 2-1 ratio in the number of men to women within the top 4% of the population or so. If we hypothesize that a man needs to be 2.5 standard deviations above the norm to be a successful scientist, we have N(-2.5) = 0.0062. But since 2.5 male standard deviations would equal 3 female standard deviations, we have N(-3) = 0.0013. Voila, a 5-1 ratio in the number of men to women.

My own limited evidence might undercut the differences-in-variability theory; I looked at my most recent multivariable class and calculated about the same average and standard deviation in grades for men and women. However, that's for a preselected class of students whose abilities lie in a specific range, so that isn't completely compelling evidence either. The point is that (1) Summers's theory (which surely is not original) deserves serious, dispassionate inquiry, and that (2) a fair number of Summers's critics either suffer from innumeracy or are willfully misinterpreting his thesis. But I guess nowadays it's not possible to make comments about an aggregate of people without people thinking that it applies to them as individuals.

One final (facetious) thought: the flip side of Summers's theory is that men are also five times as likely as women to be dumb as rocks (at least mathematically). Why is there not a humongous kerfluffle about this insinuation?

The second thing that I think one has to recognize is present is what I would call the combination of, and here, I'm focusing on something that would seek to answer the question of why is the pattern different in science and engineering, and why is the representation even lower and more problematic in science and engineering than it is in other fields. And here, you can get a fair distance, it seems to me, looking at a relatively simple hypothesis. It does appear that on many, many different human attributes-height, weight, propensity for criminality, overall IQ, mathematical ability, scientific ability-there is relatively clear evidence that whatever the difference in means-which can be debated-there is a difference in the standard deviation, and variability of a male and a female population. And that is true with respect to attributes that are and are not plausibly, culturally determined. If one supposes, as I think is reasonable, that if one is talking about physicists at a top twenty-five research university, one is not talking about people who are two standard deviations above the mean. And perhaps it's not even talking about somebody who is three standard deviations above the mean. But it's talking about people who are three and a half, four standard deviations above the mean in the one in 5,000, one in 10,000 class. Even small differences in the standard deviation will translate into very large differences in the available pool substantially out. I did a very crude calculation, which I'm sure was wrong and certainly was unsubtle, twenty different ways. I looked at the Xie and Shauman paper-looked at the book, rather-looked at the evidence on the sex ratios in the top 5% of twelfth graders. If you look at those-they're all over the map, depends on which test, whether it's math, or science, and so forth-but 50% women, one woman for every two men, would be a high-end estimate from their estimates. From that, you can back out a difference in the implied standard deviations that works out to be about 20%. And from that, you can work out the difference out several standard deviations. If you do that calculation-and I have no reason to think that it couldn't be refined in a hundred ways-you get five to one, at the high end. Now, it's pointed out by one of the papers at this conference that these tests are not a very good measure and are not highly predictive with respect to people's ability to do that. And that's absolutely right. But I don't think that resolves the issue at all. Because if my reading of the data is right-it's something people can argue about-that there are some systematic differences in variability in different populations, then whatever the set of attributes are that are precisely defined to correlate with being an aeronautical engineer at MIT or being a chemist at Berkeley, those are probably different in their standard deviations as well. So my sense is that the unfortunate truth-I would far prefer to believe something else, because it would be easier to address what is surely a serious social problem if something else were true-is that the combination of the high-powered job hypothesis and the differing variances probably explains a fair amount of this problem.

Naturally, in this New York Times piece the authors get from "there's a bigger spread in the mathematical abilities of men than there is in women" to "women aren't as good as men".

Granted, Larry Summers's evidence for his proposition consist of one study of twelfth-graders and a bunch of back-of-the-envelope calculations, but it's clear that he's put some amount of thought into his theory. Just looking at a table of values for the cumulative distribution function N, N(-2) = 0.0228 and N(-1.6) = 0.0548. So a 20% difference in the standard deviations for men and women would lead to approximately a 2-1 ratio in the number of men to women within the top 4% of the population or so. If we hypothesize that a man needs to be 2.5 standard deviations above the norm to be a successful scientist, we have N(-2.5) = 0.0062. But since 2.5 male standard deviations would equal 3 female standard deviations, we have N(-3) = 0.0013. Voila, a 5-1 ratio in the number of men to women.

My own limited evidence might undercut the differences-in-variability theory; I looked at my most recent multivariable class and calculated about the same average and standard deviation in grades for men and women. However, that's for a preselected class of students whose abilities lie in a specific range, so that isn't completely compelling evidence either. The point is that (1) Summers's theory (which surely is not original) deserves serious, dispassionate inquiry, and that (2) a fair number of Summers's critics either suffer from innumeracy or are willfully misinterpreting his thesis. But I guess nowadays it's not possible to make comments about an aggregate of people without people thinking that it applies to them as individuals.

One final (facetious) thought: the flip side of Summers's theory is that men are also five times as likely as women to be dumb as rocks (at least mathematically). Why is there not a humongous kerfluffle about this insinuation?