Summers Vindicated (again)

For the past week or so the newspapers have been trumpeting a new study showing no difference in average math ability between males and females.  Few people who have looked at the data thought that there were big differences in average ability but many media reports also said that the study showed no differences in high ability.

The LA Times, for example, wrote:

The study also undermined the assumption -- infamously espoused by former Harvard University President Lawrence H. Summers in 2005 -- that boys are more likely than girls to be math geniuses.

Scientific American said:

So the team checked out the most gifted children. Again, no difference. From any angle, girls measured up to boys. Still, there’s a lack of women in the highest levels of professional math, engineering and physics. Some have said that’s because of an innate difference in math ability. But the new research shows that that explanation just doesn’t add up.

The Chronicle of Higher Education said:

The research team also studied if there were gender discrepancies at the highest levels of mathematical ability and how well boys and girls resolved complex problems. Again they found no significant differences.

All of these reports and many more like them are false.  In fact, consistent with many earlier studies (JSTOR), what this study found was that the ratio of male to female variance in ability was positive and significant, in other words we can expect that there will be more math geniuses and more dullards, among males than among females.  I quote from the study (VR is variance ratio):

Greater male variance is indicated by VR > 1.0. All VRs, by state and grade, are >1.0 [range 1.11 to 1.21].

Notice that the greater male variance is observable in the earliest data, grade 2.  (In addition, higher male VRS have been noted for over a century).  Now the study authors clearly wanted to downplay this finding so they wrote things like "our analyses show greater male variability, although the discrepancy in variances is not large."  Which is true in some sense but the point is that small differences in variance can make for big differences in outcome at the top.  The authors acknowledge this with the following:

If a particular specialty required mathematical skills at the 99th percentile, and the gender ratio is 2.0, we would expect 67% men in the occupation and 33% women. Yet today, for example, Ph.D. programs in engineering average only about 15% women.

So even by the authors' calculations you would expect twice as many men as women in engineering PhD programs due to math-ability differences alone (compare with the media reports above).  But what the author's don't tell you is that the gender ratio will get larger the higher the percentile.  Larry Summers in his infamous talk, was explicit about this point:

...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...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.

If you do the same type of calculation as the authors but now look at the expected gender ratio at 4 standard deviations from the mean you find a ratio of more than 3:1, i.e. just over 75 men for every 25 women should be expected at say a top-25 math or physics department on the basis of math ability alone (see the extension for details on my calculation).  Now does this explain everything that is going on?  I doubt it.  As Summers also pointed out it takes more than ability to become a professor at Harvard and if there are variance differences in characteristics other than ability (and there are) we can easily get a even larger expected gender ratio.

Does this mean that discrimination is not a problem?  Certainly not but we need the media and academia to accurately present the data on ability if we are to understand how large a role other issues may play.

Addendum: Andrew Gelman points out that perhaps alone among the media, Keith Winstein at the WSJ reported the story correctly.

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The authors show variance ratios of 1.11 to 1.21, I take a VR of 1.16.  If we set the female variance to 1 this implies the standard deviation for female ability is 1 and for male ability 1.077.  Using an online calculator for the Normal distribution you can find that given their standard deviation .0102% of males have ability of 4 or greater (4 female sds) but given their sd only .0032% of females can be expected to have the same level of ability, thus a gender ratio of 3.18.

Note that we are assuming that mathematical ability is normally distributed - we know the data fit this distribution around the mean but we don't know much about what happens at the very top.