Tuesday, July 08, 2008

Summers part 29,476   posted by Herrick @ 7/08/2008 09:43:00 PM

Slate has been having a debate on sex differences. Along the way, they hit on a key Summers issue: The apparent higher male variability of math scores. Shaffer, the author, refers to the classic Feingold piece, a cross-cultural meta-study of the variability of mental abilities across genders. Shaffer makes the common claim that there are data on both sides--sometimes women have a higher variance, and sometimes the men do. But is the difference statistically significant?

I did a simple analysis of Feingold's data from 54 math tests from 20 countries, and 19 tests of spatial ability from 9 countries. I ran least squares and least absolute deviation tests.

Here are the p-values for the restriction that men and women have equal variability:

Math, least squares: p<0.1%
Math, least absolute deviation: p<5%
Spatial, least squares: p<10%
Spatial, least absolute deviation: p=11% (but only 19 observations!)

OK, so it's reasonable to conclude that men have higher variability in this cross country sample, and that the cases of greater female variability are just flukes. But are the differences quantitatively significant, not just statistically significant?

Feingold, the author of the study, says no. He notes: "The median V[ariance]R[atio] of 1.09 indicated greater male variability [on math tests]," then claims that "the magnitude of the gender difference was trivial." Not so. Excel will show you that three or four standard deviations above the mean--Larry Summers territory to be sure--that's enough to get you a 2-to-1 one ratio. And that's with no difference in means whatsoever.

This paper (Table 1, page 10) works out the rough gender ratios you'd expect to see under various assumptions for means and variances. The bottom line is no surprise: with small differences in means plus small differences in variance, you can get big results: 4 to 1 ratios are easy to come by, and 10 to 1 are plausible. Yes, yes, further research is needed, but most of the research is pointing in the same direction. And we Bayesians know what to do when research mostly points in one direction....

(Oh, and the median male/female variance ratio for spatial ability in Feingold's data is 1.14. And yes, none of this gets at genes v. culture. But let's start with the journalism before we head to causation.)

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