Friday, October 27, 2006

Tail effects   posted by the @ 10/27/2006 05:49:00 PM

There's another article in Science about women and science.

It appears to consist predominately of (1) rebuttals to straw-men arguments and (2) Lewontin-like claims that we're all the same despite our differences. A great deal of the text deals with describing (without much detail) male-female differences on a variety of criteria.

The magnitude of each gender difference was measured using the d statistic (6), d = (MM - MF)/sw,where MM is the mean score for males, MF is the mean score for females, and sw is the pooled within-sex standard deviation. The d statistic measures the distance between male and female means, in standard deviation units.

They list a variety of metrics on which the sex-difference (measured in d) is small. They fail to mention the male advantage in spatial ability, but do mention the male advantage(?) in aggression. While focusing on differences in measures of ability among children, they relegate discussion of tail effects to the supplemental online text. There they mention tail effects as an effect of differences in variance, but ignore the fact that mean differences also cause tail effects.

Rather than dig any deeper into this paper, I will present what they chose to ignore: the theoretical effects of small differences in mean and variance between males and females will produce large differences at the tails of a normally distributed trait.

This table presents the percentage of females above a +3 SD threshold as various differences in mean (pink) and SD (orange) in standardized units units. Thus, if the mean and SD are equal (0,0) then women make up 50% of the population above 3 SD on this imaginary trait. But if d=0.3 and males have a variance that is 0.06 SD units greater than women, then the female percentage above 3 SD will be 17.1%. A d of 0.3 is labeled "small", and an SD difference of 0.06 (women SD = 0.97, men SD = 1.03) would be hard to establish in small samples. Nonetheless, it would produce exactly the kind of large differences in male:female ratios among the most talented individuals that we observe in math-heavy disciplines.

 3 0.00 0.02 0.04 0.06 0.08 0.10 0 50.0% 45.1% 40.3% 35.6% 31.2% 27.1% 0.05 45.9% 41.1% 36.4% 31.9% 27.8% 24.0% 0.1 41.9% 37.2% 32.7% 28.5% 24.6% 21.1% 0.15 37.9% 33.4% 29.2% 25.2% 21.7% 18.5% 0.2 34.1% 29.9% 25.9% 22.3% 19.0% 16.1% 0.25 30.6% 26.5% 22.9% 19.5% 16.6% 14.0% 0.3 27.2% 23.5% 20.1% 17.1% 14.4% 12.1% 0.35 24.1% 20.6% 17.6% 14.9% 12.5% 10.4% 0.4 21.2% 18.1% 15.3% 12.9% 10.8% 9.0% 0.45 18.6% 15.8% 13.3% 11.1% 9.3% 7.7% 0.5 16.2% 13.7% 11.5% 9.6% 8.0% 6.6%