Small Differences in Variability of Ability Translate Into Big Differences 3-4 Std. Deviations from Mean
Results of a statistical experiment, to follow up on these two CD posts(link and link):
1. Generate a random sample of 1,000,000 observations in EVIEWS, distributed normally with zero mean, and variance of 1, representing female mathematical ability on a standardized test.
2. Generate a sample of 1,000,000 observations, distributed normally with zero mean, and variance of 1.21, to represent male mathematical abililty, with greater variability according to the table above for Grade 8.
3. Then look at the upper tail of each distribution and compare the M/F ratio for the "super-genius" level, many standard deviations above the mean.
4. For 3 standard deviations above the mean, the M/F ratio is 2.4 (3,188 "males" vs. 1337 "females").
5. For 4 standard deviations above the mean, the M/F ratio increases to 3.8 (111 males to 31 females).
6. For 5 standard deviations above the mean, there are 3 males and o females.
Assume that to be successful and get tenure in the math department at Harvard or MIT, you have to be 3-4 standard deviations above average, which is what Larry Summers said - "We're talking about people who are three and a half, four standard deviations above the mean...."
In that case, wouldn't we expect females to be under-represented, as the experiment above demonstrates, where "men" represent 77% of those observations 4 or more standard deviations above the mean?
See Alex Tabarrok's related discussion here on Marginal Revolution.