Reading http://www.scientificamerican.com/article/how-to-raise-a-genius-lessons-from-a-45-year-study-of-supersmart-children/ and it talks about how the top 1% in intelligence are responsible for a lot of our science and culture. Some controversial previous discussion has focused on how differences in variance and means between different groups might affect questions like this; Larry Summers got fired over such discussion, for example.
The Wikipedia article https://en.wikipedia.org/wiki/Sex_differences_in_intelligence summarizes the situation.
The top 1% are 2.33 standard deviations to the right of the mean in a Gaussian distribution, which is 135 IQ in the overall population:
> qnorm(.99)
[1] 2.326348
> qnorm(.99, mean=100, sd=15)
[1] 134.8952
> pnorm(135, mean=100, sd=15)
[1] 0.9901847
What if you have a subpopulation with a 10% smaller variance? It turns out to reduce the number of people in that subpopulation in the top 1% by more than a factor of 2:
> pnorm(135, mean=100, sd=15*.9)
[1] 0.9952372
What if instead you have a subpopulation whose mean is shifted down by one standard deviation, without changing the variance — an effect size of 1? This reduces the number of people in that subpopulation in the top 1% by a factor of 20:
> pnorm(135, mean=85, sd=15)
[1] 0.9995709
This is true even though the probability is quite reasonable that an event from this group will be higher than an event drawn from the entire population.
The situation becomes more extreme as you go to more extreme quantiles. Consider the top 0.01%:
> qnorm(.9999)
[1] 3.719016
> qnorm(.9999, mean=100, sd=15)
[1] 155.7852
> pnorm(156, mean=100, sd=15)
[1] 0.9999055
> pnorm(156, mean=100, sd=15*.9)
[1] 0.9999832
> pnorm(156, mean=85, sd=15)
[1] 0.9999989
That is, at IQ 156 and above, where we find 0.01% of the population, a hypothetical population with a 10% lower standard deviation will be underrepresented by almost 6:1, and a hypothetical population with a 1-SD-lower mean will be underrepresented by more than 90:1; getting the same result by tweaking the variance requires a 21% smaller standard deviation, 11.9 IQ points.
A hypothetical group with a 1-SD-lower mean but also a 21% larger standard deviation (of 19 IQ points) matches and slightly exceeds the overall population at this level:
> pnorm(156, mean=85, sd=15/.79)
[1] 0.9999077
In this context it is worth pointing out that tutoring improves student performance by two standard deviations (Anania (1982, 1983) and Burke (1984)): http://changelog.ca/quote/2012/09/23/tutoring_two_sigma.
So what if we have a hypothetical group with an unchanged variance but a mean two standard deviations higher?
> pnorm(156, mean=130, sd=15)
[1] 0.9584818
If student performance is equivalent to IQ, 4.2% of them will exceed the performance of 99.99% of the rest of the population. This is a larger difference than has ever been suggested for (mean or variance) intelligence differences due to race or sex.
(In the paper described, tutoring also substantially reduced the variance of achievement scores, but this is probably because the achievement scores had a ceiling; all three of the distribution curves from the experimental data intersect at the X-axis on the right, and the “tutorial” group’s distribution is noticeably skewed to the left.)