And what I find frustrating in this discussion is that people don't like to think in term of distributions. The 78k female aren't less capable in any way than the 120k males, there is just fewer of them. But then a journalist will summarize that as "women scoring less than males in math at SAT". And then we go into a sterile flame war of blanket statements about gender, etc. The debate on the Damore memo in that respect was depressing, particularly from educated engineers who should know better.

As a mathematical comment, if these are two distributions are gaussians, and one has 120k above some cutoff, the other 78k, then the distributions above this cutoff are not the same.

They will typically have slightly different means (not far above the cutoff) and quite strongly different tails.

This is a great point and I wish these kinds of discussions always had someone to point out that kind of important detail. I think the parent's idea would be better expressed by saying that we can't say anything about individuals in either population without more information. You don't know where in the tail that individual lies. Assuming a continuous distribution it could be anywhere. So while it might be more likely for a random individual from population A to outperform a random individual from population B, there is no way to know a priori if it will happen for specific individuals. Obvious, I know, but it's the kind of thing that people often miss when they get into a heated discussion.

Being that someone appears to be my regular job :)

It's absolutely true that, if all you know about an individual is membership of one of these populations, then you know very little. Any new information about the actual person will quickly swamp this signal. If you've actually met A & B (or seen their work) then group membership is irrelevant. This is the situation in hiring people.

But it's also true that these distributions lead you to make predictions about the tails. While any one individual is vanishingly unlikely to win a Nobel prize, you can predict that (if they are fairly awarded) there should be a huge slant in how many go to each group. And I really mean huge, this ratio (if things are Gaussian, etc) grows exponentially in how far along the tail you go. Seeing such a skew is thus no evidence that hiring/prize-awarding is biased.

Right, but if you define the skill as "good enough", i.e. score of 700 and above, then my point stands.

My point is really that individuals aren't defined by the distribution, just the number of individuals is defined by the distribution. But many people take these distributions personally.

Edit, too late: of course 120:78 is not a 2:1 ratio; males are outnumbering females better than 3 to 2.

Would the drop out rate of men (assuming higher) provide an additional control? If low achieving men selected out of the data set before it was collected at the SAT level, there could be a long tail of underachieving men. This would certainly effect averaging, and if number of dropouts were greater than the 90th percentile, there could be some selection/survivor bias in play on how males appear as a group.

From some quick googling, it sounds like less than half of high-school students take the SAT. In which case the bottom end of the scores will probably tell you nothing about the population.

"3.8 million students took the PSAT -- the practice version of the SATs ... 1.7 million students from the class of 2015 took the SAT"

If we want to look at the long tail of intellectual ability, the opposite side would be intellectual disability. Either measured by IQ, diagnosed as neurodevelopmental disorders or now days more commonly both.

Both side of the bell curve has poor definitions and IQ tests are far from perfect as it is said to optimize the accuracy between IQ 70 and 100. That said, we do see more men in practically every kind of type of data set for both side of the bell curve.

To cite a professor of both biology and neurology, no society on earth like to talk or even think about the mentally disabled. The "downsides" of greater variability is a group that most people treat as they don't exist, so I would not be surprised if many people really believed that the only explanation for a different top 1% is discrimination.

Dropout rate is not relevant, because we're only looking at high achievers and we believe their dropout rate is negligible. There are roughly equal numbers of boys and girls graduating high school (technically, there are more girls!), but there are many more high-achieving boys than girls. Boys are much more likely to excel.

This is not a statement about the group average, but about the group variance. The SAT is not a good tool to use to assess group averages; I already mentioned this.