Depends entirely on the distribution around the mean, not just the mean.
Of course.
Let's do a super oversimpified demonstration. Imagine we only have two cohorts, A and B.
We can just use the
2023-24 data from AAMC instead of oversimplified made up numbers.
A has a mean of 50, with a standard deviation of 25. That gives us about 2/3 of the cohort falling between 25 and 75.
B has a mean of 55, with a standard deviation of 10. That gives us 2/3 falling between 45 and 65.
Your hypothetical has difference in μ being significantly bigger than difference in σ. But the MCAT distributions are not like that.
For applicants, Asian MCAT is at a mean of 509.1 with a σ of 9.3.
Black applicants have MCAT at a mean of 497.5 with a σ of 10.0. I.e. standard deviations are similar, but means are not.
If the threshold is set at 60, you will end up with a higher percentage of cohort A being above the threshold than of cohort B, even though B has a higher mean. The same threshold can produce different results based on the standard deviation.
But that is because your distributions have different σs while having similar μs.
Let's apply that to actual data. If we set 500 as cutoff and assume normality, 83.6% of Asian applicants will be above the cutoff, but only 40.1% of blacks will. That's less than half! If we increase the cutoff to 505, 67.0% of Asians are above the cutoff, but only 22.7% of blacks are. That's only about a third. So proportions will change, but the order is not going to flip given these parameters.
Now if you lowered that threshold to 50, you would flip that result - you'd end up with quite a bit more of B in your result than A.
ETA: If you were morbidly inclined, you might consider it score gerrymandering - adjust the aggregate threshold until you get a mix that suits your objectives.
As I have shown, that does not really work unless means are really close and st. devs. are significantly different.
Which is why the admins resort to shady practices like Harvard College assigning Asians poor personality scores in order to limit their numbers.