I think usually it boils down to teachers in advanced schools were the kind of pupils who got most things easily and have no clue how to explain things more than <list of set comprehension>. Got me to think that these universities are more magnets from strong ex-pupils to similar minded new pupils that will fit in without much pedagogical efforts.
The average IQ of math majors is absurdly high, more than two sigma above the mean[1]. In an organization like that the smartest person is likely going to be a breathtaking five sigma above the general population mean[2]. I get the impression that people in math departments like it that way, so weeding out persons of more ordinary intelligence is more of a feature than a bug. I doubt many would admit to that, but it fits the observables.
[2] Assuming gaussian distribution with no skew, which is probably a bad assumption, but I'm spitballing here. Suffice to say math departments at elite colleges have many extremely bright people.
I wouldn't say feature, but they're probably not naturally inclined toward spending more time raising the average level instead of enjoying the few genius newcomers.
There's a small sadness in this but I can't word it out perfectly.
ps: I mean I would understand very much that these people just want "new buddies to play hyperstimulating math games with no drag whatsoever" but from school I expect just a few pointers. The rest is on me (us).
IQ has no skew by definition. That said, I think there are a number of IQ boosting mutations and innate mathematical aptitude is likely correlated to possession of one or more of these rare(ish) mutations.
I misread the point they were trying to make, I took the 5 std to be a sign they were pulling from general population IQ. On second reading it’s clear that is not their intended point.
In any case the resulting intermediate distribution from selection criteria bias would be a Chi distribution (I think - it has been a while.) 5 standard deviations on a normal distribution is 175 IQ or top 1/700K which is insane.
A few additional biases, Physics majors have higher average IQs, and not everyone smart enough to do the math has the desire or the opportunity.
The point I was making is that IQ score has been normalized, when imagining the intermediate distribution of IQ it would make more sense to look at composite probabilities of factors that led to the IQ score and consider the probability that those same factors lead to a sufficient innate ability in math.
Let's say every mathematician draws an IQ from the IQ distribution and a luck score from the luck distribution. Then, the highest sum of numbers wins a Fields medal.
That would be distributed according to the distribution of the maximum of N Gaussian-distributed random variables, and according to this[0] stack overflow answer, it's skewed downwards, below its mean. (Its mean, of course, is much higher than the mean IQ+Luck).
The more selection criteria depends on cut-off (which it is in the university setting), the more likely selection from normal distribution is going to be skewed. Then Fields medalists are selected from long tail of already skewed population, all bets are off.
