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Comment by KK7NIL | Hacker News Reader

The problem is that intellectual productivity is generally not possible to measure directly, so you instead end up with indirect measurements that assume a Gaussian distribution.

IQ is famously Gaussian distributed... mainly because it's defined that way, not because human "intelligence" (good luck defining that) is Gaussian.

If you look at board game Elo ratings (poor test for intelligence but we'll ignore that), they do not follow a Gaussian distribution, even though Elo assumes a Gaussian distribution for game outcomes (but not the population). So that's good evidence that aptitude/skill in intellectual subjects isn't Gaussian (but it's also not Pareto iirc).

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All polygenic traits would be Gaussian by default under the simplest assumptions.

E.g. if there are N loci, and each locus has X alleles, and some of those alleles increase the trait more than others, the trait will ultimately present in a Gaussian distribution.

i.e. if there are lots of genes that affect IQ, IQ will be a Gaussian curve across population.

Very interested point, this is a close corollary to the central limit theorem, no?

Doesn't this assume a linear relationship between relevant alleles and the given trait though?

The missing assumptions are that the number of genes is large, independently distributed (i.e. no correlations among different genes), and identically distributed. And the whopper: that nurture has no impact.

You can weaken some of those assumptions, but there are strong correlations amongst various genes, and between genes and nurture. And, one "nurture" variable is overwhelmingly correlated to many others: wealth.

Unpacking wealth a little, for the sake of a counterexample: one can consider it to be the sum of a huge number of random variables. If the central limit theorem applied to any sum of random variables, it should be Gaussian, right? Nope, it's much closer to a Pareto distribution.

In summary: the conclusion of the central limit theorem is very appealing to apply everywhere. But like any theorem, you need to pay close attention to the preconditions before you make that leap.

"Number of genes is large" is what I said, that's not a missing assumption, I said that explicitly.

The nurture/nature relationship to IQ has been well-studied for many decades. There are easy and obvious ways to figure this out by looking at identical twins raised in different homes, adopted children and how much they resemble their birth parents vs adopted parents, etc. Idealists always like to drag out nurture effects on IQ like it's some kind of mystery when it's a well-studied and well-solved empirical question.

It easily includes nature impact for the same reasons: an incredible amount of nuture items are both Gaussian distributed and the population sampled is large.

Wealth being distributed as Pareto would imply its effects on nuture are not Pareto since the effects of wealth are not proportional to wealth. At best there’s diminishing returns. Having 100x the wealth won’t give 100x intelligence, 100x the lifespan, etc. And once you realize this, it’s not far till the math yields another Gaussian.

It does. A lognormal distribution would model that better which gives a nice right tail so maybe it is a useful toy model.

A long right tail Gaussian fits the Elo ratings of active chess players very well, as I discussed in adjacent comments here.

Isn't that just because there is a practical limit to how bad at chess someone can be? That is to say, making utterly random moves.

But there is no limit to how good they can be.

So of course the right tail is longer; the left tail is cut off!

Do you have a reference for Elo ratings not being Gaussian? A casual search shows lots of graphs and discussions saying it is.

Look at my reply to bhouston.

Elo ratings for active players are close to Gaussian, but not quite, they show a very clear asymmetry, especially for OTB old school Elo (compared to online Glicko-2).

The active players restriction is a big one and one I didn't assume I in my original statement.

> so you instead end up with indirect measurements that assume a Gaussian distribution.

100%. I was going to write something similar.

> If you look at board game Elo ratings (poor test for intelligence but we'll ignore that), they do not follow a Gaussian distribution, even though Elo assumes a Gaussian distribution for game outcomes (but not the population). So that's good evidence that aptitude/skill in intellectual subjects isn't Gaussian (but it's also not Pareto iirc).

Interesting, yeah, Elo is quite interesting. And one can view hiring in a company as something like selecting people for Elo above a certain score, but with some type of error distribution on top of that, probably Gaussian error. So what does a one sided Elo distribution look like with gaussian error in picking people above that Elo limit?

Lichess has public population data (they use a modified version of Glicko-2 which is basically an updated version of Elo's system): https://lichess.org/stat/rating/distribution/blitz

It's basically a Gaussian with a very long right tail.

Big caveat here is that these are the ratings of weekly active players. If we instead include casual players, I suspect we'd have something resembling a pareto distribution.

The big caveat is that it's trivial to measure the AIC, BIC and other quality of fit measurements for a distribution. If you think it's so and so distribution, go for it. In my experience in this specific case of chess rankings and in the broader case of test scores, skew-normal and log-normal have worse fits than plain Guassian.

I have no idea why you would believe increasing the population would make this Gaussian distribution look Pareto, when the exact opposite is true - increasing populations make things look more Gaussian - in all natural circumstances.

I was conjecturing that the distribution would be closer to Pareto for everyone (including people who've never learned how to play chess), hence why I said that "active players" is a big caveat.

> increasing populations make things look more Gaussian - in all natural circumstances.

This is just not the case, there's plenty of "natural circumstances" where populations have non-Gaussian distributions.

Perhaps you meant a specific type of population, like chess ratings? I'd be interested in seeing what you find there, but all I've found shows significantly distorted tails (not to mention a skew from 1500).

Good question - do the bad players play less because they are bad, or are they bad because they play less?

> Good question - do the bad players play less because they are bad, or are they bad because they play less?

Both for sure. If you don't practice you will never rise much about bad. But if you are bad and not progressing you won't play much because it isn't rewarding to lose.

One needs to almost figure out those with low ELO ratings, what is their history compared to the number of games played and see if they were following an expected ELO progression.

I wonder if you can estimate with any accuracy where a player will eventually plateau given just a small-ish sampling of their first games. Basically estimate the trajectory based on how they start and progress. This would be interesting. Given how studied Chess is, I expect this is already done to some extent somewhere.

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