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Comment by OscarCunningham | Hacker News Reader

With fat-tailed distributions, increasing the sample size actually makes it more likely that you'll get one element much greater than the others.

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Yes, but if you have many fat tailed distributions, increasing the sample size makes it increasingly less likely that it's _the same person at the end each time_

This would imply that there's a single factor upstream leading to his exceptional ability in several measures downstream.

https://medium.com/@casciari/messi-is-a-dog-18b8f32766c8

https://www.youtube.com/watch?v=mDsztJOgqUg

TL;DR / TL;DW: Messi is a dog. The ball is his.

Recommended. For further context, it's the narration of an article written in Spanish by H. Casciari, translated to English.

It's a lovely but also profound literary article in the subject of Messi never losing sight of the ball, while also a critique on modern day analysis of the game and its influence on player behavior.

In Spanish: https://www.youtube.com/watch?v=Mg9fDOvE80g

Isn't the surprising part that it's fat-tailed rather than normal?

Why would it be? All careers with leverage seem to follow a Pareto distribution. It would surprise me if any sport, art, or creative endeavor was normally distributed.

You're right. I was thinking the "initial talent" might be normal in the way IQ scores are, but the parent was talking about actual performance, where history, training, etc all come into play.

And empirically, you have Gretzky in hockey, Jordan in basketball....

IQ is normally distributed because it's graded on a curve to make that true by definition. I'm not sure how it's distributed on any sort of 'natural' scale.

No, it's scaled but linearly and not redistributed.

There's still studies of relatively raw distributions and it still looks normal. The biggest deviations from normality are due to an excess mass in the lower range due to disease, and a tendency for scores to spike a tiny bit at certain numbers, probably due to people administering the test fudging a bit sometimes for various reasons.

What would such a natural scale look like?

I'm reminded of the paper 'Measuring universal intelligence: Towards an anytime intelligence test' https://www.sciencedirect.com/science/article/pii/S000437021....

Thank you for the pointer.

Maybe a cross-species IQ score would be interesting. Chimpanzees and corvids both can solve many IQ-test problems, but also are not experiencing the Flynn effect as far as we know. So we could measure IQ in multiples of a chimp's IQ, just like we measure cars in multiple's of a workhorse's sustainable pulling output. (Which is actually a less objective measure, when you think about it; we were breeding workhorses to increase that very target at the time the concept of "horsepower" was invented!)

That would be a great start. And it would also force us to acknowledge that a lot of the things that we are doing with these animals are utterly un-ethical.

I don’t even think initial talent is normally distributed. Take Usain Bolt, for example, he’d be a far outlier in a Pareto distribution for sprinters just on the basis of his natural heritage.

I suppose the distribution of aptitude would be a normal distribution in a world where careers were assigned randomly at birth and advancing in your career was on a strict seniority basis. So, for example, you'd see it in career paths that are mostly followed as generational family businesses.

keep in mind that even if football ability were (untrainable and) normally distributed, the distribution of ability among international players would be taken from one of the tails of that bell.

Also a good point. Do you know what the exact distribution would be when sampling randomly and rejecting all values below some threshold? You can tell it would at least be Pereto-ish in general shape.

I can't do much with https://en.wikipedia.org/wiki/Truncated_normal_distribution but maybe you can?

This sounds like nonsense. We're talking about multivariate distributions, and you haven't defined a norm by which ordinal comparisons can be made between sample points.

All norms on finite-dimensional vector spaces are equivalent so the choice of norm shouldn’t matter for tails really.

Norms are equivalent in the sense they differ by less than a constant in both directions, not in the sense that outliers are the same in all norms.

Norms are for dealing with magnitude not direction (and were brought up by my parent commenter). If you care about direction specify an angle, quadrant, cone or other subregion that allows you to take the limit to infinity which then doesn’t depend on the norm. Note this is the same in the univariate case where we talk about left and right tail if we need to distinguish.

> All norms on finite-dimensional vector spaces are equivalent

How are you going to tell me that an L1 norm is equivalent to an L2 norm, for example?

This is a standard result which you can look up in most text books, or here on StackExchange: https://math.stackexchange.com/questions/57686/understanding...

In the end it doesn’t whether we go to infinity in one norm or the other.

Note that I am talking about finite dimensions, so I guess you didn’t mean the L^p norms or \ell^p for integrable functions or sequences but the finite-dimensional p-norms.

This theorem is completely irrelevant - the equivalence relation described by the theorem does not imply an equivalence between ordinal relationships imposed by different choice of norm. Also, "tails" isn't the same as "at infinity".

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