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Comment by hughes | Hacker News Reader

Smaller countries can also get disproportionate representation in the olympics. When there's one team or competitor per country, the smallest country has the highest per-capita representation. While they have a smaller pool of athletes to draw from, I'd bet it changes the potential outcome where a straight up per-capita medal count is also somewhat misleading.

I wonder what medals per competitor per capita would look like?

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Small samples have high variance, while sample mean approaches the mean of the true population as you increase the sample size.

This is a fun website, but the Olympics would need a lot more events for it to mean something. If each athlete were only allowed to compete in a single event, then it might be insightful for countries at least as successful at capturing medals as, say, Great Britain.

I'll be Phelps' hometown has a crazy medal count per capita.

They also have more variability. A tiny nation that gets one medal gets a huge per-capita frequency. You'd want to regularize it a bit.

Definitely. A similar example: at one point studies started coming out that showed the best performing schools were small. Cue a lot of noise on how small schools are better, if you want your kids to be successful, send them to a small school, etc. I think these findings even showed up in government reports/recommendations.

Problem is, the worst performing schools were also small. Almost all of the effect could be explained by higher variance.

Unfortunately I don't remember where where I read about this, wish I could provide sources.

EDIT: found it. It wasnt government, it was the Gates foundation, who spent billions into supporting smaller schools.

http://press.princeton.edu/chapters/s8863.pdf

Oh my God. They wasted a billion dollars and made education worse because they didn't understand variance.

This is a beautiful article. Though I've read a number of reports that the higher male variance isn't all it was thought to be, but can't find it again. There should be done interesting studies about the variance in different species whose gender differentiation is done by a different mechanism, namely where the female has the two different chromosomes.

While the research wasn't performed by the government, I think its the basis for the move toward small "schools-within-schools" as an educational "reform" in a number of public schools.

How would one regularize to account for such variability (in a way that doesn't discount and favor the performance of small and large countries respectively)?

Genuine question as I'm admittedly not very familiar with the mathematical/statistical aspects of this. I am however mildly satisfied with my home country's per-capita performance in this and prior summer olympics, and curious to learn if/why I maybe shouldn't be :)

You could use a Bayesian model to incorporate prior information about the distribution of global medal rates into a conclusion (a posterior) about a particular country. This explanation will be simple and empirical, but ideally you'd integrate over all country-level data and the prior at once.

Count data is nice to model with a Poisson distribution, and a conjugate Gamma prior distribution. We'll assume a lot of things about the distribution of medal rates and medals which probably aren't true in reality.

For the global prior, eyeballing the medals-per-capita list for all countries, it has mean value around 3E-7 and standard deviation about 5E-7. That corresponds very roughly to the distribution Gamma(0.25, 1E6) for the prior. A Gamma with parameters (a, b) has mean value a/b and standard deviation sqrt(a)/b, so I just matched the numbers.

For a country with N medals and P population, we want to update our posterior distribution of their medal-per-capita rate based on a Gamma(0.25, 1E6) prior and a Poisson distribution. The updated posterior distribution is conveniently, Gamma(0.25+N, 1E6+P). (That's the conjugacy property).

So Grenada's medal rate, with N=1, P=100,000, has posterior distribution Gamma(1.25, 1.1E6). This has mean value 1.25/1.1E6 = 1.1E-6. Compare that mean value to the raw medal-per-capita rate of 1E-5 - in other words, if this model is to be believed, we should heavily discount Grenada's great performance. On the other hand Australia's medal rate barely changes from 9.2E-7 to 9.7E-7 by using the prior, close to Grenada's Bayesian rate. So it takes a larger population into account quite naturally.

There's a few things there too, consider for every Grenada with one medalist how many small countries have none which can kind of add to that pool of participants.

The other thing is that in most sports each country is limited to the same number of participants. In swimming for example if the US had the 4 of the best 10 swimmers in an event only two get to compete at the Olympics which leaves them at a slight disadvantage.

mmm, I think we should get rid of the variance and filter out populations under a million.

so, New Zealand on top! sounds good :)

I think 10 million is the clear winner. Any thing less than that is clearly not even a country, more a large city. And cities can't perform in the Olmypics. So who does that put on top?

Aussie! Aussie! Aussie!

Bending the rules until we win since before bowling under arm was officially banned.

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