Olympic medal tally

Australia media like to talk about the "medals per capita" medal tally during the olympics, while such a thing is hardly found in the US.  I've long thought that a real measure of success needs to be somewhere in between total medals and medals per person.  Talking to Jordan, I mentioned that one really needs to plot medals won against population, and look at the trend and any outliers from it, and promised to do so once the olympics ended.

Looking at the figures, I realised that it is better to work with total medals than golds only, simply because the numbers are a little too small otherwise so random fluctuations are large.

I only considered countries that won at least two medals, because otherwise the small countries that randomly pick up one medal every few olympics would be credited with more than their due the times that they do.

I looked at both population and GDP, and the latter was a better predictor of medal count, so I used that as my x value.  In fact, among countries with equal GDP, the less populous one is likely to win more medals.

So, here's a plot of medal count vs GDP (PPP) for all countries with at least two medals.  I excluded countries not listed in the GDP ranking I used: Cuba and North Korea (isolated communist countries) and Puerto Rico (not an independent country).  The axes are logarithmic.

It appears that there is a GDP-dependent upper bound on the number of medals a country can win, though there is no lower bound.  I found this bound by fitting a power law (a straight line on a log-log graph) through the top 10 countries (top 10 in the sense I describe below: the process was slightly iterative).  This bound becomes a target, the maximum number of medals one can expect a country to win.  I then ranked countries by what percentage of this target they achieved.

The top 20 are:
1 Russia 151%
2 Jamaica 144%
3 Great Britain 122%
4 Australia 95%
5 United States 90%
6 Hungary 87%
7 Kenya 86%
8 China 85%
9 Georgia 84%
10 Ukraine 83%
11 New Zealand 81%
12 Mongolia 78%
13 Germany 73%
14 Belarus 70%
15 Azerbaijan 70%
16 France 64%
17 Kazakhstan 64%
18 South Korea 61%
19 Netherlands 61%
20 Italy 57%

The conclusion for Australia: we're already achieving as highly as can be reasonably expected, and despite the AOC's best efforts we're not going to lift our medal count significantly at the next few games.  The most medals we could have managed was 37 and we got 35.  The only countries that beat their targets were Russia, presumably due to its history of valuing sporting excellence during the Cold War, Jamaica, with one supreme athlete who boosts the figures in a way difficult for a larger country, and the host nation.  As for the colour of the medals, the data are consistent with each country's medals being distributed randomly between gold, silver and bronze.  The exception is that it appears that the United States, China and Great Britain were good enough that they were 'saturating' in events, i.e. winning gold in events where they would also have won lesser medals had they been able to enter enough competitors (though for some reason Russia doesn't display the same effect).