Let's assume human talent for any given skill follows a statistical distribution (like a bell curve). So any random sample of a significant number of people would follow that distribution. Let's further assume that human talent is equally distributed across the planet. If it really was completely equal, then the people from one nation would form a random sample, and the distribution of talent within that country should conform to that same distribution as the world population. With me so far?
So, the point of the Olympics, it seems, is to
This brings up the topic of who "won" the Olympics. Typically, this is figured by gold medal count (since no one cares about silver or bronze). This year, the US "won" the Olympics by bringing home the most gold medals (46), compared with second-place China (38). But is this really the best indicator of "winning"? If skill was distributed equally across the globe, then you would expect those nations with more people to have more athletes at the top of the bell curve, and therefore more gold medals. So really, there's nothing unexpected about the top two gold medal winners also being the third and first most populous countries. This would even happen if, on average, Americans and Chinese fell below the curve on athletic talent.
With this in mind, let's take a look at the gold medal winnings. The Olympics web site has a chart of all medaling nations and how many medals each one won, comparing that with Wikipedia's list of countries by population allows us to calculate how many medals were won per 100 million residents of each country. This way, we can figure out which countries really did earn more medals than would be expected for equal talent distribution.
Sorted this way, we find the real winners of the Olympics:
