As I was watching the Jamaican sprinters sweep the 200m dash, I started wondering how such a relatively small and not wealthy by any stretch of imagination country could achieve such dominance. Is there a correlation between the population of the country and its haul of medals? Almost certainly. Perhaps a more incendiary and more interesting question is “does money buy medals?” Not in a literal sense, of course, but in a statistical sense. Is there a correlation between the per-capita medal count and per-capita income? There should be. Money buys equipment, coaching and medical staff, transportation, etc.
As I embarked on this project, I expected to find a significant positive correlation. But what I found was even more shocking. Medal count per person grows as the square of per-capita income. The graph below shows the medal count (obtained from http://www.london2012.com and a Wikipedia article) divided by the population of each country (obtained from Wikipedia) vs. the purchasing parity GDP per capita (obtained from the CIA column of this Wikipedia article). Only populous (>50,000,000) countries are included since statistical trends are more clear in large samples and the fluctuations that obscure these trends are smaller. The straight line is the quadratic fit.
Why does the medal haul grow faster than linearly with the resources? I have an explanation for this striking phenomenon which assumes that each sport has an entrance threshold and that the distribution of these entrance thresholds is roughly uniform. If some contry has a GDP per capita that is greater than the entrance threshold for a particular sport, it enters competition. It follows that the number of competitors is inversly proportional to the entrance threshold of a sport. I further assume that all competitors are equally likely to get a medal once they enter competition. Therefore the number of medals each competitor wins is inversely proportional to the number of competitors and consequently it is proportional to the entrance threshold of the sport. The final logical step is to notice that a country with per-capita income competes in all sports whose entrance thresholds are Thus the total number of medal is proportional to
Thus the quadratic dependence of the medal haul on the GDP per capita comes from the fact that richer contries enter more sports and it is easier to win medals in more expensive sports since not as many countries can enter.