## The Olympics and Power Law Distributions

22 08 2008

I find amusing all the discussion around ranking countries in the Olympics Medal Standings based on the overall total or the number of golds. This may be relevant for China or the US, as holding the top position is a strong statement in world sports dominance. In the case of Brazil and Canada, as of this writing, it may mean a jump from #26 to #16 and from #17 to #13, respectively, on the stands, which may look like a big deal, but in a cold analysis, you’re just seeing a Power Law distribution effect, the math pattern behind the long tail.

When you are in the long tail, you’re merely comparing peanuts. One extra gold medal may make you go up several positions, but a jump from 40th place to 20th does not mean that you improved 100%. Using the gold-medal-first rank, Brazil was #52 in Sydney (2000) and #16 in Athens (2004) and Canada #21 and #24. The variation there does not mean that those countries became much better or worse in a 4-year span. It just means that they both are in that majority where sports excellence is the exception, not the rule. Nothing to be ashamed of.

Our brains are used to normal distributions and linear relationships and we tend to interpret logarithmic relationships in a linear way. I remember a speaker making a joke about a supposedly dumb statement by a US presidential candidate around the lines of “silly person was astonished to learn that half of the US population was below average in performance criteria X”. The underlying assumption was that “average” always marks the middle point of a distribution. Of course, that only occurs in perfectly normal distributions, with mirrored tails on both ends.

Inspired by Clay Shirky in his excellent book “Here comes everybody”, I plotted the medal stands and got the following curve:

The speaker above was probably thinking about median, not average. Start paying attention to published stats around you, and you will notice how often numbers are over-extended, converting subtle differences in absolute rankings. I think I mentioned this in a previous post: numbers don’t lie, but they can easily mislead.