# Medals per capita is biased

By John Pickering 14/08/2012

The Games are over, let the analysis begin.

We’ve had some fun with ranking countries’ performance at the London Olympics according to medals per million (medals per capita) or medals per 100 billion of Gross Domestic Product (see my tables below).  As I predicted a few posts back, the medals per capita will be one by a country with very low population and few medals – Grenada is the winner here.  It seems obvious when we think about it that a country with a population of just 100,000 (0.1 Million) may end up with a very high medals per million score if they win just 1 or 2 medals (even though that is still a difficult feat).  What is not so easy to see is that countries with very high populations have a “limit” for their performance that is very much lower.  With just ~900 medals on offer and a population of over 1340 million China’s possible maximum medals per million score is just 0.67 (compared with Grenada’s 9000).  It is this breadth of this range of possible values that causes the bias in the ranking system.

I like to visualise data.  The two graphs below show the bias for the “Official Rankings” (you know, the ones that rank according to number of golds first, silvers second and bronzes third) and for the medals per capita.  The bias is obvious because the points on the graph are not scattered without any discernible pattern all over the graphs. The “Official Rankings” obviously are biased towards countries with greater populations, the Medals per capita is biased towards countries with lesser populations.  Obviously, dividing by population does not remove the bias, merely shifts the bias. Note, that the scales on the “y” axis are what we call “log scales”.  This enables us to see all the data more easily (ie countries with 100,000 and 1.3 billion can be displayed on one graph). What is not shown on the graph is the 122 countries ranked 80th equal who won no medals at all.

Later this week, once I am happy with my grant writing and get my head around some data I am trying to analyse I shall attempt to put together an equation which will better help us answer the important question of the day – “Which is the greatest olympic nation?”

Top graph: The Official Rankings verse Population (note the log scale).
Bottom graph: The Ranking of number of medals won per million population v population