Punching above your weight

Tonight there's a programme on Prime about Paul Simon's Graceland. It's one of my favourite albums so I mustn't miss it.

The Olympics are finally over. To sum them up in two words: bloody great. I wish I could have caught the closing ceremony - I've seen a clip of it involving the Beatles' I Am the Walrus. What a song.

One phrase I hear a lot in New Zealand is "punching above our weight". Which countries did that in the Olympics? Whose performances were the most impressive when you take population into account? It's not as easy to measure as you think. Just dividing medals by population is misleading; it's much easier for a smaller nation to vastly outperform its expected medal quota than a larger nation. To see what I mean, imagine tossing 100 coins. Getting 60 heads is unusual but by no means impossible: you'll get 60 or more heads about 3% of the time. But if you toss 1000 coins, getting 600 or more heads is ridiculously unlikely. In fact getting 530 or more heads from 1000 coins is just as impressive as 60+ from 100 (they both have a probability of about 3%).

I tried to use that principle to obtain a ranking of countries' performances at the Games relative to their expected medal haul given their populations. I also employed the 4-2-1 points system (four points for gold, two for silver, one for bronze). There were 302 gold, 304 silver and 356 bronze medals awarded in total. That works out to be 2172 points. Each country's "expected haul" is their share of the 2172 total as a fraction of the world population. For instance China has 19.16% of the world's population so you could expect it to receive 19.16% of the points, i.e. 416. In fact China "only" received 228 points.

I realise there are all kinds of flaws to my method. I had to tweak the parameters a bit, and at any rate the allocation of medals is hardly random. Then most events limit the number of athletes per country (in some cases to just one) so even a theoretical supreme sporting nation wouldn't take home all the medals.

For all their flaws, these are the two tables I came up with. First, the twenty best performing countries:
1. Grenada
2. Jamaica
3. New Zealand
4. Hungary
5. Bahamas
6. Great Britain
7. Australia
8. Trinidad & Tobago
9. Netherlands
10. Croatia
11. Cuba
12. Denmark
13. Belarus
14. Lithuania
15. Slovenia
16. Czech Republic
17. Kazakhstan
18. Russia
19. Georgia
20. South Korea

And now the ten worst performers:
1. India
2. Pakistan
3. Nigeria
4. Bangladesh
5. Indonesia
6. Philippines
7. Vietnam
8. Congo
9. Myanmar (Burma if you prefer)
10. China (yes, number two on the actual medal table).

Seven of those countries on the "worst" list didn't win a single medal. Obviously they all have substantial populations. India's 1210 million people managed two silver and four bronze, while Indonesia (238 million) got one silver and one bronze. Pakistan (180 million) were the biggest population of those who scored "nul points". At the other end, New Zealand (4.4 million) clearly did punch above their weight; Usain Bolt led the Jamaican charge on the track (their population is 2.7 million); Grenada's solitary medal was gold in the men's 400m and with a population of only 105,000 that was enough for top spot.