## 08 December 2009

### Distribution of Putnam scores

The distributions of Putnam exam scores are interesting. See, for example, the 2001 distribution. It takes a bit of number-crunching to get an actual distribution of scores from the data; they report the "rank" of the people getting each score. The rank corresponding to a given score is, I assume, A+(B+1)/2 where A is the number of people scoring higher than that score and B is the number of people scoring that particular score. For example, in 2001 -- which happens to be one of the years in which I took the Putnam -- the table begins

 Score 101 100 86 80 79 77 73 72 71 70 69 68 Rank 1 2 3 4.5 6 7.5 9 11 14 16.5 19 23.5 Number 1 1 1 2 1 2 1 3 3 2 3 6

where the first two rows are provided by the organizers, and the third row can be worked out by working left to right. For example, once we know 17 people got 70 or better, the fact that the score 69 corresponds to rank 19 means that the people scoring 69 must have been the 18th, 19th, and 20th-best; so there were three of them. (Incidentally, most increasing sequences of half-integers, when interpreted as sequences of ranks, don't appear to correspond to legitimate score distributions; the number of people getting certain scores ends up negative if you're note careful.)

Anyway, if you crunch the numbers on a typical Putnam score distribution you observe two things:
- the scores follow, roughly, a power law; the number of people scoring 10n decays like some power of n, for integer n.
- once you remove this decay (which I haven't actually done; I've just eyeballed it), there are "spikes" at multiples of 10. For example, the number of people scoring 18, 19, 20, 21, 22, 23 in 2001 were 8, 23, 99, 60, 39, 11. Twenty-four people scored 50; seven scored each of 49 and 51.

I can't explain the first one (and it may just be an artifact of the way I'm doing the plotting; lots of things look close to linear when plotted on a logarithmic scale). But the second one is actually easy to explain; Putnam problems are worth ten points each, and most scores are 0 or 10 with a smattering of 1, 2, 8, or 9. Scores between 3 and 7 on a problem are exceedingly rare. So to get a score of, say, 55, one has to get five problems right and have made a bit of progress on three to five more, which is less likely than straight-out solving five or six problems (for 50 or 60, respectively).

Incidentally, I haven't looked at the problems from the 2009 Putnam, because I have work to do.