A couple of nights ago I presented goal distributions for all twenty teams in the 2008-09 English Premier League season, in an attempt to calculate the exponent that would be used for my expansion of the Pythagorean. I realized after my calculation that I needed to consider the sum of the squares of both the error from the goals scored data and the goals allowed data, instead of separately like I had done before. Fortunately that correction was easy to fix, so now I'd like to present the alpha and gamma terms for all 20 teams in last year's Premiership:
The mean of the exponent is 1.5394 (median 1.5002), with a standard deviation of 0.1457. The numbers that I've seen on the web for the exponent term are right in the 1-sigma range of this estimate. It was also close to my guess of 1.5, which was more of a gut instinct than anything else.
I suppose that if I want to make a stronger claim that this curve-fit is the right one, I would perform a chi-square goodness-of-fit test, but I'll leave that for later or as an exercise for someone more enterprising.