# Pythagorean exponents in the 2008-09 English Premier League

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:

 Club  Alpha_GF Alpha_GA Exponent Arsenal 2.7560 1.5805 1.5030 Aston Villa 2.2955 2.1485 1.4849 Blackburn 2.0192 2.6941 1.4604 Bolton 2.0961 2.3750 1.3561 Chelsea 2.6428 1.2474 1.4092 Everton 2.3008 1.6886 1.4051 Fulham 1.9156 1.7100 1.3650 Hull City 1.9233 2.3686 1.8368 Liverpool 2.9020 1.3778 1.4974 Manchester City 2.4678 2.2673 1.6112 Manchester United 2.4407 1.1003 1.6722 Middlesborough 1.6117 1.9444 1.6419 Newcastle United 1.9951 2.4601 1.8405 Portsmouth 1.9960 2.3075 1.4803 Stoke City 1.8209 2.3877 1.5949 Sunderland 1.7685 2.2380 1.5820 Tottenham 1.9768 1.9093 1.6944 West Brom 1.8442 2.9093 1.3816 West Ham 2.0579 2.0006 1.5762 Wigan 1.7025 1.9550 1.3951

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.

My solution approach is described in this document.  I implemented it using a script in Scilab.  If you'd like a copy of the script I can send it to you, but you will have to download Scilab.