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.

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