## A Pythagorean Exponent for the NHLMarch 17, 2015

Posted by tomflesher in Sports.
Tags: , , , , ,
The optimal exponent turned out not to be 2 in just about any sport; in baseball, for example, the optimal exponent is around 1.82. This is found by setting up a function – in the case of the National Hockey League, that formula would be $\frac{GF^x}{GF^x + GA^x}$ – with a variable exponent. This is equivalent to $(1 + (\frac{GA}{GF})^x)^{-1}$. Set up an error function – the standard is square error, because squaring is a way of turning all distances positive and penalizing bigger deviations more than smaller deviations – and minimize that function. In our case, that means we want to find the x that minimizes the sum of all teams’ $((1 + (\frac{GA}{GF})^x)^{-1}) - \frac{W}{W+L})^2$. Using data from the 2014 season, the x that minimizes that sum of squared errors is 2.113475, which is close enough to 2.11 that the sum of squared errors barely changes.