## A Pythagorean Exponent for the NHL March 17, 2015

Posted by tomflesher in Sports.
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A Pythagorean expectation is a statistic used to measure how many wins a team should expect, based on how many points they score and how many they allow. The name ‘Pythagorean’ comes from the Pythagorean theorem, which measures the distance between the two short sides of a right triangle (the hypotenuse); the name reflects the fact that early baseball-centric versions assumed that Runs^2/(Runs^2 + Runs Allowed^2) should equal the winning percentage, borrowing the exponent of 2 from the familiar Pythagorean theorem (a^2 +b^2 =c^2).

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.

Porting that exponent into the current season, there are a few surprises. First of all, the Anaheim Ducks have been lucky – almost six full wins worth of luck. It would hardly be surprising for them to tank the last few games of the season. Similarly, the Washington Capitals are on the precipice of the playoff race, but they’re over four games below their expected wins. With 11 games to go, there’s a good chance they can overtake the New York Islanders (who are 3.4 wins above expectation), and they’re likely to at least maintain their wild card status.

On the other end, somehow, the Buffalo Sabres are obscenely lucky. The worst team in the NHL today is actually 4 games better than its expectation. Full luck standings as of the end of March 16th are behind the cut.

 Team Lucky Wins Anaheim Ducks 5.9 New York Rangers 0.1 St. Louis Blues 0.8 Tampa Bay Lightning -1.2 Montreal Canadiens 1.9 Nashville Predators 1.2 New York Islanders 3.4 Pittsburgh Penguins -0.3 Chicago Blackhawks -1.6 Detroit Red Wings 0.6 Washington Capitals -4.4 Minnesota Wild -1.3 Boston Bruins -0.4 Vancouver Canucks 3.2 Los Angeles Kings -3.5 Calgary Flames -1.3 Winnipeg Jets -1.5 Ottawa Senators -3.8 Florida Panthers 1.2 San Jose Sharks -0.5 Colorado Avalanche -0.7 Dallas Stars -1.1 Philadelphia Flyers -3.2 New Jersey Devils -0.7 Columbus Blue Jackets 2.8 Toronto Maple Leafs -1.3 Carolina Hurricanes -2.4 Arizona Coyotes 2.7 Edmonton Oilers -2.1 Buffalo Sabres 4.0