• Chris Ramondelli

Does the Pythagorean Expectation work in the NHL?

Updated: Sep 26, 2019

Three Questions to answer

1. Can the Pythagorean Expectation Formula be used to predict win percentage in hockey?

2. What was the lowest Winning Percentage of teams that made the playoffs?

3. What Combination of Goals For and Goals Against will get you in the playoffs?


The Pythagorean Win-Loss formula has been around ever since the baseball statistician Bill James introduced it during the early 1980s. This formula provides a winning percentage (WP) a team is supposed to have at any point in time based on their Runs Scores (RS) and Runs Against (RA). Bill James believed the exponent to be 2, but later analysis shows that for baseball the exponent used should be approximately 1.8.

This formula has been studied in some other sports as well. Schatz (2003) applied the model to football and Oliver (2004) applied the model to basketball. Cochran and Blackstock (2009) applied the formula to hockey, as well as Chris Apple and Marc Foster (2002,2010).


The model used for the Pythagorean Expectation in this analysis is:

This model is from Dayaratna and Miller (Hockey Research Journal 2012/2013). The most recent season they studied was the 2010-2011 season where the exponent was 2.1, which is the exponent I have used for this analysis as well.


Data from the 2008 season up until the 2019 season (excluding 2012/2013 lockout season) was used to see what the lowest win total of a playoff team was each season as well as goal differential for that team and the lowest goal differential any of the playoff teams that season.

Data was also gathered from the previous 3 seasons to test the Pythagorean Expectation formula on the final standings for all teams.

All the Data is available in the Excel Document Below


Note: These Results are not taking into account points a team gets for an overtime loss. The results look specifically at wins.

We will revisit the 3 questions from the beginning of the article:

1. Can the Pythagorean Expectation Formula be used to predict win percentage in hockey?

Using the past three seasons, each team’s winning percentage and Pythagorean Expectation was calculated. The differences between the predicted Pythagorean Expectation and the Actual Winning Percentages were calculated and averaged out to show these results:


  • The Average difference in the predicted vs actual winning percentages never was above a tenth of a percent (.01%).

  • Using the calculated Pythagorean expectation never yielded over a game difference in win totals

2. What was the lowest Winning Percentage of teams that made the playoffs?

Looking at every season from 2008 until present, the following was collected:

Takeaways from the table:

  • Over the past 10 seasons the Lowest Win Total for a playoff team averaged 40.1, less than a 50%-win percentage (48.9%)

  • 6 out of the 10 seasons had a team make the playoffs winning less than 50% of their games

  • The Average Goal differential of the lowest win total team was 2.7

  • The Worst Goal differential for a playoff team in the given season averaged out to -4.2

  • There were plenty of teams that made the playoffs with a negative goal differential

3. What Combination of Goals For and Goals Against will get you in the playoffs?

From the table in question 2, we see that the average winning percentage for the teams with the lowest win total each season was 0.48902. They won around 48.9% of the games during the regular season. Using the Pythagorean Expectation Formula, I ran a GRG Nonlinear Optimizer that is an optimizer for problems are a smooth nonlinear.

The optimizer was run for different constraints on goals for and goals against. The main constraint of the optimizer is that the Pythagorean Expectation had to equal to 0.48902. This Formula assumes the exponent is still the 2.1 that was introduced previously.


  • In order to make the playoffs (The last possible spot), a team should have a goal differential no worse than -6

  • If a team scores 220 goals, they must not give up more than 225 goals to have a good shot at making the post-season


  • The Pythagorean Expectation Formula is a good way to predict a team’s winning percentage in hockey.

  • The Worst Goal differential of teams in the playoffs on average don’t exceed -4.2

  • In order to make the postseason, a team should have better than a -6 goal differential



  • Schatz, Aaron. “Pythagoras on the Gridiron.” Football Outsiders. 14 July 2003. http://www.footballoutsiders .com/stat-analysis/2003/pythagoras-gridiron.

  • Oliver, Dean. Basketball on Paper: Rules and Tools for Performance Analysis. Dulles, Va.: Potomac Books, 2004.

  • Cochran, James J., and Rob Blackstock.“Pythagoras and the National Hockey League.” Journal of Quantitative Analysis in Sports. 5.2 (2009).

  • Apple, Chris, and Marc Foster.“Glancing into the crystal ball: Playoffs projections based on Pythagorean Performance.” Sports Illustrated. 12 January 2002. 2002/01/09/just_stats/.

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