Pythagoras wasn’t quite around for tailgating, fight songs, cheerleaders, and Super Bowls, but Bill James was. James is a historian, mathematician, and statistician whose work has greatly influenced the evolution of baseball (see sabermetrics), and has had an impact on other sports as well. One of James’ most widely cited formulas is the Pythagorean Expectation, a formula that has been adopted by Pro-Football-Reference.com and FootballOutsiders.com, among others, and can be used to project the 2013 Philadelphia Eagles’ record with a relatively fair degree of accuracy.

But first, a clarification. If you can recall what you learned in Trigonometry, you’ll realize that James’ formula is not related to triangles. The true Pythagorean theorem is a^{2} + b^{2} = c^{2}, where a and b are the sides of a triangle forming a right angle, and c is the hypotenuse. In contrast, James’ formula is used to determine win expectation based on runs scored and runs allowed. It bears resemblance to the Pythagorean theorem, hence the name:

Runs Scored^{2} / (Runs Scored^{2} + Runs Allowed^{2})

The result of the formula is a ratio, or percentage, that is multiplied by the number of games in a season. Historically, that product is eerily close to a team’s actual number of wins (especially in baseball, where the sample size is much larger), but many times not quite the same. James theorized that this “pythagorean expectation” value is equal to the number of games a team should win, based solely on runs scored and allowed, when luck is controlled. Additionally, James found that the difference between actual wins and Pythagorean wins can be used to project the number of wins in the following season with some degree of accuracy. Again, all of this can be, and has been, applied to football. In fact, Pythagorean wins within a season more accurately predict Super Bowl winners than can actual wins.

So let’s apply this to the Philadelphia Eagles. Interestingly, it turns out the “2”, or “squared” exponent in the formula is not the best exponent to use when applying the Pythagorean Expectation formula to football. It changes slightly as each new season progresses. Some have argued the best exponent is 2.37, others have said it is 2.67, and still others have claimed that it is 2.535. The differences in results among the three are relatively minimal, tenths of points when determining wins. When applying this formula to the Eagles, I chose to use the 2.535 exponent.

Thanks to the info on Pro-Football-Reference.com, I compiled the Eagles’ points scored, points allowed, wins and losses from 1933 to present. I then applied the following formula to determine the Pythagorean Expectation for each season:

[Points Scored^{2.535} / (Points Scored^{2.535} + Points Allowed^{2.535})] x Games Played

In the last five years, results indicate the Eagles performed to expectation in 2012, 2010, and 2009 but were perhaps unlucky and did not perform as the points scored and allowed would seem to indicate in 2011 and 2008:

Year | PF | PA | Wins | Losses | Ties | Pyth Exp | Diff |

2012 | 280 | 444 | 4 | 12 | 0 | 3.8 | 0.2 |

2011 | 396 | 328 | 8 | 8 | 0 | 9.9 | -1.9 |

2010 | 439 | 377 | 10 | 6 | 0 | 9.5 | 0.5 |

2009 | 429 | 337 | 11 | 5 | 0 | 10.4 | 0.6 |

2008 | 416 | 289 | 9 | 6 | 1 | 11.5 | -2.5 |

According to Football Outsiders, the difference (Diff) column can be used to indicate improvement or regression the following year. If greater than 1, the team is more likely to regress and if less than -1, the team is more likely to improve (note that’s not what happened from 2011 to 2012). I indicated that as follows (if likely to regress, then -1; if likely to improve, then 1, else 0):

Year | PF | PA | Wins | Losses | Ties | Pyth Exp | Diff | Regr/Impr |

2012 | 280 | 444 | 4 | 12 | 0 | 3.8 | 0.2 | 0 |

2011 | 396 | 328 | 8 | 8 | 0 | 9.9 | -1.9 | 1 |

2010 | 439 | 377 | 10 | 6 | 0 | 9.5 | 0.5 | 0 |

2009 | 429 | 337 | 11 | 5 | 0 | 10.4 | 0.6 | 0 |

2008 | 416 | 289 | 9 | 6 | 1 | 11.5 | -2.5 | 1 |

Next, linear regression can be used to determine the strength of the relationship between these variables and the record we want to project for the following year (I call this Year + 1), and provides us with a formula that we can use to project the actual number of wins for the following season. First I checked the strength and statistical significance of the correlation between the Pythagorean Expected wins, the Difference, and next year’s wins. The correlation coefficient was r^{2} = .41, which is fairly strong (The p-value was extremely less than .001, making it also statistically significant). Then, I checked if the Regr/Impr values made an impact, and it turns out they do. When including these values, the r^{2} increases from .41 to .49 (and again, the p-values are less than .001). As a result, the formula produced by the linear regression that we can use to project wins for the following year is:

Year + 1 Wins = 2.22 + .67(Pyth Exp) + 1.91(Diff) +2.92(Regr/Impr)

When this is applied to Eagles’ seasons from 1933 to present, the results correlate well with a standard error within 2.36 games. Here are the results since 1996:

Year | Year+1 Wins | Year+1 Pyth Exp |

2012 | 5.2 | |

2011 | 4 | 8.2 |

2010 | 8 | 9.5 |

2009 | 10 | 10.4 |

2008 | 11 | 8.2 |

2007 | 9 | 9.1 |

2006 | 8 | 9.0 |

2005 | 10 | 6.5 |

2004 | 6 | 9.7 |

2003 | 13 | 9.1 |

2002 | 12 | 9.3 |

2001 | 12 | 10.7 |

2000 | 11 | 9.1 |

1999 | 11 | 5.2 |

1998 | 5 | 5.4 |

1997 | 3 | 5.8 |

1996 | 6 | 7.7 |

As you can see, the formula projects the Eagles to finish the 2013 season with a 5-11 record. According to results past, however, this may not be the case. For example, when using 2011 data, the Eagles were projected to win 8.2 games in 2012, when in fact they won four. Yet when using 2009 data to project 2010 wins, the expected results (10.4) are very similar to actual (10). Regardless of these results, fans will undoubtedly have their own predictions that make sense, no math included. Sometimes it’s simply more fun to make predictions with a beer in hand, while tailgating and singing fight songs. But based on this, how do you think the Eagles will fair in 2013?

**Topics:** Bill James, NFL, Philadelphia Eagles, Pythagorean Expectation