Applying the Pythagorean Theorem

If you've read many baseball blogs over the past few years, you're likely familiar with Bill James's Pythagorean Theorem of Baseball. He created a formula that compares a team's runs scored to runs allowed that produces a fairly accurate prediction of that team's win-loss record. It's a simple formula based on a simple assumption - the greater the gap between a team's runs scored and its runs allowed, the more games you would expect them to win.

The same idea translates well to basketball. A team that scores 70 ppg while allowing 60 ppg should win more often than one that scores 65 ppg and allows 60 ppg. Ken Pomeroy provides a good explanation of the formula, since I'm rushing through the details to get to the results.

The formula looks like this -

**Expected Winning Percentage**= Points Scored^10 / (Points Scored^10 + Points Allowed^10)

Then you multiply the team's expected winning percentage by its games played to see how often they're expected to win and lose.

Let's see how accurate it is by looking at last year's final Big Ten standings.

Team | Pts For | Pts All | Exp W% | Exp W | Exp L | Act W | Act L | Diff |

Illinois | 1115 | 979 | .786 | 12.6 | 3.4 | 13 | 3 | +0.4 |

Michigan St | 1141 | 1041 | .714 | 11.4 | 4.6 | 12 | 4 | +0.6 |

Wisconsin | 1093 | 916 | .854 | 13.7 | 2.3 | 12 | 4 | -1.7 |

Iowa | 1136 | 1113 | .551 | 8.8 | 7.2 | 9 | 7 | +0.2 |

Michigan | 1076 | 1046 | .570 | 9.1 | 6.9 | 8 | 8 | -1.1 |

Northwestern | 966 | 979 | .467 | 7.5 | 8.5 | 8 | 8 | +0.5 |

Indiana | 1028 | 1065 | .413 | 6.6 | 9.4 | 7 | 9 | +0.4 |

Purdue | 979 | 981 | .495 | 7.9 | 8.1 | 7 | 9 | -0.9 |

Ohio State | 1015 | 1109 | .292 | 4.7 | 11.3 | 6 | 10 | +1.3 |

Minnesota | 1066 | 1169 | .284 | 4.6 | 11.4 | 3 | 13 | -1.6 |

Penn State | 893 | 1110 | .102 | 1.6 | 14.4 | 3 | 13 | +1.4 |

The projections resemble the final standings fairly well - most teams finished within a win and a half of their expected win total. Teams that finish far from their projection are usually termed "lucky" or "unlucky" because of their performance in close games. Penn State, for example, was typically an easy victory for opponents, but had more wins than expected by sneaking in 2 wins of three points or less.

Another explanation for wide differences between wins and expected wins is blowout wins and losses. A win by 30 is the same as a win by 3 in the standings, but a blowout win will inflate a team's expected winning percentage. This partly explains Wisconsin's underperformance, as they won six games by at least 15 points.

Let's see how this year's group is shaping up, as sorted by expected winning percentage.

Team | Pts For | Pts All | Exp W% | Exp W | Exp L | Act W | Act L | Diff |

Illinois | 983 | 806 | .879 | 11.4 | 1.6 | 13 | 0 | +1.6 |

Michigan St | 873 | 732 | .853 | 10.2 | 1.8 | 10 | 2 | -0.2 |

Wisconsin | 815 | 760 | .668 | 8.0 | 4.0 | 8 | 4 | 0.0 |

Ohio State | 850 | 812 | .612 | 8.0 | 5.0 | 7 | 6 | -1.0 |

Iowa | 808 | 810 | .494 | 5.9 | 6.1 | 4 | 8 | -1.9 |

Minnesota | 811 | 820 | .472 | 6.1 | 6.9 | 7 | 6 | +0.9 |

Indiana | 752 | 761 | .470 | 5.6 | 6.4 | 7 | 5 | +1.4 |

Purdue | 783 | 803 | .437 | 5.2 | 6.8 | 3 | 9 | -2.2 |

Northwestern | 735 | 782 | .350 | 4.2 | 7.8 | 5 | 7 | +0.8 |

Michigan | 737 | 878 | .148 | 1.9 | 11.1 | 3 | 10 | +1.1 |

Penn State | 694 | 877 | .088 | 1.1 | 10.9 | 1 | 11 | -0.1 |

Iowa has clearly been the Big Ten's tough luck loser this year (or just a plain disappointment, depending on your perspective), with five losses by five points or less. Their other three losses were somewhat "expected," as they came to teams above them on this list. Given that they've scored just about as many points as they've allowed, you'd expect them to be right at .500.

I was surprised to see Purdue so far below their projection, but beating Michigan and Penn State by a combined 56 points helped close their gap between points scored and allowed.

This Wednesday's game between Iowa and Minnesota should be one worth watching. Each has essentially performed like a .500 team in the conference (based on points scored and allowed), and each will be playing with no margin for error in order to keep their tournament hopes alive (as noted in today's Sunday-*NY Times*-length read from the Big Ten Wonk).