## Recreational Math – Pythagoras and Baseball?

*February 19, 2010 at 10:29 am* *
2 comments *

Since Spring Training began this week, I thought I would put together another baseball related article. The reference to Pythagoras has nothing to do with right triangles (How far is it directly from home plate to second base?), but instead a formula for predicting winning percentages based on a formula that has a resemblance in places to the Pythagorean theorem.

### Pythagorean Record

Pythagorean Record was created by Bill James as a way to predict winning percentage based on Runs Scores (RS) and Runs Allowed (RA). Here it is:

RS^{2} / (RS^{2} + RA^{2})

The resemblance of the denominator to the Pythagorean theorem led to its name.

### 2009 Season

Here are the actual and predicted wins for the 30 major league teams last season.

Team | RS | RA | Actual Wins | Pythagorean Wins | Difference |

Arizona | 720 | 782 | 70 | 74 | -4 |

Atlanta | 735 | 641 | 86 | 92 | -6 |

Baltimore | 741 | 876 | 64 | 68 | -4 |

Chicago | 724 | 732 | 79 | 80 | -1 |

Chicago | 707 | 672 | 83 | 85 | -2 |

Cincinnati | 673 | 723 | 78 | 75 | 3 |

Cleveland | 773 | 865 | 65 | 72 | -7 |

Detroit | 743 | 745 | 86 | 81 | 5 |

Florida | 772 | 766 | 87 | 82 | 5 |

Houston | 643 | 770 | 74 | 67 | 7 |

Kansas City | 686 | 842 | 65 | 65 | 0 |

Milwaukee | 785 | 818 | 80 | 78 | 2 |

New York | 671 | 757 | 70 | 71 | -1 |

Oakland | 759 | 761 | 75 | 81 | -6 |

Pittsburgh | 636 | 768 | 62 | 66 | -4 |

San Diego | 638 | 769 | 75 | 66 | 9 |

San Francisco | 657 | 611 | 88 | 87 | 1 |

Seattle | 640 | 692 | 85 | 75 | 10 |

Tampa Bay | 803 | 754 | 84 | 86 | -2 |

Texas | 784 | 740 | 87 | 86 | 1 |

Toronto | 798 | 771 | 75 | 84 | -9 |

Washington | 710 | 874 | 59 | 64 | -5 |

Boston | 872 | 736 | 95 | 95 | 0 |

Colorado | 804 | 715 | 92 | 90 | 2 |

Los Angeles | 883 | 761 | 97 | 93 | 4 |

Los Angeles | 780 | 611 | 95 | 100 | -5 |

Minnesota | 817 | 765 | 87 | 86 | 1 |

New York | 915 | 753 | 103 | 97 | 6 |

Philadelphia | 820 | 709 | 93 | 93 | 0 |

St. Louis | 730 | 640 | 91 | 92 | -1 |

There were a few teams whose differences stand out: Toronto underachieved by 9 games & Cleveland by 7. Three teams outperformed their Pythagorean Record by at least 7 games: Houston (+7), San Diego (+9), & Seattle (+10).

The correlation coefficient between Actual Wins & Pythagorean Record was 0.91.

### A Red Sox Example

Would the Red Sox be better served by trying to increase offensive output by 10% or by trying to improve defensive efficiency by 10%?

If they increased RS by 10% last season, they would have scored 959 runs, and their Pythagorean Record would have been 102 wins. That’s an improvement of 7 wins.

If they decreased RA by 10% last season, they would have allowed 662 runs, and their Pythagorean Record would have been 103 wins. That’s an improvement of 8 wins.

Considering that defense & pitching was easier to acquire this off season, it seems clear that trying to improve RA was the way to go. Note that if both totals decreased by 10%, the Pythagorean Record would remain at 95 wins, so it is imperative that while the defense improves, the offense stays close to previous output. Time will tell.

-George

*I am a mathematics instructor at College of the Sequoias in Visalia, CA. Each Friday my blog contains an article on recreational mathematics. Let me know if there are other topics you’d like me to address. You can reach me **hrough the contact page on my website – http://georgewoodbury.com.*

Entry filed under: Math, Recreational Math. Tags: amatyc, baseball, bill james, education, george woodbury, ictcm, Math, pythagorean, red sox, sabermetrics, statistics, stats, woodbury.

1.luckytoilet | February 19, 2010 at 9:32 pmThis is interesting, although I’m not sure why would baseball scores have any correlation to a geometry theorem. Would you care to elaborate on that part?

2.georgewoodbury | February 19, 2010 at 9:44 pmThey named it that because the denominator is the sum of two squares, like a^2 + b^2 from the Pythagorean theorem.