## Recreational Math – Pythagoras and Baseball?

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:

RS2 / (RS2 + RA2)

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.

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• 1. luckytoilet  |  February 19, 2010 at 9:32 pm

This 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 pm

They named it that because the denominator is the sum of two squares, like a^2 + b^2 from the Pythagorean theorem.