# Pythagorean Expectation Calculator & Formula

In baseball, one way to predict a team's expected number of wins and losses is with a simple formula called the *Pythagorean Expectation*, invented by the sabermetrician Bill James.

The Pythagorean win/loss formula uses the number of runs scored (RS), number of runs allowed (RA), and number of games (G) to predict how many games a team should have won. The original forumula for win percent (W%) and total wins was

W% = RS^{2}/(RS^{2} + RA^{2}) and

Wins = (G)(W%)

The presense of the sum of squares in the denominator is what prompted James to call this the Pythagorean formula. James later revised the equation for W% to

W% = RS^{1.83}/(RS^{1.83} + RA^{1.83}).

He noted that an exponent of 1.83 predicted the actual number of wins more closesly than an exponent of 2. This has led other sabermetric analysts to find an exponent x such that the equation

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

predicts the percentage of wins as accurately as possible. To date, one of the most widely used values of x is

x = [(RS + RA)/G]^{0.285}

which was developed by David Smyth. This value of x is not a fixed constant but rather a function of RS, RA, and G. Either x = 1.83 or [(RS + RA)/G]^{0.285} will provide a good prediction for the actual number of games won. The following table shows win/loss stats for arbitrarily selected teams and years:

Team and Year | RS | RA | G | Actual Wins | Pythag. Exp. * | Pythag. Exp. ** |

Chicago Cubs, 1992 | 593 | 624 | 162 | 78 | 77 | 77 |

Detroit Tigers, 2005 | 723 | 787 | 162 | 71 | 75 | 75 |

Florida Marlins, 1997 | 740 | 669 | 162 | 92 | 88 | 89 |

Florida Marlins, 2009 | 772 | 766 | 162 | 87 | 82 | 82 |

New York Mets, 2003 | 642 | 754 | 161 | 66 | 69 | 69 |

Texas Rangers, 1977 | 767 | 657 | 162 | 94 | 92 | 92 |

* x = 1.83 ** x = [(RS+RA)/G]^{0.285}

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