# Pythagorean Expectation Calculator & Formula

Pythagorean Expectation Calculator
Runs Scored RS =
Runs Allowed RA =
Number of Games G =

* x = 1.83   ** x = [(RS+RA)/G]0.285

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 formula for win percent (W%) and total wins was

W% = RS2/(RS2 + RA2) and
Wins = (G)(W%)

The presence 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% = RS1.83/(RS1.83 + RA1.83).

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

RSx/(RSx + RAx)

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

When a team's actual number of wins is above the Pythagorean wins, the team is said to have been lucky that year. Conversely, when a team has fewer wins than the Pythagorean predicted number of wins, the team is said to have been unlucky.