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 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% = RS^1.83/(RS^1.83 + RA^1.83).

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

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

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.

© Had2Know 2010