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 YearRSRAGActual WinsPythag. Exp. *Pythag. Exp. **
Chicago Cubs, 1992593624162787777
Detroit Tigers, 2005723787162717575
Florida Marlins, 1997740669162928889
Florida Marlins, 2009772766162878282
New York Mets, 2003642754161666969
Texas Rangers, 1977767657162949292

     * 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