# Pythagorean Theorem of Baseball

### From BR Bullpen

The **Pythagorean Theorem of Baseball** is a creation of Bill James which relates the number of runs a team has scored and surrendered to its actual winning percentage, based on the idea that runs scored compared to runs allowed is a better indicator of a team's (future) performance than a team's actual winning percentage. This results in a formula which is referred to as **Pythagorean Winning Percentage**.

## [edit] The Formula

There are two ways of calculating Pythagorean Winning Percentage (W%). The more commonly used, and simpler version uses an exponent of 2 in the formula.

W%=[(Runs Scored)^2]/[(Runs Scored)^2 + (Runs Allowed)^2]

More accurate versions of the formula use 1.81 or 1.83 as the exponent.^{[citation needed]}

W%=[(Runs Scored)^1.81]/[(Runs Scored)^1.81 + (Runs Allowed)^1.81]

**Expected W-L** can then be obtained by multiplying W% by the team's total number of games played, then rounding off. Expected W-L for each team is published by ESPN on their website.

## [edit] Justification

The rationale behind Pythagorean Winning Percentage is that, while winning as many games as possible is still the ultimate goal of a baseball team, a team's run differential (once a sufficient number of games have been played) provides a better idea of how well a team is actually playing. Therefore, barring personnel issues (injuries, trades), a team's actual W-L record will approach the Pythagorean Expected W-L record over time, not the other way around. The average difference between the actual and the Expected W-L is a bit more than 3 games at the end of a season (although a recent exception is the 2005 and 2007 Arizona Diamondbacks, who both beat their expected W-L by 11 games), as did the 2012 Baltimore Orioles. Deviations from expected W-L are often attributed to the quality of a team's bullpen, or more dubiously, "clutch play"; many sabermetrics advocates believe the deviations are the result of luck and random chance.

Because of this, expected W-L makes for a good predictor of performance in mid-season. If a team has a 40-25 record, but a Pythagorean winning percentage at or below .500, it should not be surprising when this team's record drops as they start losing close games. In fact, expected W-L correctly predicted the fate of the 2005 Washington Nationals. On July 5 of that season, Washington was 19 games over .500 and 4 1/2 games ahead of second place, but had a Pythagorean W% of exactly .500. They went 30-49 the rest of the season to finish at .500 (four games ahead of their final expected W-L).

Nevertheless, given that advocates of the theorem point to teams that exceed their predicted number of wins as having done so due only to random chance, it is questionable whether the theorem provides anything indicative with respect to an individual team during a given season, as opposed to being a construct that shows the general relationship between scoring runs and preventing runs in winning baseball games. For instance, in the Nationals example above, another outcome from a different season might have shown the Nationals improving their run differential in the second half making their first half run differential the outlier.

## [edit] Other Sports

Pythagorean Winning Percentage has also been used in basketball. The exponent used varies between 13 and 16.5.