How to use the pythagorean expectation calculator
- Enter the points or runs scored and allowed for the season so far.
- Enter the games played and choose an exponent for your sport, Pythagenpat, or your own.
- Add the team's actual wins to compare.
- Read the expected win percentage and wins, and the gap between expectation and actual.
Formula
Expected win fraction = PF^x / (PF^x + PA^x)
Expected wins = expected win fraction x games
Pythagenpat exponent = ((PF + PA) / games)^0.287
Commonly used exponents: baseball 1.83, basketball 13.91, football 2.37, hockey 2.15
PF is points or runs for, PA is points or runs against and x is the exponent. The exponents are fitted values, not laws, and you can enter your own.
A baseball and a basketball season
A baseball team that scores 800 runs and allows 700 over 162 games, with an exponent of 1.83, has an expected win fraction of 800^1.83 / (800^1.83 + 700^1.83) = 56.08 percent, which is 90.85 wins. If the team really won 92, it did 1.15 wins better than expected. With Pythagenpat the exponent is 1.8941 and the expectation 56.29 percent, or 91.19 wins.
A basketball team scoring 9,020 points and allowing 8,700 over 82 games, using 13.91, has an expectation of 62.30 percent, or 51.09 wins; 53 actual wins is 1.91 above that.
What the formula says
A team that scores more than it allows should win more than it loses, and the Pythagorean expectation turns the ratio of points for to points against into a win fraction. It was invented by Bill James for baseball, named for its similarity to the Pythagorean theorem. The exponent controls how quickly a scoring edge becomes wins, and it is much higher in basketball, where a small edge is spread over many possessions.
For other baseball numbers use the baseball stats calculator. Related sports tools include the Poisson scoreline calculator, the league table generator and the fantasy football points calculator.
Luck, skill and what the gap means
A team whose wins differ from the expectation was probably lucky or unlucky in close games, or it may be strong in ways that totals do not show, such as a good bullpen. Over a long season, the formula's expectation is a better guide to a team's strength than its record, but it is not a forecast: injuries, trades and schedule all change future results.
Assumptions and limits
- Points or runs for and against are totals for the same games, both above zero.
- The exponents listed are commonly used fitted values and are editable through the custom option.
- Pythagenpat uses the scoring per game to set the exponent and is mainly used for baseball.
- The result is a model of past scoring, not a prediction of wins in future games.
Frequently asked questions
What is the Pythagorean expectation?
It is a formula that estimates a team's winning percentage from the points or runs it scored and allowed. The ratio of the two, raised to a sport-specific power, gives the expected win fraction.
What exponent should I use?
The commonly used values are 1.83 for baseball, 13.91 for basketball, 2.37 for football and 2.15 for hockey. Pythagenpat adjusts the exponent to the scoring level, which helps when scoring is unusual.
Why does a team win more than the formula says?
It may have won a lot of close games, which tend to even out over time. A large gap is a sign that the team's record may overstate its strength.
Does the formula work for any sport?
It works best in sports with many points and a steady scoring pattern. With a custom exponent you can fit it to any league where you have data.