Pythagorean Expectation Calculator (Expected Wins)
The winning percentage and number of wins a team 'should' have from the runs or points it scored and allowed, and how many it gained or lost to luck.
Results
What this tool does
Bill James noticed that a baseball team's record follows runs scored squared over the sum of runs scored squared and runs allowed squared, a shape that reminded him of Pythagoras. Later work found that an exponent around 1.83 fits baseball better, about 2.37 suits the NFL and about 13.9 suits basketball, where scores are high and close. Teams that win many more games than their run differential predicts usually owe it to close games and bullpen luck, and they tend to fall back the next season, which is why analysts trust the expected record more than the real one early in a season.
Formula
win % = RS^x ÷ (RS^x + RA^x)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
rs | Runs or points scored | — |
ra | Runs or points allowed | — |
gp | Games played | — |
xp | Exponent (2 classic, 1.83 baseball, 2.37 NFL, 13.91 NBA) | — |
wa | Actual wins | — |
WP | Expected winning percentage | % |
EW | Expected wins | — |
EL | Expected losses | — |
LK | Wins above expectation (luck) | — |
RD | Run or point differential | — |
Worked example
- Runs or points scored750
- Runs or points allowed680
- Games played162
- Exponent (2 classic, 1.83 baseball, 2.37 NFL, 13.91 NBA)1.83
- Actual wins88
- Expected winning percentage54.5 %
- Expected wins88.2
- Expected losses73.8
- Wins above expectation (luck)-0.2
- Run or point differential70
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.