Pythagorean Expectation Calculator (Expected Wins)
Enter points or runs scored and allowed to estimate win percentage with the Pythagorean expectation, then see expected wins over a season. Use it for the Pythagorean win formula and to see how many wins a run differential predicts.
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Formula
Win % = RS^x / (RS^x + RA^x), where RS is runs or points scored and RA is allowed.
| Sport | Exponent x |
|---|---|
| Baseball | 1.83 |
| Basketball | 13.91 |
| Hockey | 2.15 |
| Football | 2.37 |
Expected wins = win % x games in the season
Worked example
A baseball team scores 800 and allows 700 in 162 games.
- 800^1.83 / (800^1.83 + 700^1.83) = 0.5608 (56.08%)
- Expected wins = 0.5608 x 162 = 90.85, expected losses 71.15
The exponents are empirical fits, so the result is an estimate and real records differ because of luck and context.
When to use it
Use it to check whether a team's record is lucky or unlucky. If actual wins are far above the Pythagorean figure, the team may have won a lot of close games and could regress. If they are below, results might improve.
Typical values (rules of thumb)
- A team that scores and allows the same amount is expected to win 50%.
- In baseball roughly 10 extra runs scored per season is worth about 1 win.
- Pythagorean win totals usually land within a few wins of the real total over a full season.
Common mistakes
- Using runs per game instead of totals; either works if both sides use the same basis.
- Using the baseball exponent for another sport. Choose the right sport or enter a custom exponent.
- Reading the number as a prediction. It is a model, not a guarantee.
How it differs from nearby tools
The win percentage calculator uses the actual record. The Elo calculator updates ratings game by game. This page estimates the record from scoring.
Frequently asked questions
What is Pythagorean expectation?
Bill James's estimate of win percentage from scoring: RS^x / (RS^x + RA^x). A baseball team scoring 800 and allowing 700 has 800^1.83 / (800^1.83 + 700^1.83) = 56.08%, about 90.8 wins in 162 games.
Which exponent should I use for each sport?
The presets are common empirical fits: baseball 1.83, basketball 13.91 (Daryl Morey), hockey 2.15 and football 2.37. Different studies give slightly different values, so choose Custom to test your own.
How accurate is it?
It is a model, so treat the result as an estimate. Baseball teams usually finish within a few wins of it. A team that wins well above its expectation was often lucky in close games.
What do expected wins tell me?
It is win percentage times games. A basketball team scoring 110 and allowing 105 per game has 65.64%, which is 53.8 expected wins over 82 games. Compare that with the real record to see over- or under-performance.
How many wins does 800 runs scored and 700 allowed predict?
Win % = 800^1.83 / (800^1.83 + 700^1.83) = 56.08%, which is 90.85 wins over 162 games. Round that to about 91 wins.
How many wins does a basketball team with 9,000 points for and 8,500 against expect?
With the basketball exponent 13.91, win % = 9000^13.91 / (9000^13.91 + 8500^13.91) = 68.89%. Over 82 games that is 56.49 wins, about 56.
How many wins does a football team with 450 points for and 350 against expect?
Using the football exponent of 2.37, win % = 450^2.37 / (450^2.37 + 350^2.37) = 64.47%. Over 17 games that is 10.96 wins, or about 11.