Pythagorean Expectation: Estimating Baseball Wins From Runs Scored and Allowed
October 29, 2026
Bill James observed that a baseball team's win percentage is closely related to the ratio of runs scored to runs allowed. His Pythagorean expectation is: win% = RS^x ÷ (RS^x + RA^x), originally with an exponent of 2, which later analysis refined to about 1.83 for baseball.
A team scores 800 runs and allows 700. Using 1.83: 800^1.83 ÷ (800^1.83 + 700^1.83) = 1.2767 ÷ 2.2767 = 0.561 (the ratio (800/700)^1.83 = 1.2767). Over 162 games, 0.561 × 162 = 90.8 expected wins.
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Equal runs scored and allowed (750 and 750) gives 0.500, or 81 wins. A team scoring 850 and allowing 650 has a ratio of 1.308 and 1.308^1.83 = 1.634, giving 0.620, or 100.5 wins. A team scoring 700 and allowing 800 expects 0.439, or 71.2 wins. As a rule of thumb, about 10 extra runs of run differential equals about one extra win. The team above with +100 run differential (800 − 700) is worth about 10 wins above .500 (91 wins), consistent with the rule.
| Runs scored | Runs allowed | Expected win % | Expected wins (162 games) |
|---|---|---|---|
| 800 | 700 | 0.561 | 90.8 |
| 750 | 750 | 0.500 | 81.0 |
| 850 | 650 | 0.620 | 100.5 |
| 700 | 800 | 0.439 | 71.2 |
Luck and Forecasting
If a team's actual wins differ a lot from its Pythagorean wins (by 5 or more), part of the difference is typically luck in close games, and teams tend to regress toward the expected record the next year. A team that wins 95 games with a Pythagorean record of 88 may be overrated, while one with 80 actual wins and 87 expected may be better than its record. Front offices and analysts use the expectation to evaluate trades and player contributions in terms of runs and wins.
Other Sports
The same idea works with different exponents: about 2.37 for the NBA (points for and against) and about 2.37 to 2.7 for the NFL, and roughly 2 for the NHL. The exponent depends on how much randomness affects scoring. Soccer analysts apply it to goals using an exponent around 1.7 to 2. Use the exponent estimated from historical data for the sport and league.
Limits
The formula treats all runs equally, so it ignores timing (bullpen and clutch hitting), the distribution of runs across games, and the strength of the schedule. It is a descriptive estimate, not a guarantee. This is a simplified illustration.
Frequently Asked Questions
What is the Pythagorean expectation?
Win% = RS^x ÷ (RS^x + RA^x), with x about 1.83 in baseball.
How many runs equal a win?
About 10 runs of run differential per win.
What does a gap between actual and expected wins mean?
Often luck, with regression expected next year.
Does it work in other sports?
Yes, with different exponents.
What does it ignore?
Timing of runs, bullpen use, and schedule strength.