Bill James’s insight
In the late 1970s, baseball analyst Bill James noticed that a team’s record follows closely from the runs it scores and allows. He called the formula the Pythagorean expectation, because it resembles the Pythagorean theorem:
Win% = RS^x ÷ (RS^x + RA^x)
RS is runs (or points) scored, RA is runs allowed and x is an exponent fitted to each sport.
Exponents by sport
| Sport | Typical exponent |
|---|---|
| Baseball (MLB) | 1.83 (James’s original 2) |
| Basketball (NBA) | 13.91 to 16.5 |
| American football (NFL) | 2.37 |
| Ice hockey (NHL) | 2.0 to 2.15 |
Basketball’s exponent is large because many points are scored, so a small edge in points per game translates into many wins.
Example
A baseball team scores 800 runs and allows 700 over 162 games:
- expected win percentage: 800^1.83 ÷ (800^1.83 + 700^1.83) ≈ 56.1%;
- expected record: about 91 wins and 71 losses.
If the team actually won 88 games, it under-performed its run differential by about 3 wins.
What luck tells you
Teams that win many more games than their Pythagorean expectation usually have an unusual record in close games or a great bullpen. Close-game records are mostly unrepeatable, so those teams tend to regress the next season. Teams below their expectation often improve. Analysts use the gap to separate skill from luck, especially in the first half of a season.
A rule of thumb
In baseball, about 10 extra runs of run differential are worth one extra win. The example team’s +100 run differential is worth about 10 wins above .500.
Limits
- The formula works best over a full season. After a few games it is noisy.
- Blowouts inflate run differentials without adding wins.
- Sports with frequent ties, such as soccer, need adapted models that treat draws separately.
- Variants such as Pythagenpat adjust the exponent to the scoring environment: x = ((RS + RA) ÷ games)^0.287.