Pythagorean expectation: what a team's scores say about its record
Winning percentage tells you what happened. But a team that wins lots of close games and loses a few blowouts may have a better record than its play deserves. Pythagorean expectation is a simple formula that estimates the winning percentage a team "should" have from the points (or runs, or goals) it scored and allowed.
It was introduced for baseball by the writer and analyst Bill James, and versions of it are now used across many sports.
The formula
Expected win % = Scoredx ÷ (Scoredx + Allowedx)
"Scored" and "allowed" are season totals. The exponent x depends on the sport. James's original version used 2, which is where the name comes from: the formula looks like the Pythagorean theorem, a² + b².
Which exponent to use
| Sport | Common exponent | Why |
|---|---|---|
| Baseball | 2, or 1.83 for a closer fit | Low-scoring with plenty of randomness |
| American football | About 2.37 | Fewer games, larger scoring margins |
| Ice hockey | Close to 2 | Low-scoring, similar to baseball |
| Basketball | About 14 (13.91 is often quoted) | High-scoring, so small margins are decisive |
The exponent rises with scoring. In basketball, where teams score over 100 points a game, a small edge in points per game turns into a big edge in wins. In baseball, a team can outscore opponents over a season and still lose a lot of games.
Worked example: baseball
A team scores 800 runs and allows 700 over a 162-game season.
- With x = 2: 800² ÷ (800² + 700²) = 640,000 ÷ 1,130,000 = .566, or about 92 wins.
- With x = 1.83: the same sum gives .561, or about 91 wins.
A simpler way to work it out is to take the ratio of scored to allowed, raise it to the exponent, and divide by one plus itself. Here (800 ÷ 700)1.83 ≈ 1.277, and 1.277 ÷ 2.277 ≈ .561.
Worked example: basketball
A team averages 112 points a game and allows 108. With x = 14: (112 ÷ 108)14 ≈ 1.66, and 1.66 ÷ 2.66 ≈ .624. Over an 82-game season that's about 51 wins. A margin of just 4 points a game is worth about 10 wins above an average team.
Lucky and unlucky teams
The interesting part is comparing a team's real record with its Pythagorean record.
- Real record much better than expected: the team has probably won an unusual number of close games. Close games are partly luck, so this often evens out, and the team may fall back the following season.
- Real record much worse than expected: the team has probably lost a lot of close games. It may be better than its record suggests.
A gap of a few wins is normal over a baseball season. A gap of eight or ten is worth a closer look.
Limits
- Small samples. Early in a season, totals are too small to say much.
- Blowouts. One huge win can inflate the "scored" total. Some analysts cap margins for this reason.
- Not a law of nature. The exponents are fitted to past seasons. They describe the typical relationship, not every team.
- Draws. The basic formula assumes no draws, so it fits football leagues less neatly.
Using it for your own league
If your club or office league records scores, you can try this yourself. Total each team's points for and against, start with an exponent of 2, and compare the expected winning percentage with the real one. For actual records, the winning percentage guide and Win% do the counting for you.