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Pythagorean expectation calculator: expected wins from points, runs or goals

Bill James noticed that a baseball team’s winning percentage tracks the ratio of runs scored to runs allowed, and wrote the relationship in a form that resembles the Pythagorean theorem. The same idea has been applied to other sports with exponents fitted to their scoring.

Its common use is as a check on a record. A team that wins well beyond what its scoring margin implies is usually read as having overperformed, and MLB’s glossary uses exactly that reading to explain why such a team should not be expected to repeat its record. Enter the real wins too, and the calculator shows the gap.

By Reign Creative LLCPublished More in the sports hubs

Calculate expected winning percentage

Use totals from the same games: runs for baseball, points for basketball, goals for hockey.

Each preset is the exponent a cited source publishes. Choose Custom for any other.

To compare the real record with the expectation.

With actual wins, shows where the record ends up at the expected rate.

Expected winning percentage.561

Expected wins
90.8of 162
As a record
91–71
Exponent used
1.83

The formula

Expected winning percentage

win% = scoredx ÷ (scoredx + allowedx)

Expected record

expected wins = win% × games played

The exponent x sets how strongly the scoring margin translates into wins. Bill James’s original used 2. Later work fitted exponents to each sport’s results; the presets in the calculator are the ones a cited source publishes:

Exponents the calculator offers
PresetExponentSource
Baseball — Bill James’s original (2)2MLB.com glossary — Bill James’s original formula
Baseball — Baseball-Reference (1.83)1.83MLB.com glossary — the exponent Baseball-Reference uses
Basketball — Basketball-Reference (14)14Basketball-Reference glossary — its Pythagorean wins formula
Hockey — James’s formula as applied by Ryder (2)2Ryder (2004), Hockey Analytics — James’s formula interpreted for hockey

There is no American football preset: we could not open a published source for the exponent usually quoted for it, so if your reference gives one, enter it as a custom exponent.

Worked examples

Two illustrative seasons — not real teams: a baseball club that scored 800 runs and allowed 700 over 162 games, and a basketball team that averaged 112 points and allowed 108 over 82 games (9,184 and 8,856 in total). The hockey row is Alan Ryder’s own example: four goals for and three against.

Expected records from the same formula with different exponents
PresetScored – allowedExpected win%Expected wins
Baseball — Bill James’s original (2)800 – 700.56691.8 of 162
Baseball — Baseball-Reference (1.83)800 – 700.56190.8 of 162
Basketball — Basketball-Reference (14)9,184 – 8,856.62551.2 of 82
Hockey — James’s formula as applied by Ryder (2)4 – 3.640—

The basketball team outscores its opponents by less than 4% yet projects to win well over half its games: with an exponent of 14, small margins carry a lot of weight. In baseball, the 100-run margin is worth about ten wins above an even 81–81 record with either exponent.

How to read it, and its limits

  • It describes what a season’s results imply about a team’s strength. It does not forecast any game and is not betting guidance.
  • A record well above the expectation is usually read as overperformance that is unlikely to last; MLB’s glossary uses the 2018 Seattle Mariners to make the point.
  • Draws, overtime and shootouts complicate the formula in sports that have them.
  • Early in a season the margin rests on few games, so the expectation is noisy too.

For how simulators turn ratings into results — and why a good team still loses — read understanding probability in sports simulations.

162-0 Baseball Draft & GM Team asks a related question from the other direction: it projects the record a roster you draft would post over 162 games, from roster quality, position fit, hitting, pitching, speed, defence and coaching.

Reign Creative LLC makes 162-0 Baseball Draft & GM Team, so we have a commercial interest in it. It is not affiliated with, or endorsed by, any league, team, players’ association or athlete.

Good to know

Frequently asked questions

Which exponent should I use?
The one fitted to your sport and your data source. Bill James’s original exponent was 2; Baseball-Reference uses 1.83 for baseball, Basketball-Reference uses 14 for basketball, an exponent it fitted to basketball results, and Alan Ryder’s 2004 hockey analysis applies James’s formula with an exponent of 2 to goals. For any other sport, use the exponent your reference publishes.
What does it mean when a team beats its Pythagorean record?
That it has won more games than its scoring margin would normally produce. MLB’s glossary uses the 2018 Seattle Mariners: 89-73 in reality, against a Pythagorean record of 77-85 as calculated by Baseball-Reference.
Does it work for American football or soccer?
The idea can be applied to any sport with a scoring margin, but draws and overtime complicate it, and we have not been able to open a published source for an American football or soccer exponent to verify one. Use the custom exponent if your source gives one.
Is the result a prediction?
No. It describes what a set of results implies about a team’s underlying strength. The rest-of-season figure only shows what the record would be if the team kept winning at that rate — a hypothetical, not a forecast of any game, and not betting guidance.

Sources

  1. Pythagorean Winning Percentage (opens in a new tab)MLB.com Glossary — accessed
  2. Glossary — Pythagorean wins (opens in a new tab)Basketball-Reference.com (Sports Reference) — accessed
  3. Win Probabilities (2004) — the Pythagorean relationship applied to hockey (opens in a new tab)Alan Ryder, Hockey Analytics — accessed

League, governing-body and team names appear here only to identify the formulas and rules being used. Reign Creative LLC's sports games and calculators are not affiliated with, endorsed by or licensed by any league, team, players' association or athlete, and all names and marks belong to their owners.

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