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Pythagorean Expectation Calculator

Estimate the winning percentage and win-loss record a team's runs scored and allowed suggest, and compare it with the real record.

Level: Grade 6

  • Common Core standard 6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.
  • Common Core standard 6.EE.A.2.cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
Printable worksheet

Pythagorean Expectation Calculator

Inputs
Exponent

The original formula squares runs, and 1.83 fits modern baseball a little better. Other sports need their own exponent.

Compares the real record with the expected one.

Result

Expected winning percentage

0.565

Scoring 750 and allowing 650 projects to a .565 record, about 91.5–70.5 over 162 games. The team actually won 90, 1.5 fewer than expected.

As written in standings
.565
Expected wins
91.5
Expected losses
70.5
Run differential
100
Actual wins minus expected wins
-1.5Won fewer than the runs suggest.

See the step-by-step solution ↓

Step-by-step solution

  1. Here's the plan: raise both totals to the power k = 1.83. The expected winning percentage is the runs-scored part's share of the two added together

    pct=RS1.83RS1.83+RA1.83

  2. Raise runs scored to the exponent

    7501.83=182,541.71

  3. Do the same for runs allowed

    6501.83=140,485.48

  4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win

    182,541.71182,541.71+140,485.48=0.565097

  5. Multiply the expected percentage by games played to get expected wins

    0.565097×162=91.545721

  6. Subtract expected wins from actual wins to see how far the real record is off

    90−91.545721=−1.545721

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out what a team's record should be, based on the runs it has scored and allowed. A win-loss record depends partly on how a team's runs are spread out. Winning several one-run games and losing a few blowouts can give a better record than the run totals deserve.

The Pythagorean expectation, created by baseball writer Bill James, estimates the winning percentage those run totals suggest. Its name comes from the squares in the original formula, which look like the Pythagorean theorem.

Enter a team's run totals and games played to get the expected winning percentage and record. Add actual wins, and the calculator shows how far the real record sits above or below what's expected. That gap is a common way to measure luck in close games.

Where it's used

Why this math matters outside the classroom:

  • Judging a hot start. A team that has scored 300 runs and allowed 100 projects to 300² ÷ (300² + 100²), which is .900.
  • Squared exponents. With 750 runs scored and 650 allowed, squaring gives 562,500 ÷ 985,000, an expected .571 record.
  • Season previews. Baseball writers compare last season's expected and actual records to flag teams likely to regress.
  • Fantasy and sim leagues. Simulation league commissioners use expected records to judge which teams were helped by close-game luck.
  • Coaching evaluations. A youth team that scored 120 runs and allowed 120 should play about .500 ball, whatever its record.

Grade level and standards: Grade 6; Common Core State Standards for Mathematics 6.EE.A.1, 6.EE.A.2.c. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Enter runs scored and runs allowed for the season so far.
  2. Choose the exponent, the power each total is raised to: 1.83 for a fine-tuned baseball estimate, 2 for the original formula, or your own for another sport.
  3. Enter games played, and actual wins if you want the comparison.
  4. Read the expected winning percentage, the expected record and the gap from the real record.

What the inputs mean

Exponent
Squaring runs is the original version. Later analysis found that an exponent near 1.83 matches baseball records slightly better. Higher-scoring sports need much larger exponents, often in the teens for basketball points.
Actual wins (optional)
Optional. A positive gap means the team has won more often than its run totals predict.

Formula and method

expected pct = RS^k ÷ (RS^k + RA^k)
expected wins = expected pct × games
luck = actual wins − expected wins

RS is runs scored, RA is runs allowed and k is the exponent. A team that scores as many as it allows is expected to win half its games (.500). The further apart the totals get, the closer the estimate moves toward 0 or 1.

Expected wins turn the percentage into a number of wins for the games played. They usually include part of a win, which is fine for an estimate.

Worked example

A team scores 750 runs and allows 650 over 162 games, and has won 90. Use the 1.83 exponent.

  1. 750^1.83 = 182,541.71 and 650^1.83 = 140,485.48.
  2. Expected percentage: 182,541.71 ÷ (182,541.71 + 140,485.48) = 0.565097.
  3. Expected wins: 0.565097 × 162 = 91.545721, so about 91.5–70.5.
  4. Actual minus expected: 90 − 91.545721 = −1.545721.

Result: The runs suggest a .565 team, about 91.5 wins. So the club has won roughly 1.5 games fewer than expected.

Understanding the result

Most teams finish within a few wins of their Pythagorean record. A gap of five or more often comes from an unusual record in one-run games, or a strong or weak bullpen (the relief pitchers). It tends not to repeat the next season.

Early in a season, run totals are a steadier guide to how good a team is than the win-loss record. That's why many fans look at the expected record when judging a fast or slow start.

Assumptions and limitations

  • Runs scored and allowed cover the same games as the games-played figure.
  • The exponent is a fixed value. The calculator doesn't adjust it for how much scoring is typical in that league or season.
  • It's an estimate. It ignores schedule strength, injuries and roster changes. And it says nothing about which games remain.
  • Exponents for other sports vary between sources. Treat any custom value as a choice you're making, not a rule.

Frequently asked questions

Why is it called Pythagorean?

The original formula squares runs scored and runs allowed, and adds the squares. That looks like a² + b² in the Pythagorean theorem. There's no triangle involved.

Should I use 2 or 1.83?

Either gives a reasonable estimate. The squared version is easier to work by hand. 1.83 is the more common choice in modern baseball reference work, because it fits real records slightly better.

Does beating the expectation mean a team is lucky?

Partly. Close-game results are mostly random over a season. But a dominant bullpen can help a team win more one-run games than average. A large gap is a reason to look closer, not proof of luck.

Can I use this for football or basketball?

Yes. Use points instead of runs, and a custom exponent that suits the sport. Scoring is higher in those sports, so they need bigger exponents than baseball.

Sources

How this calculator is checked

Method: Bill James's Pythagorean expectation as described in the MLB glossary, with exponent 2 (original) or 1.83 (common refinement), or a user-chosen exponent. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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