Pythagorean Win Expectation Calculator
Pythagorean win expectation estimates how many games a team should win based on how many runs it scores and allows, rather than its actual record. Devised for baseball, it captures the idea that a team's run differential predicts its winning percentage more reliably than win-loss results alone, which can be skewed by luck in close games. The classic formula squares runs scored, then divides by the sum of runs scored squared and runs allowed squared. This calculator applies that formula and returns an expected winning percentage, along with projected wins over a season length you choose. Enter 750 runs scored and 650 runs allowed over a 162 game season and the tool returns an expected win rate of about 0.571 and roughly 92.51 projected wins. Comparing this expectation with a team's real record shows whether it has been lucky or unlucky in one-run games, which analysts use to forecast regression toward the expected level. Because the inputs are season run totals and the exponent is fixed at two in this classic version, the calculation is fully deterministic. The complete formula and a worked example that reconciles exactly to the calculator above appear in full below for you to follow by hand.
Pythagorean win expectation predicts win rate from runs: win% = RS^2 / (RS^2 + RA^2). With 750 runs scored and 650 allowed, the expected win rate is about 0.571, roughly 92.51 wins over 162 games.
Pythagorean win expectation formula
expected win% = RS^2 / (RS^2 + RA^2)
RS = runs scored, RA = runs allowed
projected wins = expected win% x season games
Runs scored squared, divided by the sum of runs scored squared and runs allowed squared, gives the expected winning percentage. Multiplying by the number of games projects the season win total.
Worked example
A team that scores 750 runs and allows 650 over a 162 game season.
- RS^2 = 750^2 = 562,500
- RA^2 = 650^2 = 422,500
- Expected win% = 562,500 / (562,500 + 422,500) = 562,500 / 985,000 = 0.5711
- Projected wins = 0.5711 x 162 = 92.51
These are the calculator's default inputs, so the result above matches the widget exactly.
Pythagorean Win Expectation Calculator: frequently asked questions
What is Pythagorean win expectation?
It is an estimate of how many games a team should win based on runs scored and runs allowed, rather than its actual record. The classic formula divides runs scored squared by the sum of runs scored squared and runs allowed squared.
Why is it called Pythagorean?
Because the formula resembles the Pythagorean theorem, with squared terms in the numerator and denominator. The name stuck even though it has nothing to do with right triangles; it simply uses squares of the run totals.
What does it tell you about a team?
It tells you the winning percentage a team's run differential implies. If a team's real record is well above its Pythagorean expectation, it may have been lucky in close games and could regress; if below, the reverse.
Does the exponent have to be 2?
The classic version uses an exponent of 2, which this calculator applies. Analysts have proposed other exponents that fit historical data slightly better, but 2 remains the standard, simple form and is accurate enough for most uses.
Can it be used for other sports?
Yes, with points scored and allowed in place of runs, the same idea applies to many sports, though the best-fitting exponent varies by sport. This calculator uses the classic squared form, which is closest to its baseball origin.
Official sources
- Physical activity and health guidance: US Centers for Disease Control and Prevention (CDC). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.