Pythagorean Win Expectation formula (Bill James Sports Analytics)

A scoring-based estimate, not a measurement of luck

The classic squared-runs model uses exponent2. Other exponents must be fitted or justified for the sport and data; no fixed exponent is universally correct. A fixed1.83 is not the same as a context-dependent Pythagenpat calculation.

The output is a model estimate based on aggregate scoring. Actual-minus-estimated wins is a residual, not proof of luck, true talent or inevitable future regression. Schedule, scoring distribution and model error can contribute.

Use scoring totals and game counts from the same period. This example excludes draws, requires nonnegative scoring totals with at least one positive, a positive exponent, positive whole-number games and whole-number wins between0 and games.

Generalized Pythagorean Win Percentage = Scored^k / (Scored^k + Allowed^k)

B4 is an assumed positive exponent; the example uses2. Validate model fit before interpreting predictions.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<0,B3<0,B2+B3=0,B4<=0,B5<=0,B5<>INT(B5),B6<0,B6>B5,B6<>INT(B6)),"Invalid inputs",ROUND((B2 ^ B4) / ((B2 ^ B4) + (B3 ^ B4)), 4)))

Actual-minus-model wins

Subtract the unrounded expected win count from observed wins. This residual has no automatic causal interpretation.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<0,B3<0,B2+B3=0,B4<=0,B5<=0,B5<>INT(B5),B6<0,B6>B5,B6<>INT(B6)),"Invalid inputs",ROUND(B6 - (B5 * ((B2 ^ B4) / ((B2 ^ B4) + (B3 ^ B4)))), 1)))

Model-estimated wins

Multiply games by the unrounded model proportion, then round the displayed count.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<0,B3<0,B2+B3=0,B4<=0,B5<=0,B5<>INT(B5),B6<0,B6>B5,B6<>INT(B6)),"Invalid inputs",ROUND(B5 * ((B2 ^ B4) / ((B2 ^ B4) + (B3 ^ B4))), 1)))

Further reading

• Microsoft Support — POWER function
• Research on improvements to Pythagorean win estimates