Pythagorean expectation

The Pythagorean expectation is the formula famous baseball analyst Bill James developed to calculate the home runs scored or allowed. The fans can compare this result with competitive indexes and understand their favourite team: under or over-estimated. The calculation formula is

Win% = (Runs Scored)² / ((Runs Scored)² + (Runs Allowed)²)

This formula was originally developed with an exponent of 2, but it has since been improved upon with different exponents, such as 1.83, to provide more precise results.

For example, the NY Yankees scored 673 home runs and 698 allowed in the 2023 season. Include the result in the formula, and we get:

Win% = (673)² / ((673)² + (698)²) = -.481

Based on the 162-game schedule, the team had to win 78 games, but the Yankees won four games more despite this index. Bettors sometimes calculate the live Pythagorean at the moment of the game they’re interested in.

Baseball Prospectus modification

The enthusiasts always tried to modify the formula. For example, Baseball Prospectus uses the logarithm of the ratio of the home runs scored plus allowed to total games, then multiples it by 1,5 and adds the 0,45. The full calculation is as follows.

First, let’s find the exponent:

Exponent = 1.50 * log((Runs Scored + Runs Allowed) / Games Played) + 0.45

Logarithm base 10 is used in this formula. By customising the exponent, Davenport’s formula can better estimate a team’s winning percentage based on their run environment. This modified Pythagorean expectation formula reduces the mistake caused by the set exponent.

After that, we need to move the exponent here to find the Win%:

Win% = (Runs Scored)^Exponent / ((Runs Scored)^Exponent + (Runs Allowed)^Exponent)

Thus, if we calculated the 2023 New York Yankees percentage via this formula, it should be:

Exponent = 1.50 * log((673+698) / 162) + 0.45 = 1.8405

Win% = (673)^1.8405 / ((673)^1,8405 + (698)^1,8405) = 0,483 or 78 wins again.

Second and other orders

The Sabmetricians use different methods to calculate the percentage. The classic formula considers the first order of home runs. To filter out the high chance of luck in the calculation, the enthusiasts exploit the current statistics (singles, doubles, and triples). These figures help to calculate the required number of home runs to reach the targeted results.