The so-called "Pythagorean Formula" was invented by sabermetrics pioneer Bill James in the early 1980's and is used to predict a baseball team's winning percentage on the basis of its runs scored (RS) and runs allowed (RA).
The exponent in James' original formula was lambda = 2, which reminded him of the Pythagorean theorem from Euclidean geometry, thus the name stuck. From a statistical perspective, the Pythagorean Formula is a logistic regression model where the response variable is a team's log-odds and the predictor variable is the logarithm of (RS/RA). Fitting a logistic regression model to a historical data set spanning the MLB seasons 1901-2013 gives a best fitting exponent of about 1.86.
Confidence intervals for a baseball team's winning percentage can be obtained by using a Scheffe-type simultaneous prediction band based on a fitted linear regression model that approximates the above logistic regression model. The formula for the confidence interval
is given below.
