A Derivation of Bill James Pythagorean Won-Loss Formula

What is Sabermetrics? The term sabermetrics, coined by noted baseball analyst Bill James, comes from the acronym for the Society for American Baseball Research, or SABR. James unofficially defined sabermetrics as the search for objective knowledge about baseball. Wolframs defines sabermetrics as the study of baseball statistics.

Bill James: Godfather of Sabermetrics Bill James is a baseball historian, writer, and statistician, who was one of the first supporters/pioneers of sabermetrics and has been the most influential sabermetrician since the discipline began. He started his work in sabermetrics in the early 1970s, and, though unpopular at the time, his work and influence have spread and many of his ideas and statistical inventions are in common use in baseball (as well as other sports) today He is currently the Senior Operations Advisor for the Boston Red Sox, and in 2006 was named one of Time Magazines 100 Most Influential People

James Pythagorean Won-Loss Record This formula gives what a baseball teams overall winning percentage SHOULD have been, based on the number of runs scored and runs allowed. Statistically speaking, it gives an expected value for a teams winning percentage as a function of the teams runs scored and runs allowed. The formula was named Pythagorean W-L because it reminded James of the Pythagorean theorem.

The Pythagorean formula is often used in the middle of a baseball season to estimate how a team will finish the season, or at the end of the season for a reasonable guess at next years W-L record. Here are a couple of interesting examples from this season: On August 4 th, the 2008 Texas Rangers were (W-L%.522), with a Pythagorean expectation of (W-L%.478). They finished the season at (W-L%.488). On July 20 th, the 2008 Cleveland Indians were (W-L%.443), with a Pythagorean expectation of (W-L%.505). They finished the season at (W-L%.500). On July 20 th, the 2008 Toronto Blue Jays were (W-L%.490), with a Pythagorean expectation of (W-L%.531). They finished the season at (W-L%.531). The lesson here is that a teams luck will usually catch up with them over the course of a 162 game season. Of course, there are always exceptions: On July 20 th, the 2008 Anaheim Angels were (W-L%.612), with a Pythagorean expectation of (W-L%.541). They finished the season at (W-L%.617).

Bill James discovery of this formula was, by his own admission, lucky. In response to an that I sent him asking about his methods for deriving the formula, he responded: Mostly luck. I had been experimenting with the data and had several other good formulas for data within 1 standard deviation of the mean. However, many of them were complicated, and they returned absurd answers in extreme cases. But one day, as I was walking across campus at the University of Kansas, it hit me: it was a simple relationship of squares. This presented a much better fit to the data, and was much more elegant. James Derivation of the Pythagorean Formula

James formula for predicting a baseball teams winning percentage worked beautifully, despite the fact that its derivation had little basis in statistical theory. A paper published by Steven J. Miller (then an Associate Professor of Mathematics at Brown University) showed that, under reasonable statistical assumptions about a baseball teams runs scored and runs allowed, James Pythagorean Formula can be shown to follow mathematically. James Derivation of the Pythagorean Formula

Runs scored and runs allowed can be approximated by continuous random variables In order to obtain a simple closed form for expressions for the probability of scoring more runs than allowing in a game, we assume that the runs scored and runs allowed are drawn from continuous and not discrete distributions. This allows us to replace discrete sums with continuous integrals... Of course assumptions of continuous run distribution cannot be correct in baseball, but the hope is that such a computationally useful assumption is a reasonable approximation to reality. Runs scored and runs allowed can be modeled by continuous Weibull distributions [The Weibulls flexible shape parameters] make it much easier to fit the observed baseball data with a Weibull distribution than with some of the better known distributions. Further, the exponential decays too slowly to be realistic; it leads to too many games with large scores. By choosing our parameters appropriately, a Weibull has a much more realistic decay... Runs scored and runs allowed are statistically independent In a baseball game, runs scored and runs allowed cannot be entirely independent, as games do not end in ties... Modified chi-squared tests do show that, given that runs scored and runs allowed must be distinct integers, the runs scored and runs allowed per game are statistically independent. Assumptions

The Weibull Distribution

Remark on the Weibull Distribution parameters

Statement of the Theorem:

Since a teams winning percentage is the probability that they will score more runs than they allow, we want to find P(X>Y), where X is runs scored and Y is runs allowed. Since this probability depends jointly on X and Y, we use a joint probability density function: The Joint PDF

Independence of Random Variables

Expected Values of the RVs X and Y