Transcript Slide 1

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.”
• Wolfram’s www.mathworld.com 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 Magazine’s
100 Most Influential People
James’ Pythagorean Won-Loss Record
This formula gives what a baseball team’s 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 team’s winning percentage as a function
of the team’s 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 year’s W-L record. Here are a couple of interesting examples from this
season:
On August 4th, the 2008 Texas Rangers were 59-54 (W-L% .522), with a
Pythagorean expectation of 54-59 (W-L% .478). They finished the season at 7983 (W-L% .488).
On July 20th, the 2008 Cleveland Indians were 43-54 (W-L% .443), with a
Pythagorean expectation of 49-48 (W-L% .505). They finished the season at 8181 (W-L% .500).
On July 20th, the 2008 Toronto Blue Jays were 48-50 (W-L% .490), with a
Pythagorean expectation of 52-46 (W-L% .531). They finished the season at 8676 (W-L% .531).
The lesson here is that a team’s luck will usually catch up with them over the course
of a 162 game season. Of course, there are always exceptions:
On July 20th, the 2008 Anaheim Angels were 60-38 (W-L% .612), with a Pythagorean
expectation of 53-45 (W-L% .541). They finished the season at 100-62 (W-L% .617).
James’ Derivation of the
Pythagorean Formula
Bill James’ discovery of this formula was, by his own admission,
lucky. In response to an email 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 team’s 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 team’s runs scored and runs allowed, James’
Pythagorean Formula can be shown to follow
mathematically.
Assumptions
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 Weibull’s 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.”
The Weibull Distribution
Remark on the Weibull Distribution
parameters
Statement of the Theorem:
The Joint PDF
Since a team’s 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:
Independence of Random Variables
Expected Values of the RVs X and Y