Streamlining the "Pythagorean Theorem of baseball"
Mathematicians test simplified formula to predict winning baseball
percentages
College Park, MD (March 29, 2004)--Is your local Major League Baseball
team better than its record suggests? Math researchers are considering
alternatives to the Pythagorean Theorem of Baseball, devised by baseball
statistician Bill James. Introduced in the 1980s, the "theorem" predicts
the winning percentage of a baseball team based on how many runs the
team scores--and how many runs it allows.
Websites, including ESPN's, often include the Pythagorean
prediction of the winning percentage of a team during
the season. Fans compare the Pythagorean Theorem to
the actual winning percentage, in an effort to determine
if a team is under- or over-achieving.
When a team scores fewer runs than it allows, the
Pythagorean model predicts that the team should have
a losing record. For the 2001 season, the New York
Mets allowed more runs than they scored and had a winning
record; they did much better than the Pythagorean model
predicted. So they can be considered an overachieving
team. Because the Colorado Rockies scored more runs
than they allowed but had a losing record, they were
possibly an underachieving team.
Now, Michael Jones and Linda Tappin of Montclair State
University in New Jersey have devised mathematically
simpler alternatives to the Pythagorean Theorem of
Baseball.
To predict the winning percentage of a team, one new
model simply uses a little addition, subtraction, and
multiplication. It starts with the total runs scored
by the team in all its games (Rs), and subtracts the
runs it allows (Ra), and then multiplies it by a number
called "beta" (B) which is chosen to produce
the best results. For the 1969-2003 seasons, the optimal
values of B range from 0.00053 to 0.00078, with an
average of 0.00065.
Adding 0.5 to the result gives the predicted winning
percentage of the team. The resulting formula looks
like this:
The estimated winning percentage, P = 0.5 + B*(Rs-Ra)
Because they only use addition, multiplication, and
subtraction, these formulas are known as "linear
functions"-the simplest kind of equations in mathematics.
In contrast, the original Pythagorean Theorem of Baseball
is more complex. It uses exponents: Runs scored and
runs allowed are squared-raised to the second power.
The resulting formula is:
P=[Rs2/Ra2+Rs2)]
The equation gets its name because of its similarity
to the Pythagorean Theorem in geometry, which relates
the lengths of the sides in a right triangle as
a2 + b2= c2, where a and b are the shorter sides and
c is the longest side (the hypotenuse).
Because the Pythagorean theorems use exponents, these
formulas are "nonlinear" equations, which
are generally more complex than linear formulas.
So was the original Pythagorean Equation of Baseball
needlessly complicated? Does the linear equation do
just as good a job?
For the baseball seasons between 1969-2003 the linear
formula works almost as well in its predictions as
the original Pythagorean theorem, Jones and Tappin
reported at this winter's Joint Mathematics Meetings
in Phoenix. The one real exception is the 1981 season
when there was a baseball strike.
While Tappin and Jones have only analyzed whole seasons
with their new formula, they are exploring how well
it works for seasons-in-progress. If their formula
meets with continued success, you may soon find it
on your favorite sports website.
Experts
Michael A. Jones
Associate Professor
Department of Mathematical Sciences
Montclair State University
Upper Montclair, NJ 07043
973-655-5448
Linda Tappin
Associate Professor of Statistics
Department of Mathematical Sciences
Montclair State University
Upper Montclair, NJ 07043
973-655-7242
Contact
Ben Stein
American Institute of Physics
301-209-3091
Mike Breen
American Mathematical Society
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