How the Pythagorean Theorem says the Blue Jays will win the American League East Division!

I’ll bet you might still remember the Pythagorean Theorem from grade 9 math class.  You know the one that related the sides of the any right angle triangle with a2 + b2 = c2, where c is the “hypotenuse” while a & b are the lengths of the other two sides.  You’ve probably even know how to use it to solve “find the length of the missing side” problems.  What you may not know is that it is actually able to help you make remarkably accurate predictions about how many games a baseball team will win based on how many runs they scored and how many runs they had scored against them.

The Pythagorean winning percentage is a formula introduced by well known baseball statistician Bill James that can be used to predict the winning percentage of a baseball team over the course of a season using only two variables: the number of runs a team scores and the number of runs scored by their opponents.    The name “Pythagorean winning percentage” comes from the form of the original equation.

The Pythagorean winning percentage

Win % =  〖(Runs For)〗^2/(〖(Runs For)〗^2+〖(Runs Against)〗^2 )

It should be noted that the original formula has been modified slightly in recent years as computer technology has found that the formula

Win % = 〖(Runs For)〗^1.82/(〖(Runs For)〗^1.82+〖(Runs Against)〗^1.82 )

does a better job of predicting reality.

The Blue Jays are currently in a battle for the American League East title with the Baltimore Orioles and Boston Red Sox.  I decided to give the Pythagorean winning percentage a try.  I took the mid season statistics from  The results are shown in the table below.

Team RS RA Predicted Winning %
Toronto 478 393 0.588158533
Boston 512 439 0.56953688
Baltamore 452 424 0.529063869
note: RS = “runs scored”, RA = ” runs against”

As you can see the good news (if you’re a Blue Jays fan like me!) is that the formula predicts that Toronto will win the American League East and make the playoffs for the second year in a row!

The success of James’s formula for baseball inspired statisticians to develop “Pythagorean winning percentages “for other sports.  This is where things depart significantly form the Pythagorean analogy, for example the commonly used formula for NBA basketball is,

Win % = 〖(Runs For)〗^13.91/(〖(Runs For)〗^13.91+〖(Runs Against)〗^13.91 )

For NFL football it is,

Win % = 〖(Runs For)〗^2.37/(〖(Runs For)〗^2.37+〖(Runs Against)〗^2.37 )

I decided to see just how accurate the NBA formula is, so I used the statistics from the 2015- 2016 season for the Eastern conference teams.  Turns out to be pretty accurate … see for yourself!

Predicted winning percentage vs actual winning percentage from 2015-2016 NBA season (Eastern Conference teams) source:

Team Points For Points Against Predicted Winning % Actual Winning %
Cleveland 8552.6 8060.6 0.695112411 0.695
Toronto 8421.4 8052.4 0.65095737 0.683
Atlanta 8429.6 8134.4 0.621484626 0.585
Boston 8667.4 8405 0.605305808 0.585
Charlotte 8478.8 8257.4 0.590986968 0.585
Miami 8200 8052.4 0.562831313 0.585
Indiana 8372.2 8241 0.554707213 0.549
Detroit 8364 8314.8 0.520504777 0.537
Washington 8536.2 8577.2 0.48334346 0.5
Chicago 8331.2 8454.2 0.449210042 0.512
Orlando 8372.2 8503.4 0.446136819 0.427
New York 8068.8 8290.2 0.406963061 0.39
Milwaukee 8118 8462.4 0.359405211 0.402
Brooklyn 8085.2 8692 0.267638599 0.256
Philadelphia 7986.8 8823.2 0.200149761 0.122

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