Pythagorean Expectation in Baseball

• Originally developed by Bill James
• Simple formula: \(\text{Winning Percentage} = \dfrac{Runs Scored^2}{Runs Scored^2 + Runs Allowed^2}\)
• Gives an indicator of how lucky a team was
  • Above expectation: lucky
  • Below expectation: unlucky
• Not a perfect estimator, but usually within 3 or 4 games for the majority of teams
• There are usually a few outliers far from their expectation, like in any distribution
  

• Data from Lahman DB (http://www.seanlahman.com/baseball-archive/statistics)

runs <- read.csv("runs.csv"); runs$Pythag <- (runs$R^2)/(runs$R^2 + runs$RA^2)
plot(runs$Pythag, runs$WPct, xlab = "Pythagorean Expectation",
     ylab = "Winning Percentage", main = "WPct vs Pythag since 1955; r = 0.94")
fit <- lm(runs$WPct ~ runs$Pythag); abline(fit, col = "red", lwd = 2)

• There are close or equivalent forms that some people prefer
• One example: \(\dfrac{Wins}{Losses} = \dfrac{Runs Scored^2}{Runs Allowed^2}\)
• Dispute over the exponent
  • 2 was James's original choice, and it's the most simplistic
  • 1.83 has been shown to be more accurate in general
  • Other versions (Pythagenport, Pythagenpat) change the exponent for each team
• In the app, I give the user the option of which exponent to use
• The user also inputs the runs scored and runs allowed numbers (default is 700 for both)

• Panel for explanation and user input; separate panel for displaying the user's choices and results
• Winning percentage as well as record over a 162-game season are calculated and displayed
• Sample calculation below for a team that scored 750 runs and allowed 700, using 2 as the exponent

RS <- 750; RA <- 700; exponent <- 2
WPct <- (RS^exponent)/(RS^exponent + RA^exponent)
wins <- WPct * 162; losses <- (1 - WPct) * 162
wins; losses; format(round(WPct, 3), nsmall = 3) # show to 3 decimal places
## [1] 86.58
## [1] 75.42
## [1] "0.534"