from Amazon
Thursday, January 18, 2007
Do Umpires Have Their Own Strike Zone?
Yes. Here’s how to tell.
DanAgonistes asks: “ It would be interesting to see whether there is any trend that holds over from year to year “.
We don’t need year-to-year data. We just follow the same process we follow for everything else. Figure out what a random distribution would look like. Figure out the actual distribution. The difference is the umpires’ inherent “skill”. Convert that measure into a correlation.
Using the wonderfully-compiled umpire data here:
http://www.baseballprospectus.com/statistics/sortable/index.php?cid=131616
I added the “strikes + balls” of each umpire. (Balls in Play were excluded. And, it seems that the data is somewhat off, as I expected strikes+balls+battedBall to equal pitches. It doesn’t.)
I took the 64 umpires with the most strikes+balls (which I’ll now call pitches, which I hope is not confusing.) The strike percentage, or strikes per pitch, was .534. I figured for each umpire what one standard deviation would be, if all calls followed a binomial around this population mean. And then, figured how many standard deviations they were above the population mean, which is their z-score. I then took the standard deviation of the z-score.
(A similar process was followed here in more detail: http://www.tangotiger.net/dipsbands.html )
If we get a z-score of 1.00, then we know it was all random, and umpires call it by the book. The z-score was a high 1.65. This translates into a correlation coefficient of r = .63 (or 1 - 1/1.65^2).
Another way to get to that number is to note that with an average of 8000 pitches, the random standard deviation is .0056. The actual observed standard deviation was .0092.
Regression toward the mean = var(luck) / var(observed)
where var = variance, and is standard deviation squared.
So, regression toward the mean = (.0056/.0092)^2 = .37
Note also that r = 1 minus regression toward the mean, making r = .63
Now that we have our r of .63, and we know the number of pitches was 8000, we can create our regression toward the mean equation:
regression toward the mean = 4700 / (4700 + pitches)
If you have an umpire with 8000 pitches, you regress his actual observed strike percentage 37% toward the mean. Greg Gibson had 9037 pitches, meaning his regression toward the mean is 34%. His observed strike percentage is 51.9%, of which we regress 34% toward .534, making Gibson’s “skill” strike percentage as .524.
McClelland is at .520 and Eddings at .550.
If anyone wants to do a year-to-year correlation, I would bet that if the average number of pitches in the sample was around 8000, the correlation would be an r = .63.