Tuesday, July 2, 2013

More fun with binomial stats: Dunn, Davis, and Anderson

I mentioned in my last post that it is possible to use binomial theorem to compare player statistics year-to-year (between any two samples actually) and assess whether a player’s current numbers are in line with previous performance or well beyond our expectations, indicating that something – a new mental approach, a lingering injury, a mechanical adjustment – has changed and substantially affected his or her abilities.

Since it’s fun* and easy,** let's apply this to a few more cases and see what we can see.

*You bet your sweet ass it’s fun.
**Again, see my previous post for a slightly more detailed explanation of just what we’re up to here.

Case 1: Adam Dunn
Because I’m not sad enough (as a White Sox fan that is), let’s take a look and see what happened to the Big Donkey during the 2011 season that saw him post one of the worst seasons by an everyday player in the history of the game. Ooof.

To be brief, let’s just look at home runs. In 2011, Dunn had 11 in 496 plate appearances. Now if we compare that to his career numbers of 427 HRs in 7514 PAs, we can get some appreciation for whether “normal” Adam Dunn would have been capable of the numbers that 2011 “zombie” Adam Dunn produced.

In a word: No. In two words: Hell. No. For this comparison, the p-value is 0.0010. That’s basically the equivalent of my computer grabbing me by the collar and screaming “WTF, are you crazy!?*” A p-value of 0.001 means we can be 99.9% sure that zombie Adam Dunn is not the same person as normal Adam Dunn.

*Or, since I have a Mac maybe it would be the equivalent of my computer spitting its fair-trade espresso in my face at the ridiculousness of such a suggestion.

So what happened? I have my own theories, but it could have been any number of things. Adjusting from playing in the field every day to mostly DHing could have thrown off Dunn’s routine, he could have had a difficult time getting along with his brash new manager, or he could have confused himself with a comic book character and rushed back to quickly after an appendectomy. Whatever the case was, the fact that it statistically shouldn’t have happened is of very little consolation.

Case 2: Chris Davis
Davis, who mans first base for the Baltimore Orioles, has been on fire since the season began. Although he trails Miguel Cabrera and Mike Trout in fWAR, he leads all of baseball in homeruns with 30 (as of Sunday). And besides, that’s some pretty good company to be in regardless. Davis began his career with the Rangers and though he flashed plenty of power in the minors (and some with the big club), he never could get himself entrenched in the show.

When he got traded to the Orioles in 2011 it didn’t generate too much noise and his 33 HR campaign last year was real, but not quite spectacular. This year he has been simply sublime – he’s just three shy of last year’s total as of Sunday. So does it make sense? Actually, it kinda does – to the extend that any stretch of baseball with an ISO of nearly .400 can make sense.

Comparing “monster” 2013 Davis to his “regular” 2012 self, we get a p-value just under 0.15 which, depending on what parameters you set for yourself beforehand, is generally taken to mean that monster Davis is really no different than regular Davis. Even if we look at his career stats (minus an awful 2010 season that probably played no small role in convincing the Rangers to move him) versus 2013, we get a p-value just over 0.05 which is typically the threshold for significance. So even though it was unlikely, Davis’s power surge was nowhere near as anomalous as Dunn’s outage.

Case 3: Brady Anderson
Anderson emerged from nowhere to hit 50 HRs for the 1996 Baltimore Orioles.* It was just about as outlier of a season as you can get. Comparing that season with Anderson’s career, we get a p-value of 1 preceded by ten zeroes and a decimal place. We’re almost a billion orders of magnitude more certain about this comparison than we were about Adam Dunn’s weird 2011 year. Whoa.

*His career year was probably overshadowed by this.

Over these three examples, we’ve seen how binomial theorem can be used to identify outlier seasons – whether for identifying player talent, better approaches/technique, or other reasons (injury, chemistry, etc). Although we’ve only applied it to home runs, it can obviously work for pretty much any statistic with an outcome that can be evaluated strictly as success/failure.

Ed Note: All stats accurate as of before Mass Sunday morning 6/30/12.

Ed Note II: it has since been brought to my attention that we should be using Poisson statistics since we can never know the "true talent" of Player X. In these cases, the p-values change but the conclusions remain the same.


  1. What exactly is it that you're saying is binomially distributed (i.e. home runs with plate appearances as trials or something else)?

    1. in this case yes, HR as "successes" with PA as "trials"

    2. Interesting. What do you think of using a Poisson distribution (which is also the same as assuming that the inter-arrival times are distributed as a continuous exponential distribution)? The Poisson would indicate a variance higher than the binomial distribution, which I think might actually line up better with observations. Variation from year to year is pretty high. This paper (not even sure if this was ever published in a reviewed journal in any way) discusses the Poisson model for home runs:


      Although there are many more technical discussions of models for baseball statistics. In any case I wonder if you're inflating your P-value significance with the Binomial model.

      One other note on Poisson: on a take-home final for a stat class recently I had to solve a problem related to a bivariate Poisson model. Later, I looked up that type of distribution, and the first application mentioned was scoring rates for NFL teams. :)

    3. Shiiiit, i think you're right. If I remember right you use Poisson when you don't know the "true" rate (which would apply for just about any sports situation). I just reached back and pulled out the first test I remembered from stats for this, which happened to be binomial.

      I reran the numbers and indeed the p-values do change, but the conclusions (reject or accept) remain - good thing I chose such extreme examples!

    4. Also, I saw that paper you posted - have something similar cooking but with a slight twist.


Keep it civil.