## How often should Youk take his base? June 30, 2010

Posted by tomflesher in Baseball, Economics.
Tags: , , , , , , , ,

Kevin Youkilis is sometimes called “The Greek God of Walks.” I prefer to think of him as “The Greek God of Take Your Base,” since he seems to get hit by pitches at an alarming rate. In fact, this year, he’s been hit 7 times in 313 plate appearances. (Rickie Weeks, however, is leading the pack with 13 in 362 plate appearances. We’ll look at him, too.) There are three explanations for this:

1. There’s something about Youk’s batting or his hitting stance that causes him to be hit. This is my preferred explanation. Youkilis has an unusual batting grip that thrusts his lead elbow over the plate, and as he swings, he lunges forward, which exposes him to being plunked more often.
2. Youkilis is such a hitting machine that the gets hit often in order to keep him from swinging for the fences. This doesn’t hold water, to me. A pitcher could just as easily put him on base safely with an intentional walk, so unless there’s some other incentive to hit him, there’s no reason to risk ejection by throwing at Youkilis. This leads directly to…
3. Youk is a jerk. This is pretty self-explanatory, and is probably a factor.

First of all, we need to figure out whether it’s likely that Kevin is being hit by chance. To figure that out, we need to make some assumptions about hit batsmen and evaluate them using the binomial distribution. I’m also excited to point out that Youk has been overtaken as the Greek God of Take Your Base by someone new: Brett Carroll.

I’m going to assume that the rate of hit batsmen is constant over time. This assumption is probably justified, since the number of hit batsmen per team per American League game has stayed between .21 and .25 since 1996, and the number of plate appearances per team per game has stayed between 33.98 and 34.9 over the same time period. Based on that, I feel justified in using the 2009 hit batsman rate to evaluate Youkilis’s stats this year. It’s undesirable to use this year’s rates if 2009’s will fit, since this year has a much smaller number of occurrences. Since a number of players with only a few at-bats might distort the average, I limited my sample to only players with 50 plate appearances or more, then divided the total number of HBP by the total number of plate appearances and got .00859. (For the record, the sample of all players with at least one plate appearance had a rate of .00850.)

I’m also going to assume that occurrences of hit batsmen are binomially distributed. That is, they occur at a known rate, which is equivalent to the rate of hit batsmen in 2009, and that every individual hit-by-pitch is independent of all others. (I might have to relax this assumption later, but it’s good for a first approximation.) As a result, the probability of being hit by a pitch k times in n plate appearances with a known rate of p is

$f(k;n,p) = {n\choose k}p^k(1-p)^{n-k}$

where

${n\choose k}=\frac{n!}{k!(n-k)!}$

Using R, I estimated a binomial distribution using n=313 plate appearances, k=1,2,..,10, and p=.00859 to determine the probability that he’d be hit k times. My results are:

 HBP p(HBP) Total 0 0.06721 0.06721 1 0.18225 0.24946 2 0.2463 0.49576 3 0.2212 0.71696 4 0.14851 0.86547 5 0.07951 0.94498 6 0.03536 0.98034 7 0.01344 0.99378 8 0.00445 0.99823 9 0.0013 0.99953 10 0.00034 0.99987

If Youkilis is a normal hitter, then it’s 98% likely that Youkilis would be hit less than seven times. It’s very unlikely that in those 313 plate appearances he’d be hit by chance alone 7 times.

Youkilis has company, though: the aforementioned Rickie Weeks, who’s been hit 13 times in 362 plate appearances. I re-estimated the distribution using k=1,2,…,15, n=362, p=.00859 and got the following results:

 HBP p(HBP) Total 0 4.404E-02 0.04404 1 1.381E-01 0.18216 2 2.160E-01 0.39815 3 2.245E-01 0.62269 4 1.746E-01 0.79727 5 1.083E-01 0.90556 6 5.582E-02 0.96138 7 2.459E-02 0.98597 8 9.450E-03 0.99542 9 3.220E-03 0.99864 10 9.900E-04 0.99963 11 2.731E-04 0.999903127 12 6.921E-05 0.999972337 13 1.614E-05 0.99998848 14 3.486E-06 0.999991966 15 7.007E-07 0.999992667

It’s almost impossible for Weeks to have been hit that much. Again, he’s 95% or more likely to have been hit six times or fewer, and there’s a whopping 99.99885% chance that if he’s an average hitter he’d be hit less than he has this season in as many plate appearances.

The king of hit batsmen, though, and the new Greek God of Take Your Base, is Florida Marlins pinch hitter and outfielder Brett Carroll. In 90 plate appearances this year, he’s been hit seven times! That’s as much as Youkilis, but far more efficient – he required less than one-third of the plate appearances to achieve the same number of plunks. Using his 90 plate appearances and k=1,2,..10, Carroll’s distribution is below:

 HBP p(HBP) Total 0 4.60E-01 0.4600902 1 3.59E-01 0.8188182 2 1.38E-01 0.9571132 3 3.51E-02 0.9922562 4 6.62E-03 0.99887814 5 9.87E-04 0.99986485 6 1.21E-04 0.99998594 7 1.26E-05 0.99999852 8 1.13E-06 0.999999652 9 8.93E-08 0.999999741 10 6.27E-09 0.999999747

Carroll, in 90 plate appearances, should have been hit less than twice. His rate – .078 times hit by pitch per plate appearance – is more than nine times the league’s rate. Ascend Mount Olympus, Brett, and work on getting out of the way more often.

 HBP p(HBP) Total 0 4.40400E-02 0.04404 1 1.38120E-01 0.18216 2 2.15990E-01 0.39815 3 2.24540E-01 0.62269 4 1.74580E-01 0.79727 5 1.08290E-01 0.90556 6 5.58200E-02 0.96138 7 2.45900E-02 0.98597 8 9.45000E-03 0.99542 9 3.22000E-03 0.99864 10 9.90000E-04 0.99963 11 2.73127E-04 0.999903 12 6.92103E-05 0.999972 13 1.61427E-05 0.999988 14 3.48620E-06 0.999992 15 7.00680E-07 0.999993