### How stable is a baseball player's talent?

Last post showed that players fluctuate a fair bit, year-to-year, in their talent for free-throw shooting and field goal shooting.

What about baseball? How much does batting average talent vary from year to year?

I used the same method that I did in the NBA case, but I used a lot more data: specifically, every player in MLB from 1973 (compared to 1974) to 2008 (compared to 2009). Players were included if they had at least 100 AB both years.

Going through the numbers real quick (skip to the bolded part if you want):

I calculated the Z-score for what each player's year-to-year difference would be if it were just binomial variation. I adjusted for league changes (thanks to a suggestion from Tango).

The SD of the Z-scores was 1.04875.

Since the SD (of the Z-scores) from luck should have been 1.000, the SD (of the Z-scores) from talent is the square root of [1.04875 squared minus 1 squared], which works out to 0.3135.

Assuming a typical count of 300 AB, the SD (of the difference) from luck should have been 0.232. That means the SD (of the difference) from talent is 0.3135 multiplied by 0.232, which works out to .0099. So:

The SD of the change in an MLB player's batting average talent, from year to year, is about 10 points.

(As before, keep in mind that "talent" actually refers to anything other than binomial variation -- injuries, changes to park, differences in batting opportunities with runners on base, and so on.)

I repeated the calculation, but this time only for players who had 400 AB both seasons (the average in this sample was 534 AB the first year, and 532 the second year). This time, the SD of the change was only about 7 points.

That makes sense -- players who played full-time two years in a row were probably pretty good both years, which means large changes in talent are left out of the sample. Still, it's nice to see that it worked out as expected.

Labels: baseball, distribution of talent, luck

## 1 Comments:

Also, of course, the s.d. of the binomial distribution should shrink relative to the mean (i.e., the coefficient of variation should be smaller) as there are more observations.

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