Fielding is only a small part of BABIP variance

Measures of a quarterback’s performance vary quite a bit from year to year. “The Wages of Wins” argues that’s because the results of a play depend on many of the players on the field, not just the one – the offensive line, the defensive line, the receiver and his defender, and so forth.
Today, in a blog post, David Berri argues that quarterback theory is similar to Voros McCracken’s “DIPS”, which is based on the theory that the batting average on balls in play (BABIP) for a pitcher doesn’t depend on who the pitcher is.

Berri writes,

“Why are hits per balls in play not consistent across time for pitchers? It is because how many hits a pitcher allows depends upon the ability of the eight players surrounding the pitcher on defense.”

But I don't think that's true. How many hits a pitcher allows depends only slightly on the ability of the defense. Most of the reason that BABIP is inconsistent is just random chance.

Here’s the math – hope I did it right.

The variance of team BABIP for a season is about .01^2 (or at least it was for the 1985 NL; I was too lazy to check any more than that).

The binomial variance caused by luck in 4000 balls in play (about a season’s worth), each of which has a 30% chance of being a hit, would be .0072^2.

Therefore, the variance caused by actual talent is the difference between the two. That's about .0069^2, since

.01^2 = .0072^2 + .0069^2

So the standard deviation of team BABIP defensive ability is about .0069.

A starting pitcher might give up 550 balls in play over a season. Again using the binomial approximation to normal, that would yield a standard deviation of about .0195.

So there are two forces acting on a pitcher's season BABIP: luck, with an SD of .0195, and fielding, with an SD of .0069.

Now, suppose a pitcher keeps his exact same defense from season to season. What would the correlation be between the two BABIPs?

According to this Tangotiger post, it would be

Var(fielding skill) / var(total), or .0069^2 / (.0069^2 + .0195^2).

That’s 11%.

It’s hard to agree with Berri that teammates are the cause of the inconsistency. Even with the same teammates, there’s only 11% correlation, which is pretty inconsistent.

Variance in BABIP is caused mainly by luck.