### The (fat?) tails of true talent – Part 3.2- Results for Gaussian True Talent (Pitching)

In part 3.1 I showed results for comparing a simulated true talent distribution for OBP and wOBA to the data. Here I’ll do the same for FIP-. I looked at ERA- also, but I’m not sure it adds anything over and above looking at FIP-, and I’m less sure about the statistical distribution of ERA-.

Assuming typical values for K%, BB%, and HR%, the statistical variation of FIP- has a value of about sqrt(42) per 1000 batters faced. This is something I worked out some time ago using a simulation, and the number stuck in my head. On a per PA basis, it means $\sigma_x \approx \frac{205}{\sqrt{\mathrm{PA}}}$. Using the same techniques as for OBP and wOBA, here are the contours of FIP-, for 1969-1992, and a limit of at least 350 batters faced,

That is, the mean of FIP- true talent is 99.7 and the standard deviation, 13.3. This implies you should regress by $205^2/13.3^2 \approx 240$ batters faced, or $\approx 60$ innings.

Here is the comparison of distributions and the ratio of distributions,

and finally, FIP- for the 1969-1992 time frame, and the 1993-2005 time frame, with a 550 batters-faced limit,