## Chasing the Grail, Part Two

Here are all the charts and research that didn't fit into the other post.  First, I'll start with the distributions of defensive seasons from 2002-2009 (as measured by UZR, minimum of 500 innings played) by position and then I'll provide some numbers on means and standard deviations.  Let's start at the beginning of the defensive spectrum and work our way towards the most difficult positions:

 Mean Median Std. Dev. 1B 0.11 0.45 5.23 LF -0.78 -1.25 8.79 RF 0.08 0.65 9.77 3B 1.00 0.40 8.57 CF 0.90 1.30 9.92 2B 0.36 0.10 7.64 SS 0.77 0.90 7.98 C ? ? ? Overall= 0.36 0.5 8.385781

If I squint really hard, I can almost see my theorized "trend" from 1B to SS, but I think that's just wishful thinking on my part.  There is a small increase in the mean UZR per position as you climb the defensive spectrum, but it's so small I think it's pretty negligible and can be discounted as random noise.  Maybe someone with more of a background in statistics will be able to determine if there's a statistically significant trend in the data here, but just from the eyeball test, it doesn't seem to pass.

That said, if there is no noticeable trend, then it follows that all positions follow the normal curve.  Some deviate more than others, but for the most part the positions do hold very true to the overall distribution of defensive ability.  And if that's the case, we can assume that catchers most likely fall within the same curve and will have defensive seasons that follow the curve.  So through this method, here's the breakdown of catcher UZR scores (or for UZR breakdown at any position, for that matter):

This is just a general idea of what to expect in the long run, so the distribution on a season-by-season basis could vary slightly from this.  Now, let's look at all the catchers that logged more than 500 innings last year behind the dish, as ranked by Tom Tango's Fan Scouting Report.  Players are broken up by their distance from the mean, with 3.09 being the mean and .487 being the standard deviation:

The percentages work out very close to what we'd expect from the normal distribution of defensive ability, with 72.5% of catchers falling within one standard deviation of the mean (68% expected), 25% falling between one and two standard deviations from the mean (27% expected), and 2.5% falling between two and three standard deviations from the mean (4% expected).  Now all that's left to do is to turn the Fan Scouting Report into UZR by using the standard deviations listed above.  I tried to make it as accurate as possible, weighing where players were ranked within their standard deviations as well.  Here's the final result:

Of course this is merely an estimate and shouldn't be taken as gospel, but if nothing else I had fun putting it together.  Enjoy!

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