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# Defensive Differences

I was browsing the Internet the other day and I found myself looking at a ranked list of defensive efficiency, otherwise known as DER.

DER's a stat I've talked about here in the past and I like it a lot. It's essentially BABIP from the defense's perspective.

I was highly curious, though, about how many runs the difference between a good defense and a bad defense was, given an average pitching staff. I didn't see anything on this, so I thought that I could take a stab at it with some primitive tools and Microsoft Excel.

Actually, as an aside, I think that Microsoft Excel to statistical analysis is like the printing press to literacy. People with no business in numbers or real statistical basis (like myself) can explore the tough questions in baseball, like, "how valuable can a good defense be?"

Oakland's defense this year has ranked as #1 in the league, converting 71.5% of the balls in play into outs. KC's the worst, at a clip of 66.8%. (I believe these are park adjusted, which serves our purposes quite well.)

Good Defense: .715

DIPS theory tells us that a great deal of batting average against is in the strikeouts, so we'll work from that. What's average pitching / hitting, this year? Some ratios:

CAT     FREQ

HBP/PA  0.0096
K/PA    0.1642
BB/PA   0.0817
HR/PA   0.0269
SH/PA   0.0088
SF/PA   0.0070
There's a big thing, though, that gets overlooked... average plate appearances don't hold steady when the defense struggles to convert outs. Every missed out adds another plate appearance to the opposition.

A good place to start is to figure out what percentage of plate appearance result in a "ball in play," or, essentially, any outcome besides a home run, a strikeout, a walk, or a hit by pitch. Those outcomes occur at a frequency of 28.23% of plate appearances. The other 71.77% of PAs are the dreaded "balls in play." So, our two teams, now, have the following:

HBP/PA          .0096   .0096
K/PA            .1642   .1642
BB/PA           .0817   .0817
HR/PA           .0269   .0269
TO/PA           .2823   .2823
SH/PA           .0088   .0088
SF/PA           .0070   .0070
H/BIP           .2850   .3320
The H/BIP framework doesn't help us all that much, though; we need to boil down hits into the plate appearances methodology. Plus, H/BIP doesn't count HRs, so we have to work with that, too.

This can be fairly easily derived, though, because we know the "true outcomes" percentages. It's pretty simple "plug-and-chug" math from there...

HBP/PA          .0096   .0096
K/PA            .1642   .1642
BB/PA           .0817   .0817
HR/PA           .0269   .0269
TO/PA           .2823   .2823
SH/PA           .0088   .0088
SF/PA           .0070   .0070
H/BIP           .2850   .3320
H/PA            .2314   .2652
The new H/PA figures include HRs as well, so now we have PA figures to derive a season's worth of information from.

Next step is figuring out the actual number of PAs that a team's going to get. This one starts from estimating the amount of outs that the average team gets in a season, and that data is available in these tables already. It just needs to be pulled out.

HBP/PA          .0096   .0096
K/PA            .1642   .1642
BB/PA           .0817   .0817
HR/PA           .0269   .0269
TO/PA           .2823   .2823
SH/PA           .0088   .0088
SF/PA           .0070   .0070
H/BIP           .2850   .3320
H/PA            .2314   .2652
OUTS/PA         .6773   .6436
The OUTS/PA figures are simply 1-( H/PA + BB/PA + HBP/PA ), because any play that doesn't result in one of those three things will be an out.

A quick word on this: the numbers I used for "DER" or "Defensive Efficiency" included errors, as well... I got the data from Prospectus' statistics pages. So, really, the H/PA is really H+ROE (reached on error), but I'll leave those to "hits."

The average team makes around 4,172 outs in a season, so we'll work with that figure. What do our two teams look like, then?

PA              6160    6482
H               1426    1719
BB              503     529
K               1011    1064
HR              166     174
OUTS            4172    4172
A simple model of Base Runs might be a good way to estimate how many runs these two teams would give up. We'll use the Total Bases Estimator of 1.12*H + 4*HR, as provided in the Base Runs formulas...

A = H + BB - HR
B = [1.4*TBe -.6*H -3*HR +.1*BB] * 1.0308
C = Outs
D = HR

A*(B/B+C)+D

PA              6160    6482
H               1426    1719
BB              503     529
K               1011    1064
HR              166     174
OUTS            4172    4172
BaseRuns        721     898
R/G             4.45    5.55
So, when all the number crunching is finished, we find that the difference between a great defense and a horrendous defense is worth about 1.1 runs per game. We're playing the game at the extremes here, of course, so it's rare that you'd see this type of difference in practice. To put it in perspective, though, Oakland's team ERA is 3.69 and KC's is 5.47. By FIP? Oakland's advantage diminishes greatly. Their FIP is a 4.15 and KC's is a respectable 4.78. Given that KC's park is usually more of a hitter's park than a pitcher's park, and Oakland's is more of a pitcher's park than a hitter's park, you can see where defense can have such an impact. Oakland's pitchers are better, but by how much?

A few other things:

- When evaluating pitchers, it's a good idea to look at the team context of strikeouts. A pitcher on a team with a worse defense will have more opportunities for strikeouts. This is why K/BFP is a better metric than K/9.
- Teams that strike out more batters will have LESS of a discrepancy in these numbers, and teams with below average strikeout rates will have a higher discrepancy. So, if you are building a team and your pitchers don't strike out a lot of guys, get them some defensive help.
- In the context of an average offense that scores about 800 runs, that run differential is worth around 17 wins, with these conditions.

I am surprised that the results were this drastic, to be quite honest, so I'm a bit concerned about my methodology, to be sure.

The bottom line, though, is that you can experience a much greater variance in runs scored than in the defensive aspect alone of runs allowed. So, while defense makes a difference at the margins, it just doesn't have quite the same impact as offense.