DRS team fielding seems overinflated

In a previous post, I noticed that the DRS estimates of team fielding seemed much too high in many cases. In fact, the spread (standard deviation) of team DRS was almost three times as high as other methods (UZR and OAA).

For instance, here are the three competing systems for the 2016 Chicago Cubs:

UZR: +43 runs (range)
OAA: +29 runs
DRS: +96 runs (107 - 11 for catcher framing)

Since I wrote that, the DRS people (Baseball Info Solutions, or BIS) have issued significant corrections for the 2018 and 2019 seasons (and smaller corrections for 2017). It seems the MLB feeds were off in their timing; when the camera switched from showing the batter to showing the batted ball, they skipped a fraction of a second, which is a big deal when evaluating fielders.

The corrections are a big improvement -- most of the extreme figures have shrunk. For instance, the 2018 Phillies improve from -111 runs to -75 runs. It seems that Baseball Reference has not yet updated with the new figures ... Aaron Nola's numbers remain where they were when I wrote the previous post in January. 

However, as far as I can tell, DRS numbers before 2017 remain unchanged, so the problem is still there.

------

In my previous posts, I found that the SD of BABIP (batting average on balls in play, which is where the effects of fielding should be seen) had an SD of about 35 runs. (That's about 44 plays out of 3900, at an assumed value of 0.8 runs per play.)

Again from those posts, we should expect only about 42 percent of that variation to belong to the fielders -- the rest are the result of pitchers giving up easier balls in play (48 percent) and park effects (10 percent).

In other words, we should be seeing

23 runs fielders
24 runs pitchers
11 runs park
----------------
35 runs total

That means any metric that tries to quantify the performance of the fielders should come in with an SD of about 23 runs. 

DRS, from 2003 to 2019, comes in at 41 runs (data courtesy Fangraphs). I didn't calculate the SD after subtracting off catcher, because Fangraphs doesn't provide it in their downloadable spreadsheet. 

I did figure it out for 2018, where I typed in the numbers manually from the DRS website. That season, the SD without catcher was about 85 percent of the total SD. Using the same adjustment for other years would bring the multi-year observed SD from 41 down to 35.

That SD of 35 runs happens to be the same as the SD of BABIP runs. That means DRS is effectively attributing the *entire* team BABIP performance to the fielders, and none to the pitchers or park.

By comparison, the other two metrics are more reasonable:

OAA: 18 runs
UZR: 25 runs
DRS: 35 runs

(Note: Tango tells me I need to bump up official OAA by about 7 percent to account for missing plays, so I've done that. In previous posts, I used 20 percent, which is now wrong -- first, because the data has been improved since then, and, second, because I previously forgot about regression to the mean for the missing data. I should have used 14 percent instead of 20 then, I think.)

OAA and UZR are right around the theoretical 23 runs. DRS, on the other hand, is much higher. To get DRS down to 23 runs, you have to regress it to the mean by about a third. So the 2016 Cubs need to fall from +96 to +63.

To get DRS down to the OAA level of 18 runs, you have to regress by about half, from +96 to around +49.

-----

If DRS is overinflated, does that mean it's also less accurate in identifying the good and bad fielding teams? Apparently not! Despite outsized values, In 2018, DRS predicted BABIP better than OAA did, in terms of correlations:

OAA: .58 correlation
DRS: .62 correlation

Correlations include a "built in" regression to the mean, which is why DRS could do well despite being overexaggerated.

In 2019, though, DRS isn't nearly as accurate:

OAA: .48 correlation
DRS: .33 correlation

I guess you could do more years and figure out which metric is better, and by how much. You could include UZR in there too. I probably should have done that myself, but I didn't think of it earlier and I'm too lazy to go do it now.

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And, just for reference, here are the SDs for 2018 and 2019 specifically (DRS does not include catcher):

2018 DRS:      41
2018 OAA + 7%: 20

------------------------------------

regress DRS to mean 51% to match OAA

2019 DRS:      44
2019 OAA + 7%: 19

------------------------------------

regress DRS to mean 57% to match OAA

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So I'm not sure what's going on with DRS. They seem to be double-counting somewhere in their algorithm, but I don't know how or where.

If you're using DRS, I would suggest you first regress to the mean by around a third if you want to match the theoretical SD of 23, and by around half if you want to match the OAA SD of 19. The correlations to BABIP suggest the regressed DRS could be as accurate as OAA after regressing.

Splitting defensive credit between pitchers and fielders (Part III)

(This is part 3.  Part 1 is here; part 2 is here.)

UPDATE, 2021-02-01: Thanks to Chone Smith in the comments, who pointed out an error.  I investigated and found an error in my code. I've updated this post -- specifically, the root mean error and the final equation. The description of how everything works remains the same.

------

Last post, we estimated that in 2018, Phillies fielders were 3 outs better than league average when Aaron Nola was on the mound. That estimate was based on the team's BAbip and Nola's own BAbip.

Our first step was to estimate the Phillies' overall fielding performance from their BAbip. We had to do that because BAbip is a combination of both pitching and fielding, and we had to guess how to split those up. To do that, we just used the overall ratio of fielding BAbip to overall BAbip, which was 47 percent. So we figured that the Phillies fielders were -24, which is 47 percent of their overall park-adjusted -52.

We can do better than that kind of estimate, because, at least for recent years, we have actual fielding data that can substitute for that estimate. Statcast tells us that the Phillies fielders were -39 outs above average (OAA) for the season*. That's 75 percent of BAbip, not 47 percent ... but still well within typical variation for teams. 

(*The published estimate is -31, but I'm adding 25 percent (per Tango's suggestion) to account for games not included in the OAA estimate.)  

So we can get much more accurate by starting with the true zone fielding number of -39, instead of the weaker estimate of -24. 

-------

First, let's convert the -39 back to BAbip, by dividing it by 3903 BIP. That gives us ... almost exactly -10 points.

The SD of fielding talent is 6.1. The SD of fielding luck in 3903 BIP is 3.65. So it works out that luck is 2.6 of the 10 points, and talent is the remaining 7.3. (That's because 2.6 = 3.65^2/(3.65^2+6.1^2).)

We have no reason (yet) to believe Nola is any different from the rest of the team, so we'll start out with an estimate that he got team average fielding talent of -7.3, and team average fielding luck of -2.6.

Nola's BAbip was .254, in a league that was .296. That's an observed 41 point benefit. But, with fielders that averaged .00074 talent and -0.0026 luck, in a park that was +0.0025, that +41 becomes +48.5.  

That's what we have to break down. 

Here's Nola's SD breakdown, for his 519 BIP. We will no longer include fielding talent in the chart, because we're using the fixed team figure for Nola, which is estimated elsewhere and not subject to revision. But we keep a reduced SD for fielding luck relative to team, because that's different for every pitcher.

 9.4 fielding luck
 7.6 pitching talent
17.3 pitching luck
 1.5 park
--------------------
21.2 total

Converting to percentages:

 20% fielding luck
 13% pitching talent
 67% pitching luck
  1% park
--------------------
100% total

Using the above percentages, the 48.5 becomes:

+ 9.5 points fielding luck
+ 6.3 points pitching talent
+32.5 points pitching luck
+ 0.2 points park
-------------------
+48.5 points

Adding back in the -7.3 points for observed Phillies talent, -2.6 for Phillies luck, and 2.5 points for the park, gives

 -7.3 points fielding talent [0 - 7.3]
 +6.9 points fielding luck   [+10.2 - 2.6]
 +6.3 points pitching talent
+32.5 points pitching luck
 +2.7 points park            [0.2 + 2.5]
-----------------------------------------
 41   points

Stripping out the two fielding rows:

-7.3 points fielding talent 
+6.9 points fielding luck
-----------------------------
-0.4 points fielding

The conclusion: instead of hurting him by 10 points, as the raw team BAbip might suggest, or helping him by 6 points, as we figured last post ... Nola's fielders only hurt him by 0.4 points. That's less than a fifth or a run. Basically, Nola got league-average fielding.

--------

Like before, I ran this calculation for all the pitchers in my database. Here are the correlations to actual "gold standard" OAA behind the pitcher:

r=0.23 assume pitcher fielding BAbip = team BAbip
r=0.37 BAbip method from last post
r=0.48 assume pitcher OAA = team OAA
r=0.53 this method

And the root mean square error:

13.7 assume pitcher fielding BAbip = team BAbip
11.3 BAbip method from last post
10.2 assume pitcher OAA = team OAA
10.0 this method

-------

Like in the last post, here's a simple formula that comes very close to the result of all these manipulations of SDs:

F = 0.8*T + 0.2*P

Here, "F" is fielding behind the pitcher, which is what we're trying to figure out. "T" is team OAA/BAbip. "P" is player BAbip compared to league.

Unlike the last post, here the team *does* include the pitcher you're concerned with. We had to do it this way because presumably we have data for the team without the pitcher. (If we did, we'd just subtract it from team and get the pitcher's number directly!)

It looks like 20% of a pitcher's discrepancy is attributable to his fielders. That number is for workloads similar to those in my sample -- around 175 IP. It does with playing time, but only slightly. At 320 IP, you can use 19% instead. At 40 IP, you can use 22%. Or, just use 20% for everyone, and you won't be too far wrong.

-------

Full disclosure: the real life numbers for 2017-19 are different. The theory is correct -- I wrote a simulation, and everything came out pretty much perfect. But on real data, not so perfect.

When I ran a linear regression to predict OAA from team and player BIP, it didn't come out to 20%. It came out to only about 11.5%. The 95% confidence interval only brings it up to 15% or 16%.

The same thing happened for the formula from the last post: instead of the predicted 26%, the actual regression came out to 17.5%.
  
For the record, these are the empirical regression equations, all numbers relative to league:

F = 0.23*(Team BAbip without pitcher) + 0.175*P
F = 0.92*(Team OAA/BIP including pitcher) + 0.115*P

Why so much lower than expected? I'm pretty sure it's random variation. The empirical estimate of 11.5% is very sensitive to small variations in the seasonal balance of variation in pitching and fielding luck vs. talent -- so sensitive that the difference between 11.5 points and 20 points is not statistically significant. Also, the actual number changes from year-to-year because of variation. So, I believe that the 20% number is correct as a long-term average, but for the seasons in the study, the actual number is probably somewhere between 11.5% and 20%.

I should probably explain that in a future post. But, for now, if you don't believe me, feel free to use the empirical numbers instead of my theoretical ones. Whether you use 11.5% or 20%, you'll still be much more accurate than using 100%, which is effectively what happens when you use the traditional method of assigning the overall team number equally to every pitcher.

Splitting defensive credit between pitchers and fielders (Part II)

(Part 1 is here.  This is Part 2.  If you want to skip the math and just want the formula, it's at the bottom of this post.)

------

When evaluating a pitcher, you want to account for how good his fielders were. The "traditional" way of doing that is, you scale the team fielding to the pitcher. Suppose a pitcher was +20 plays better than normal, and his team fielding was -5 for the season. If the pitcher pitched 10 percent of the team innings, you might figure the fielding cost him 0.5 runs, and adjust him from +20 to +20.5.

I have argued that this isn't right. Fielding performance varies from game to game, just like run support does. Pitchers with better ball-in-play numbers probably got better fielding during their starts than pitchers with worse ball-in-play numbers.

By analogy to run support: in 1972, Steve Carlton famously went 27-10 on a Phillies team that was 32-87 without him. Imagine how good he must have been to go 27-10 for a team that scored only 3.22 runs per game!

Except ... in the games Carlton started, the Phillies actually scored 3.76 runs per game. In games he didn't start, the Phillies scored only 3.03 runs per game. 

The fielding version of Steve Carlton might be Aaron Nola in 2018. A couple of years ago, Tom Tango pointed out the problem using Nola as an example, so I'll follow his lead.

Nola went 17-6 for the Phillies with a 2.37 ERA, and gave up a batting average on balls in play (BAbip) of only .254, against a league average of .295 -- that, despite an estimate that his fielders were 0.60 runs per game worse than average. If you subtract 0.60 from Nola's stat line, you wind up with Nola's pitching equivalent to an ERA in the 1s. As a result, Baseball-Reference winds up assigning Nola a WAR of 10.2, tied with Mike Trout for best in MLB that year.

But ... could Nola really have been hurt that much by his fielders? A BAbip of .254 is already exceptionally low. An estimate of -0.60 runs per game implies his BAbip with average fielders would have been .220, which is almost unheard of.

(In fairness: the Phillies 0.60 DRS fielding estimate, which comes from Baseball Info Solutions, is much, much worse than estimates from other sources -- three times the UZR estimate, for instance. I suspect there's some kind of scaling bug in recent BIS ratings, because, roughly, if you divide DRS by 3, you get more realistic numbers, and standard deviations that now match the other measures. But I'll save that for a future post.)

So Nola was almost certainly hurt less by his fielders than his teammates were, the same way Steve Carlton was hurt less by his hitters than his teammates were. But, how much less? 

Phrasing the question another way: Nola's BAbip (I will leave out the word "against") was .254, on a team that was .306, in a league that was .295. What's the best estimate of how his fielders did?

I think we can figure that out, extending the results in my previous post.

------

First, let's adjust for park. In the five years prior to 2018, the Phillies BAbip for both teams combined was .0127 ("12.7 points") better at Citizens Bank Park than in Phillies road games. Since only half of Phillies games were at home, that's 6.3 points of park factor. Since there's a lot of luck involved, I regressed 60 percent to the mean of zero (with a limit of 5 points of regression, to avoid ruining outliers like Coors Field), leaving the Phillies with 2.5 points of park factor.

Now, look at how the Phillies did with all the other pitchers. For non-Nolas, the team BAbip was .3141, against a league average of .2954. Take the difference, subtract the park factor, and the Phillies were 21 points worse than average.

How much of those 21 points came from below-average fielding talent? To figure that out, here's the SD breakdown from the previous post, but adjusted. I've bumped luck upwards for the lower number of PA, dropped park down to 1.5 since we have an actual estimate, and increased the SD of pitching because the Phillies had more high-inning guys than average:

6.1 points fielding talent
3.9 points fielding luck
5.6 points pitching talent
6.8 points pitching luck
1.5 points park
---------------------------
11.5 points total

Of the Phillies' 21 points in BAbip, what percentage is fielding talent? The answer: (6.1/11.5)^2, or 28 percent. That's 5.9 points.

So, we assume that the Phillies' fielding talent was 5.9 points of BAbip worse than average. With that number in hand, we'll leave the Phillies without Nola and move on to Nola himself.

-------

On the raw numbers, Nola was 41 points better than the league average. But, we estimated, his fielding was about 6 points worse, while his park helped him by 2.5 points, so he was really 44.5 points better.

For an individual pitcher with 700 BIP, here's the breakdown of SDs, again from the previous post:

 6.1  fielding talent

 7.6  fielding luck
 7.6  pitching talent
15.5  pitching luck
 3.5  park
---------------------
20.2  total

We have to adjust all of these for Nola.

First, fielding talent goes down to 5.2. Why? Because we estimated it from other data, and so we have less variance than if we just took the all-time average. (A simulation suggests that we multiply the 6.1 by, from the "team without Nola" case, (SD without fielding talent)/(SD with fielding talent).)

Fielding luck and pitching luck increase because Nola had only 519 BIP, not 700.

Finally, park goes to 1.5 for the same reason as before. 

 5.2 fielding talent

10.0 fielding luck  

 7.6 pitching talent
17.3 pitching luck
 1.5 park
--------------------
22.1 total

Convert to percentages:

 5.5% fielding talent

20.4% fielding luck
11.8% pitching talent
61.3% pitching luck
 0.5% park
---------------------
100% total

Multiply by Nola's 44.5 points:

 2.5 fielding talent 

 9.1 fielding luck
 5.3 pitching talent
27.3 pitching luck
 0.2 park
--------------------
44.5 total

Now we add in our previous estimates of fielding talent and park, to get back to Nola's raw total of 41 points:

 

-3.4 fielding talent [2.5-5.9]

 9.1 fielding luck
 5.3 pitching talent
27.3 pitching luck
 2.7 park            [0.2+2.5]
------------------------------
41 total

Consolidate fielding and pitching:

 5.6 fielding

32.6 pitching 
 2.7 park  

-------------          
41   total

Conclusion: The best estimate is that Nola's fielders actually *helped him* by 5.6 points of BAbip. That's about 3 extra outs in his 519 BIP. At 0.8 runs per out, that's 2.4 runs, in 212.1 IP, for about 0.24 WAR or 10 points of ERA.

Baseball-reference had him at 60 points of ERA; we have him at 10. Our estimate brings his WAR down from 10.3 to 9.1, or something like that. (Again, in fairness, most of that difference is the weirdly-high DRS estimate of 0.60. If DRS had him at a more reasonable .20, we'd have adjusted him from 9.4 to 9.1, or something.)

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Our estimate of +3 outs is ... just an estimate. It would be nice if we had real data instead. We wouldn't have to do all this fancy stuff if we had a reliable zone-based estimate specifically for Nola.

Actually, we do! Since 2017, Statcast has been analyzing batted balls and tabulating "outs above average" (OAA) for every pitcher. For Nola, in 2018, they have +2. Tom Tango told me Statcast doesn't have data for all games, so I should multiply the OAA estimate by 1.25. 

That brings Statcast to +2.5. We estimated +3. Not bad!

But Nola is just one case. And we might be biased in the case of Nola. This method is based on a pitcher of average talent. Nola is well above average, so it's likely some of the difference we attributed to fielding is really due to Nola's own BAbip pitching tendencies. Maybe instead of +3, his fielders were really +1 or something.

So I figured I'd better test other players too.

I found all pitchers from 2017 to 2019 that had Statcast estimates, with at least 300 BIP for a single team. There were a few players whose names didn't quite correlate with my Lahman database, so I just let those go instead of fixing them. That left 342 pitcher-seasons. I assume almost all of them were starters.

For each pitcher, I ran the same calculation as for Nola. For comparison, I also did the "traditional" estimate where I gave the pitcher the same fielding as the rest of the team. Here are the correlations to the "gold standard" OAA:

r=0.37 this method

r=0.23 traditional

Here are the approximate root-mean-square errors (lower is better):

11.3 points of BAbip this method

13.7 points of BAbip traditional

This method is meant to be especially relevant for a pitcher like Nola, whose own BAbip is very different from his team's. Here are the root-mean-squared errors for pitchers who, like Nola, had a BAbip at least 10 plays better than their team's:

 9.3 points this method

11.9 points traditional 

And for pitchers at least 10 plays worse:

 9.3 points this method

10.9 points traditional

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Now, the best part: there's an easy formula to get our estimates, so we don't have to use the messy sums-of-squares stuff we've been doing so far. 

We found that the original estimate for team fielding talent was 28% of observed-BAbip-without-pitcher. And then, our estimate for additional fielding behind that pitcher was 26% of the difference between that pitcher and the team. In other words, if the team's non-Nola BAbip (relative to the league) is T, and Nola's is P,

Fielders = .28T + .26(P-.28T)

The coefficients vary by numbers of BIPs. But the .28 is pretty close for most teams. And, the .26 is pretty close for most single-season pitchers: luck is 25% fielding, and talent is about 30% fielding, so no matter your proportion of randomness-to-skill, you'll still wind up between 25% and 30%.

Expanding that out gives an easier version of the fielding adjustment, which I'll print bigger.

------

Suppose you have an average pitcher, and you want to know how much his fielders helped or hurt him in a given season. You can use this estimate:

F = .21T + .26P 

Where: 

T is his team's BAbip relative to league for the other pitchers on the team, and

P is the pitcher's BAbip relative to league, and 

F is the estimated BAbip performance of the fielders, relative to league, when that pitcher was on the mound.

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Next: Part III, splitting team OAA among pitchers.

Splitting defensive credit between pitchers and fielders (Part I)

(Update, 2020-12-29: This is take 2. I had posted this a few days ago, but, after further research, I tweaked the numbers and this is the result. Explanations are in the text.)

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Suppose a team has a good year in terms of opposition batted ball quality. Instead of giving up a batting average on balls in play (BAbip) of .300, their opponents hit only .280. In other words, they were .020 better than average in turning (inside-the-park) batted balls into outs. 

How much of those "20 points" was because of the fielders, and how much was because of the pitcher?

Thanks to previous work by Tom Tango, Sky Andrecheck, and others, I think we have what we need to figure this out. If you don't want to see the math or logic, just head to the last section of this post for the two-sentence answer.

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In 2003, a paper called "Solving DIPS," (by Erik Allen, Arvin Hsu, Tom Tango, et al) did a great job in trying to establish what factors affect BAbip, and in what proportion. I did my own estimation in 2015 (having forgotten about the previous paper). I'll use my breakdown here. 

Looking at a large number of actual team-seasons, I found that the observed SD of BAbip was 11.2 points. I estimated the breakdown of SDs as:

 7.7  fielding talent
 2.5  pitching staff talent
 7.1  luck
 2.5  park
--------------------------
11.0  total

(If you haven't seen this kind of chart before, the "total" doesn't actually add up to the components unless you square them all. That's how SDs work -- when you have two independent variables, the SD of their sum is the square root of the sum of their squares.)

OK, this is where I update a bit from the numbers in the previous version of this post.

First, I'm bumping the SD of park from 2.5 points to 3.5 points, to match Tango's numbers for 1999-2002.  Second, I'm bumping luck to 7.3, since that's the theoretical value (as I'll calculate later).  Third, I'm bumping the pitching staff to 4.3, because after checking, it turns out I made an incorrect mathematical assumption in the previous post.  Finally, fielding talent drops to 6.1 to make it all add up.  So the new breakdown:

 6.1  fielding talent
 4.3  pitching staff talent
 7.3  luck
 3.5  park
--------------------------
11.0  total

----

We can use that chart to break the team's 20-point advantage into its components. But ... we can't yet calculate how much of that 20 points goes to the fielders, and how much to the pitchers. Because, we have an entry called "luck". We need to know how to break down the luck and assign it to either side. 

Your first reaction might be -- it's luck, so why should we care? If we're looking to assign deserved credit, why would we want to assign randomness?

But ... if we want to know how the players actually performed, we *do* want to include the luck. We want to know that Roger Maris hit 61 home runs in 1961, even if it's undoubtedly the case that he played over his head in doing so. In this context, "luck" just means the team did somewhat better or worse than their actual talent. That's still part of their record.

Similarly here. If a team gets lucky in opponent BAbip, all that means is they did better than their talent suggests. But how much of that extra performance was the pitchers, giving up easier balls in play? And how much was the fielders, making more and better plays than expected?

That's easy to figure out if we have zone-type fielding stats, calculated by watching where the ball is hit (and sometimes how fast and at what angle), and figuring out the difficulty of every ball, and whether or not the fielders were able to turn it into an out. With those stats, we don't have to risk "blaming" a fielder for not making a play on a bloop single he really had no chance on. 

So where we have those stats, and they work, we have the answer right there, and this post is unnecessary. If the team was +60 runs on balls in play, and the fielders' zone ratings add up to +30, that's half-and-half, so we can say that the 20-point BAbip advantage was 10 points pitching and 10 points hitting.

But for seasons where we don't have the zone rating, what do we do, if we don't know how to split up the luck factor?

Interestingly, it will the stats compiled by the Zone Rating people that allow us to calculate estimates for the years in which we don't have them.

------

Intuitively, the more common "easy outs" and "sure hits" are, the less fielders matter. In fact, if *all* balls in player were 0% or 100%, fielding performance wouldn't matter at all, and fielding luck wouldn't come into play. All the luck would be in what proportion the pitcher split between 0s and 100s. 

On the other hand, if all balls in play were exactly the league average of 30%, it would be the other way around. There would be no difference in the types of hits pitchers gave up, which means there would be no BAbip pitching luck at all. All the luck would be in whether the fielders handled more or fewer than 30% of the chances.

So: the more BIP are "near-automatic" hits or "near-automatic" outs, the more pitchers matter. The more BIP that could go either way, the more fielders matter.

That means we need to know the distribution of ball-in-play difficulty. And that's the data that we wouldn't have without the development of Zone ratings now keeping track of it. 

The data I'm using comes from Sky Andrecheck, who actually published it in 2009, but I didn't realize what it could do until now. (Actually, I'm repeating some of Sky's work here, because I got his data before I saw his analysis of it.  See also Tango's post at his old blog.)

Here's the distribution. Actually, I tweaked it just a tiny bit to make the average work out to .300 (.29987) instead of Sky's .310, for no other reason than I've been thinking .300 forever and didn't want to screw up and forget I need to use .310. Either way, the results that follow would be almost the same. 

43.0% of BIP:  .000 to  .032 chance of a hit*
23.0% of BIP:  .032 to  .140 chance of a hit
10.3% of BIP:  .140 to  .700 chance of a hit
 4.7% of BIP:  .700 to 1.000 chance of a hit
19.0% of BIP:          1.000 chance of a hit
---------------------------------------------
overall average: really close to .300

(*Within a group, the probability is uniform, so anything between .032 and .140 is equally likely once that group is selected.)

The SD of this distribution is around .397. Over 3900 BIP, which I used to represent a team-season, it's .00636. That's the SD of pitcher luck.

The random binomial SD of BAbip over 3900 PA is the square root of (.3)(1-.3)/3900, which comes out to .00733. That's the SD of overall luck.

Since var(overall luck) = var(pitcher luck) + var(fielder luck), we can solve for fielder luck, which turns out to be .00367.

6.36 points pitcher luck (.00636)
3.67 points fielder luck (.00367)
--------------------------------
7.33 points overall luck (.00733)

If you square all the numbers and convert to percentages, you get

 75.3 percent pitcher luck
 24.7 percent fielder luck
--------------------------
100.0 percent overall luck

So there it is. BAbip luck is, on average, 75 pitching and 25 percent fielding. Of course, it varies randomly around that, but those are the averages.

What does that mean in practice? Suppose you notice that a team from the past, which you know has average talent in both pitching and fielding, gave up 20 fewer hits than expected on balls in play. If you were to go back and watch re-broadcasts of all 162 games, you'd expect to find that the fielders made 5 more plays than expected, based on what types of balls in play they were. And, you'd expect to find that the other 15 plays were the result of balls being having been hit a bit easier to field than average.

Again, we are not estimating talent here: we are estimating *what happened in games*. This is a substitute for actually watching the games and measuring balls in play, or having zone ratings, which are based on someone else actually having done that. 

------

So, now that we know the luck breaks down 75/25, we can take our original breakdown, which was this:

 6.1  fielding talent
 4.3  pitching staff talent
 7.3  luck
 3.5  park
--------------------------
11.0  total

And split up the 7.3 points of luck as we calculated:

6.36 pitching luck
3.67 fielding luck
--------------------------
7.3  total luck

And substitute that split back in to the original:

 6.1  fielding talent
 3.67 fielding luck
 4.3  pitching staff talent
 6.36 pitching staff luck
 3.5  park
--------------------------
11.0  total

Since talent+luck = observed performance, and talent and luck are independent, we can consolidate each pair of "talent" and "luck" by summing their squares and taking the square root:

 7.1 fielding observed
 7.7 pitching observed 
 3.5 park
----------------------
11.0 total

Squaring, taking percentages, and rounding, we get

 42 percent fielding
 48 percent pitching

 10 percent park
--------------------
100 percent total 

If you're playing in an average park, or you're adjusting for park some other way, it doesn't apply here, and you can say 

 47 percent fielding
 53 percent pitching
---------------------
100 percent total

So now we have our answer. If you see a team's stats one year that show them to have been particularly good or bad at turning batted balls into outs, on average, after adjusting for park, 47 percent of the credit goes to the fielders, and 53 percent to the pitchers.

But it varies. Some teams might have been 40/60, or 60/40, or even 120/-20! (The latter result might happen if, say, the fielders saved 24 hits, but the pitchers gave up harder BIPs that cost 4 extra hits.)

How can you know how far a particular team is from the 47/53 average? Watch the games and calculate zone ratings. Or, just rely on someone else's reliable zone rating. Or, start with 47/53, and adjust for what you know about how good the pitching and fielding were, relative to each other. Or, if you don't know, just use 47/53 as your estimate.

To verify empirically whether I got this right, find a bunch of published Zone Ratings that you trust, and see if they work out to about 42 percent of what you'd expect if the entire excess BAbip was allocated to fielding.  (I say 42 percent because I assume zone ratings correct for park.)

(Actually, I ran across about five years of data, and tried it, and it came out to 39 percent rather than 42 percent. Maybe I'm a bit off, or it's just random variation, or I'm way off and there's lots of variation.)

-------

So what we've found so far:

-- Luck in BAbip belongs 25% to fielders, 75% to pitchers;

-- For a team-season, excess performance in observed BAbip belongs 42% to fielders, 48% to pitchers, and 10% to park.

-------

That 42 percent figure is for a team-season only. For an individual pitcher, it's different. 

Here's the breakdown for an individual pitcher who allows 700 BIP for the season. 

 6.1  fielding talent
 7.6  pitching talent

17.3  luck

 3.5  park
---------------------------
20.2  total

The SD of pitching talent is larger now, because you're dealing with one specific pitcher, rather than the average of all the team's pitchers (who will partially offset each other, reducing variability). Also, luck has jumped from 7.3 points to 17.2, because of the smaller sample size.

OK, now let's break up the luck portion again:

 6.1  fielding talent
 7.6  fielding luck
 7.6  pitching talent
15.5  pitching luck

 3.5  park
---------------------------
20.2  total

And consolidating:

 9.75 observed fielding
17.3  observed pitching

 3.5  park
---------------------------
20.2  total

Converting to percentages, and rounding from 31/69:

 23%  observed fielding
 73%  observed pitching

  3%  park
---------------------------
100%  total

If we've already adjusted for park, then

 24%  observed fielding
 76%  observed pitching

---------------------------
100%  total

So it's quite different for an individual pitcher than for a team season, because luck and talent break down differently between pitchers and fielders. 

The conclusion: if you know nothing specific about the pitcher, his fielders, his park, or his team, your best guess is that 25 percent of his BAbip (compared to average) came from how well his fielders made plays, and 75 percent of his BAbip comes from what kind of balls in play he gave up.

------

Here's the two-sentence summary. On average,

-- For teams with 3900 BIP, 47 percent of BABIP is fielding and 53 percent is pitching.

-- For starters with 700 BIP, 24 percent of BABIP is fielding and 76 percent is pitching.

------

Next: Part II, where I try applying this to pitcher evaluation, such as WAR.

Evidence confirming the DH "penalty"

In "The Book," Tango/Lichtman/Dolphin found that batters perform significantly worse when they play a game as DH than when they play a fielding position. Lichtman (MGL) later followed up with detailed results -- a difference of about 14 points of wOBA. That translates to about 6 runs per 500 PA.

A side effect of my new "luck" database is that I'm able to confirm MGL's result in a different way.

The way my luck algorithm works: it tries to "predict" a player's season by averaging the rest of his career -- before and after -- while adjusting for league, park, and age. Any difference between actual and predicted I ascribe to luck.

I calibrated the algorithm so the overall average luck, over thousands of player-seasons, works out to zero. For most breakdowns -- third basemen, say, or players whose first names start with "M" -- average luck stays close to zero. But, for seasons where the batter was exclusively a DH, the average luck worked out negative -- an average of -3.8 runs per 500 PA.  I'll round that to 4.

-6 R/500PA  MGL
-4 R/500PA  Phil

My results are smaller than what MGL found, but that's probably because we used different methods. I considered only players who never played in the field that year. MGL's study also included the DH games of players who did play fielding positions. 

(My method also included PH who never fielded that year. I made sure to cover the same set of seasons as MGL -- 1998 to 2012.)

MGL's study would have included players who were DHing temporarily because they were recovering from injury, and I'm guessing that's the reason for my missing 2 runs.

But, what about the 4 runs we have in common? What's going on there? Some possibilities:

1. Injury. Maybe when players spend a season DHing, they're more likely to be recovering from some longer-term problem, which also winds up impacting their hitting.

2. It's harder to bat as a DH than when playing a position. As "The Book" suggests, maybe "there is something about spending two hours sitting on the bench that hinders a player's ability to make good contact with a pitch."

3. Selective sampling. Most designated hitters played a fielding position at some time earlier in their careers. The fact that they are no longer doing so suggests that their fielding ability has declined. Whatever aspect of aging caused the fielding decline may have also affect their batting. In that case, looking at DHs might be selectively choosing players who show evidence of having aged worse than expected.

4. Something else I haven't thought of.

You could probably get a better answer by looking at the data a little closer. 

For the "harder to DH" hypothesis, you could isolate PA from the top of the first inning, when all hitters are on equal footing with the DH, since the road team hasn't been out on defense yet. And, for the "injury" hypothesis, you could maybe check batters who had DH seasons in the middle of their careers, rather than the end, and check if those came out especially unlucky. 

One test I was able to do is a breakdown of the full-season designated hitters by age:

Age     R/500PA   sample size
-----------------------------
28-32    -13.7     2,316 PA
33-37    - 6.4     4,305 PA
38-42    + 1.4     6,245 PA

(I've left out the age groups with too few PA to be meaningful.)

Young DHs underperform, and older DHs overperform. I think that's suggestive more of the injury and selective-sampling explanations than of the "it's hard to DH" hypothesis. 

----

UPDATE: This 2015 post by Jeff Zimmerman finds a similar result. Jeff found that designated hitters had a larger "penalty" for the season in cases where they normally played a fielding position, or when they spent some time on the DL.

How should we evaluate Detroit's defense behind Verlander?

Privately and publicly, Bill James, Tom Tango, and Joe Posnanski have been arguing about Baseball Reference's version of Wins Above Replacement. Specifically, they're questioning the 2016 WAR totals for Justin Verlander and Rick Porcello:

Verlander +6.6
Porcello  +5.0

Verlander is evaluated to have created 1.6 more wins than Porcello. But their stat lines aren't that much different:

            W-L   IP   K   BB   ERA
------------------------------------
Verlander  16-9  227  254  57  3.04
Porcello   22-4  223  189  32  3.15

So why does Verlander finish so far ahead of Porcello?

Fielding.

Baseball Reference credits Verlander with an extra 13 runs, compared to Porcello, to adjust for team fielding. 13 runs corresponds to 1.3 WAR -- roughly, a half-run per nine innings pitched. 

Why so big an adjustment? Because the Red Sox fielders were much better than the Tigers'. Baseball Info Solutions (who evaluate fielding performance from ball trajectory data), had Boston at 108 runs better than Detroit for the season. The 13-run difference between Porcello and Verlander is their share of that difference.

It all seems to make sense, except ... it doesn't. Posnanski, backed by the stats, thinks that even though Detroit's defense was worse than Boston's, the difference didn't affect those two particular pitchers that much. Posnanski argues, plausibly, that even though Detroit's fielders didn't play well over the season as a whole, they DID play well when Verlander was on the mound:

"For one thing, I think it’s quite likely that Detroit played EXCELLENT defense behind Verlander, even if they were shaky behind everyone else. I’m not sure how you can expect a defense to allow less than a .256 batting average on balls in play (the second-lowest of Verlander’s career and second lowest in the American League in 2016) or allow just three runners to reach on error all year (the lowest total of Verlander’s career).

"For another, the biggest difference in the two defenses was in right and centerfield. The Red Sox centerfielder and rightfielder saved 44 runs, because Jackie Bradley and Mookie Betts are awesome. The Tigers centerfield and rightfielder cost 49 runs because Cameron Maybin, J.D. Martinez and a cast of thousands are not awesome.

"But the Tigers outfield certainly didn’t cost Verlander. He allowed 216 fly balls in play, and only 16 were hits. Heck, the .568 average he allowed on line drives was the lowest in the American League. I find it almost impossible to believe that the Boston outfield would have done better than that."

------

So, that's the debate. Accepting that the Tigers' fielding, overall, was 49 runs worse than average for the season, can we simultaneously accept that the reverse was true on those days when Verlander was pitching? Could the crappy Detroit fielders have turned good -- or at least average -- one day out of every five?

Here's an analogy that says yes.

In 2015, Mark Buehrle and R.A. Dickey had very similar seasons for the Blue Jays. They had comparable workloads and ERAs (3.91 for Dickey, 3.81 for Buehrle). 

But in terms of W-L records ... Buehrle was 15-8, while Dickey went 11-11.

How could Dickey win only 11 games with an ERA below four? One conclusion is that he must have pitched worse when it mattered most. Because, it would be hard to argue that it was run support. In 2015, the Blue Jays were by far the best-hitting team in baseball, scoring 5.5 runs per game. They were farther ahead of the second-place Yankees than the Yankees were above the 26th place Reds. 

Unless, of course, Toronto's powerhouse offense just happened to sputter on those 29 days when Dickey was on the mound. Is that possible?

Yup. 

It turns out that Dickey got only 4.6 runs of support in his starts, almost a full run less than the Jays' 5.5-run average. Buehrle, on the other hand, got 6.9 runs from the offense, a benefit of a full 1.4 runs per game.

Of course, it's not really that the Blue Jays turned into a bad-hitting team, that their skill level actually changed. It's just randomness. Some days, even great-hitting teams have trouble scoring, and, by dumb luck, there happened to be more of those days when Dickey pitched than when Buehrle pitched.

Generally, runs per game has a standard deviation of about 3, so the SD over 29 games is around 0.56. Dickey's bad luck was only around 1.6 SDs from zero, not even statistically significant.

(* Note: As I was writing this post, Posnanski posted a followup using a similar run support analogy.)

------

Just as we only have season fielding stats for evaluating Verlander's defense, imagine that we only had season batting stats for evaluating Dickey's run support.

In that case, when we evaluated Dickey's record, we'd say, "Dickey looks like an average pitcher, at 11-11. But his team scored a lot more runs than average. If he could only finish with a .500 record with such a good offense, he's worse than his 11-11 record shows. So, we have to adjust him down, maybe to 9-13 or something, to accurately compare him to pitchers on average-hitting teams."

And that wouldn't be right, in light of the further information we have: that the Jays did NOT score that many runs on days that Dickey pitched. 

Well, the same is true for the Verlander/Porcello case, right? It's quite possible that even though the Tigers were a bad defensive team, they happened to play good defense during Verlander's starts, just because the sample size is small enough that that kind of thing can happen. In that light, Posnanski's analysis is crucial -- it's evidence that, yes, the Tigers fielders DID play well (or at least, appear to play well) behind Verlander, even if they didn't play well behind the Tigers' other pitchers.

Because, fielding is subject to variation just like hitting is. Some games, an outfielder makes a great diving catch, and, other days, the same outfielder just misses the catch on an identical ball. More importantly, some days the balls in play are just easier to field than others, and even the BIS data doesn't fully capture that fact, and the fielders look better than they actually played. 

(In fact, I suspect that the errors from misclassifying the difficulty of balls in play are much bigger than the effect of actual randomness in how balls are fielded. But that's not important for this argument.)

------

What if we don't have evidence, either way, on whether Detroit's fielders were better or worse with Verlander on the mound? In that case, it's OK to use the season numbers, right?

No, I don't think so. If the pitcher had better results than expected, you have to assume that the defense played better as well. Otherwise, you'll consistently overrate the pitchers who performed well on bad-fielding teams, and underrate the pitchers who performed poorly on good-fielding teams.

The argument is pretty simple -- it's the usual "regression to the mean" argument to adjust for luck.

When a pitcher does well, he was probably lucky. Not just lucky in how well he himself played, but in EVERY possible area where he could be lucky -- parks, defense, umpire calls, weather ... everything. (See MGL's comment here.)  If a pitcher pitched well, he was probably lucky in how he pitched, and he was probably lucky in how his team fielded.

You roll ten dice, and wind up with a total of 45. You were lucky to get such a high sum, because the expected total was only 35.

Since the overall total was lucky, each individual roll, taken independently, is more lucky to have been lucky than unlucky. Because, obviously, you can't be lucky with the total without being lucky with the numbers that sum to the total. We don't know which of the ten were lucky and which were not, but, for each die, we should retrospectively be willing to bet that it was higher than 3.5.

It would be wrong to say something like: "Overall for each of these dice, the expectation was 3.5. That means the first six tosses probably averaged 21. That means that the last four tosses probably scored 24. Wow! Your last four tosses were 6-6-6-6! They were REALLY lucky!"

It's wrong, of course, because you can't arbitrarily attribute all the luck to the last four tosses. All ten are equally likely to have been lucky ones.

And the same is true for Verlander. His excellent ERA is the sum of pitching and fielding. You can't arbitrarily assume all the good luck came in the rolls his pitching dice, and he had exactly zero luck in the rolls of his team's fielding dice.

-------

If that isn't obvious, try this. 

The WAR method works like this: it's taking a single game Verlander started, assigning the results to Verlander, and adjusting for what the average of what the fielders' did in ALL the games they played, not just this one.

Imagine that we reverse it: we take a single game Verlander started, assign the results to the FIELDERS, and adjust for the average of what Verlander did in ALL the games he pitched, not just this one.

One day, Verlander and the Tigers give up 7 runs, and the argument goes something like this:

"Wow! Normally, the Tigers fielders give up only 5 runs, so today they were -2. But wait!  Justin Velander was on the mound, and he's a great pitcher, and saves an average of two runs a game! If they gave up 7 runs despite Verlander's stellar pitching, the fielders must have been exceptionally bad, and we need to give them a -4 instead of a -2!"

Verlander's stats aren't just a measure of Verlander's own performance. As Tango would say, they're a measure of *what happened when Verlander was on the mound*. That encompasses Verlander's pitching AND his teammates' fielding. 

So, if the results with Verlander on the mound are better than expected, chances are that BOTH of those things were better than expected. 

------

I should probably leave it there, but if you're still not convinced, here's an explicit model.

There's a wheel of fortune with ten slots. You spin the wheel to represent a ball in play. Normally, slots 1, 2, and 3 represent a hit, and 4 through 10 represent an out. But because the Tigers fielders are so bad, number 4 is changed to a hit instead of an out.

In the long term, you expect that the Tigers' defense, compared to average, will cost a Verlander one hit for every 10 balls in play. 

But: your expectation of how many hits it actually cost depends on the specific pitcher's results.

(1) Suppose Verlander's results were better than expected. Out of 10 balls in play, he gets 8 outs. How many hits did the defense cost him?

Eight of Verlander's spins must have landed somewhere in slots 5 through 10. Out of those spins, the defense didn't cost him anything, since the defense is only at fault when the wheel stops at slot 4. 

For hits, we expect that one in four came from slot 4. For the two spins that wound up a hit, that works out to half a hit.

So, with the Tigers having given up few hits, we estimate his defense cost Verlander only 0.5 hits, not 1.0 hits.

(2) Suppose Verlander's results were below average -- he gave up 6 hits. Slot 4 hits, which are the fielders' fault, are a quarter of the 6 hits allowed. So, the defense here cost him 1.5 hits, not 1.0 hits.

(3) Suppose Verlander's results were exactly as predicted -- he gave up four hits. On average, one out of those four hits is from slot 4. So, in this case, yes, the defense would have cost him one hit per ten balls in play, exactly the average rate for the team. 

Which means, the better Verlander's stat line, the more likely the fielders played better than their overall average.

Can fans evaluate fielding better than sabermetric statistics?

Team defenses differ in how well they turn batted balls into outs. How do you measure the various factors that influence the differences? The fielders obviously have a huge role, but do the pitchers and parks also have an influence?

Twelve years ago, in a group discussion, Erik Allen, Arvin Hsu, and Tom Tango broke down the variation in batting average on balls in play (BAbip). Their analysis was published in a summary called "Solving DIPS" (.pdf).

A couple of weeks ago, I independently repeated their analysis -- I had forgotten they had already done it -- and, reassuringly, got roughly the same result. In round numbers, it turns out that:

The SD of team BAbip fielding talent is roughly 30 runs over a season.

------

There are several competing systems for evaluating which players and teams are best in the field, and by how much. The Fangraphs stats pages list some of those stats, and let you compare.

I looked at those team stats for the 2014 season. Specifically, these three:

1. DRS, from The Fielding Bible -- specifically, the rPM column, runs above average from plays made. (That's the one we want, because it doesn't include outfielder/catcher arms, or double-play ability.)

2. The Fan Scouting Report (FSR), which is based on an annual fan survey run by Tom Tango.

3. Ultimate Zone Rating (UZR), a stat originally developed by Mitchel Lichtman, but which, as I understand it, is now public. I used the column "RngR," which is the range portion (again to leave out arms and other defensive skills).

All three stats are denominated in runs. Here are their team SDs for the 2014 season, rounded:

37 runs -- DRS (rPM column)
23 runs -- Fan Scouting Report (FSR)
29 runs -- UZR (RngR)
------------------------------------
30 runs -- team talent

The SD of DRS is much higher than the SD of team talent. Does that mean it's breaching the "speed of light" limit of forecasting, trying to (retrospectively) predict random luck as well as skill?

No, not necessarily. Because DRS isn't actually trying to evaluate talent.  It's trying to evaluate what actually happened on the field. That has a wider distribution than just talent, because there's luck involved.

A team with fielding talent of +30 runs might have actually saved +40 runs last year, just like a player with 30-home-run talent may have actually hit 40.

The thing is, though, that in the second case, we actually KNOW that the player hit 40 homers. For team fielding, we can only ESTIMATE that it saved 40 runs, because we don't have good enough data to know that the extra runs didn't just result from getting easier balls to field.

In defense, the luck of "made more good plays than average" is all mixed up with "had more easier balls to field than average."  The defensive statistics I've seen try their best to figure out which is which, but they can't, at least not very well.

What they do, basically, is classify every ball in play according to how difficult it was, based on location and trajectory. I found this post from 2003, which shows some of the classifications for UZR. For instance, a "hard" ground ball to the "56" zone (a specific portion of the field between third and short) gets turned into an out 43.5 percent of the time, and becomes a hit the other 56.5 percent. 

If it turns out a team had 100 of those balls to field, and converted them to outs at 45 percent instead of 43.5 percent, that's 1.5 extra outs it gets credited for, which is maybe 1.2 runs saved.

The problem with that is: the 43.5 percent is a very imprecise estimate of what the baseline should be. Because, even in the "hard-hit balls to zone 56" category, the opportunities aren't all the same. 

Some of them are hit close to the fielder, and those might be turned into outs 95 percent of the time, even for an average or bad-fielding team. Some are hit with a trajectory and location that makes them only 8 percent. And, of course, each individual case depends where the fielders are positioned, so the identical ball could be 80 percent in one case and 10 percent in another.

In a "Baseball Guts" thread at Tango's site, data from Sky Andrecheck and BIS suggested that only 20 percent of ground balls, and 10 percent of fly balls, are "in doubt", in the sense that if you were watching the game, you'd think it could have gone either way. In other words, at least 80% of balls in play are either "easy outs" or "sure hits."  ("In doubt" is my phrase, meaning BIPs in which it wasn't immediately at least 90 percent obvious to the observer whether it would be a hit or an out.)

That means that almost all the differences in talent and performance manifest themselves in just 10 to 20 percent of balls in play.

But, even the best fielding systems have few zones that are less than 20 percent or more than 80 percent. That means that there is still huge variation in difficulty *even accounting for zone*. 

So, when a team makes 40 extra plays over a season, it's a combination of:

(a) those 40 plays came from extra performance from the few "in doubt" balls;
(b) those 40 plays came from easier balls overall.

I think (b) is much more a factor than (a), and that you have to regress the +40 to the mean quite a bit to get a true estimate. 

Maybe when the zones get good enough to show large differences between teams -- like, say, 20% for a bad fielder and 80% for a good fielder -- well, at that point, you have a system that might work. But, without that, doesn't it almost have to be the case that most of the difference is just from what kinds of balls you get?

Tango made a very relevant point, indirectly, in a recent post. He asked, "Is it possible that Manny Ramirez never made an above-average play in the outfield?"  The consensus answer, which sounds right to me, was ... it would be very rare to see Manny make a play that an average outfielder wouldn't have made. (Leaving positioning out of the argument for now.)

Suppose BIPs to a certain difficult zone get caught 30% of the time by an average fielder, and Manny catches them 20% of the time. Since ANY outfielder would catch a ball that Manny gets to ... well, that zone must really be at least TWO zones: a "very easy" zone with a 100% catch rate, and a "harder" zone with an 10% catch rate for an average fielder, and a 0% catch rate for Manny.

In other words, if Manny makes 30% plays in that zone and a Gold Glove outfielder makes 25%, it's almost certain that Manny just got easier balls to catch. 

The only way to eliminate that kind of luck is to classify the zones in enough micro detail that you get close to 0% for the worst, or close to 100% for the best.

And that's not what's happening. Which means, there's no way to tell how many runs a defense saved.

------

And this brings us back to the point I made last month, about figuring out how to split observed runs allowed into observed pitching and observed fielding. There's really no way to do it, because you can't tell a good fielding play from an average one with the numbers currently available. 

Which means: the DRS and UZR numbers in the Fangraphs tables are actually just estimates -- not estimates of talent, but estimates of *what happened in the field*. 

There's nothing wrong with that, in principle: but, I don't think it's generally realized that that's what those are, just estimates. They wind up in the same statistical summaries as pitching and hitting metrics, which themselves are reliable observations. 

At baseball-reference, for instance, you can see, on the hitting page, that Robinson Cano hit .302-28-118 (fact), which was worth 31 runs above average (close enough to be called fact).

On his fielding page, you can see that Cano had 323 putouts (fact) and 444 assists (fact), which, by Total Zone Rating, was worth 4 runs below average (uh-oh).

Unlike the other columns, UZR column is an *estimate*. Maybe it really was -4 runs, but it could easily have been -10 runs, or -20 runs, or +6 runs. 

To the naked eye, the hitting and fielding numbers both look equally official and reliable, as accurate observations of what happened. But one is based on an observation of what happened, and the other is based on an estimate of what happened.

------

OK, that's a bit of an exaggeration, so let me backtrack and explain what I mean.

Cano had 28 home runs, and 444 assists. Those are "facts", in the sense that the error is zero, if the observations are recorded correctly.

Cano's offense was 31 runs above average. I'm saying that's accurate enough to be called a "fact."  But admittedly, it is, in fact, an estimate. Even if the Linear Weights formula (or whatever) is perfectly accurate, the "runs above average" number is after adjusting for park effects (which are imperfect estimates, albeit pretty good ones). Also, the +31 assumes Cano faced league-average pitching. That, again, is an estimate, but, again, it's a pretty strong one.

For defense, comparatively, the UZR of "-4" is a very, very, weak estimate. It carries an implicit assumption that Cano's "relative difficulty of balls in play" was zero. That's much less reliable than the estimate that his "relative difficulty of pitchers faced" was zero. If you wanted, you could do the math, and show how much weaker the one estimate is than the other; the difference is huge.

But, here's a thought experiment to make it clear. Suppose Cano faces an the worst pitcher in the league, and hits a home run. In that case, he's at worst 1.3 runs above average for that plate appearance, instead of our estimate of 1.4. It's a real difference in how we evaluate his performance, but a small one.

On the other hand, suppose Cano faces a grounder in a 50% zone, but one of the easy ones, that almost any fielder would get to. Then, he's maybe +0.01 hits above average, but we're estimating +0.5. That is a HUGE difference. 

It's also completely at odds with our observation of what happens on the field. After an easy ground ball, even the most casual fan would say he observed Cano saving his team 0 runs over what another player would do. But we write it down as +0.4 runs, which is ... well, it's so big, you have to call it *wrong*. We are not accurately recording what happened on the field.

So, if you take "what happened on the field" in broad, intutive terms, the home run matches: "he did a good thing on the field and created over a run" both to the observer and the statistic. But for the ground ball, the statistic lies. It says Cano "did a good thing on the field and saved almost half a run," but the observer says Cano "made a routine play." 

The batting statistics match what a human would say happened. The fielding stats do not.

------

How much random error is in those fielding statistics? When UZR gives an SD of 29 runs, how much of that is luck, and how much is talent? If we knew, we could at least regress to the mean. But we don't. 

That's because we don't know the idealized actual SD of observed performance, adjusted for the difficulty of the balls in play. It must be somewhere between 47 runs (the SD of observed performance without adjusting for difficulty), and 30 runs (the SD of talent). But where in between?

In addition: how sure are we that the estimates are even unbiased, in the sense that they're independently just as likely to be too high as too low? If they're truly unbiased, that makes them much easier to live with -- at the very least, you know they'll get more accurate as you average over multiple seasons. But if they inappropriately adjust for park effects, or pitcher talent, you might find some teams being consistently overestimated or underestimated. And that could really screw up your evaluations, especially if you're using those fielding estimates to rejig pitching numbers. 

-------

For now, the estimates I like best are the ones from Tango's "Fan Scouting Report" (FSR). As I understand it, those are actually estimates of talent, rather than estimates of what happened on the field. 

Team FSR has an SD of 23 runs. That's very reasonable. It's even more conservative than it looks. That 23 includes all the "other than range" stuff -- throwing arm, double plays, and so on. So the range portion of FSR is probably a bit lower than 23.

We know the true SD of talent is closer to 30, but there's no way for subjective judgments to be that precise. For one thing, the humans that respond to Tango's survey aren't perfect evaluators of what they see on the field. Second, even if they *were* perfect, a portion of what they're observing is random luck anyway. You have to temper your conclusions for the amount of noise that must be there. 

It might be a little bit apples-to-oranges to compare FSR to the other estimates, because FSR has much more information to work with. The survey respondents don't just use the ball-in-play stats for a single year -- they consider the individual players' entire careers, ages and trajectories; the opinions of their peers and the press; their personal understanding of how fielding works; and anything else they deem relevant.

But, that's OK. If your goal is to try to estimate the influence of team fielding, you might as well just use the best estimate you've got. 

For my part, I think FSR is the one I trust the most. When it comes to evaluating fielding, I think sabermetrics is still way behind the best subjective evaluations.