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Last year, before his team’s magical 2008 season began, Tampa Bay Rays manager Joe Maddon gave his players t-shirts featuring an unlikely mathematical message: “9 = 8.” Nine players, playing hard for nine innings each game, could make the Rays one of the eight clubs to reach the post-season, he told the team.

After Tampa Bay’s fast start, Maddon revealed another explanation for his fuzzy math. He figured the Rays needed to improve by 27 games to win the American League East. And they could do it, he believed, if they could improve by nine victories in each of three different areas: the offense, the bullpen and the defense.

The Rays exceeded their manager’s expectations, winning 97 games, a remarkable 31-victory improvement over their last-place season in 2007. For someone who orchestrated that sort of historic turnaround, it might seem that nothing is impossible. After all, in the process of directing his club to the American League pennant, Maddon also apparently cracked the code for measuring team defense.

Origins

Since baseball’s earliest days, evaluation of defense has traditionally been based on five standard statistics: total chances, assists, putouts, errors and fielding percentage. But that has changed.

The statistical movement that transformed the way clubs evaluate offense and pitching has moved to the last frontier of objective analysis: defense. A variety of tools for measuring defensive performance are now available: Clay Davenport‘s Translations, John Dewan’s Plus/Minus system, Dan Fox‘s Simple Fielding Runs, Mitchel Lichtman’s Ultimate Zone Rating and David Pinto‘s Probabilistic Model of Range, to name just a few. Many clubs have their own proprietary systems for evaluating and valuing defense, and general managers are acquiring and paying players accordingly.

The spark ushering in this new era of analysis was lit in the Eighties, thanks in no small part to the work of Bill James, who proposed a novel idea: Much of what is perceived as being pitching is, in fact, defense. More specifically, James developed what he called a defensive efficiency record (DER), “a team statistic intended to estimate the percentage of all balls in play that a team has turned into outs.”

James’ theory, now known as simply Defensive Efficiency, leapt into the popular consciousness in 2008, when Maddon’s Rays improved from worst to first in Def Eff and in the standings, en route to the World Series. Year-over-year defensive rankings for the last 10 World Series clubs are listed below. All but one club showed significant improvement in Defensive Efficiency for the year they reached the World Series. The exception was the 2006 Cardinals, who experienced a slight drop from sixth in Def Eff in 2005 to seventh in their championship season.

A new perspective

James’ analysis was a departure from traditional fielding statistics in two distinct ways. First, he began the process by evaluating defense as a whole rather than using individual fielding statistics as a baseline. Second, he removed the subjectivity inherent in the traditional scoring model by evaluating each ball in play, regardless of how it was judged by the official scorer.

James began by calculating the number of Plays Made by a team’s defense. He did so by making two estimates.

The first estimate:

PO – K – DP – 2(TP) – OCS – A (of)

James assumed a batted ball was turned into an out every time a putout was recorded, unless 1) the putout was a strikeout, 2) two or three putouts were recorded on the same play (a double play or triple play), or 3) a runner was thrown out on the bases, either as an opponent caught stealing or via an assist from an outfielder.

The second estimate:

BFP – K – HWHBP – .71 errors

James assumed a batted ball was turned into an out every time a batter faced a pitcher, unless 1) the batter struck out, 2) the batter hit safely, 3) the batter walked, 4) the batter was hit by a pitch, or 5) the batter reached on an error.

James averaged his two estimates, giving him a reliable figure for Plays Made.

That done, James divided Plays Made by Plays Made plus Plays Not Made, and the Defensive Efficiency Record was born.

PM

PM + HHR + .71 Errors

Plays Not Made included hits, plus .71 errors, less home runs, the vast majority of which were balls defenders had no chance to field.

James found an average defensive efficiency record to be about .695, with most successful teams ranking above average. In fact, James found that his DER calculation correlated well with a club’s winning percentage. This was a marked contrast from a traditional metric, like fielding percentage, which yielded no clear pattern when compared to winning percentage.

He also reached another conclusion: Although differences in DER “might look small, they are anything but,” James wrote in his 1984 Baseball Abstract. “The effect of having a DER which is .020 below the league average would be very similar to the effect of having a team batting average which is .017 to .018 below league, which is to say that it would wipe you out of a pennant race 99% of the time.”

As calculated for Baseball Prospectus analysis today, Defensive Efficiency (Def Eff) is:

1 – ((H + ROEHR) / (PABB – SO – HBPHR))

Balls in play (hits and errors, less home runs) are divided by plate appearances, less walks, strikeouts, hit-by-pitches and home runs. The result is subtracted from one, yielding a percentage of balls in play converted into outs. Defensive Efficiency also may be approximated using the formula:

1 – BABIP

where BABIP represents batting average on balls put into play.

The best teams actually convert balls in play into outs about 73 percent of the time, a Def Eff figure of .730. The worst teams do so about 69 percent of the time (.690). James’ point on the importance of small variations in defensive efficiency still holds true.

In 2003, analyst James Click created the Park Adjusted Defensive Efficiency metric, accounting for the park factors that affect the DER calculation in each specific ballpark. Click accounted for a variety of factors that might affect a club’s defensive efficiency figure: park factor, number of balls in play allowed by the pitching staff, ground ball to fly ball ratio, lefty-righty balance of the pitching staff, and opportunities for double plays. His resulting PADE figure illustrates how much better or worse than average a team ranks in converting balls in play into outs. Click’s research has shown that a one percent difference in PADE is worth 13 runs, or 1.3 wins, over the course of a season.

According to the 2008 PADE figures, the Rays were 1.26 percent above average in turning balls in play into outs, adjusted for park figures. Only a year before, Tampa Bay had been 5.64 percent below average. Using Click’s method, the 6.9 percent defensive improvement resulted in an increase of 8.97 wins for the Rays, almost exactly what Joe Maddon wanted from his fielders.

Maddon’s new math worked, at least where defense was concerned.

Defensive Rankings for World Series teams, 2004-08
MLB rankings in parentheses. Team Defensive Efficiency statistics are available at http://www.baseballprospectus.com/statistics/sortable/index.php?cid=204024

```Tampa Bay  Def Eff      PADE
2007     .656 (30)   -5.64 (30)
2008     .710 (1)     1.26 (3)

2007        .686 (17t)    0.15 (11)
2008        .696 (9t)     0.58 (6)

2006     .684 (20)    0.94 (7)
2007     .701 (5t)    3.07 (2)

2006   .680 (25t)   -1.11 (19)
2007   .705 (1t)     3.22 (1)

2005    .698 (14t)  -0.25 (15)
2006    .702 (3)     2.01 (4)

2005    .704 (6)    1.02 (9)
2006    .697 (7)    0.85 (9)

2004    .686 (24)    - 1.64 (26)
2005    .706 (4)      0.32 (13)

2004    .695 (10)   0.06 (10)
2005    .713 (2)    2.67 (1)

2003    .700 (10)   -0.70 (17)
2004    .711 (2)     1.58 (2)

2003   .685 (23t)  -0.80 (18)
2004   .693 (14t)   0.95 (4)
```

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kgoldstein
5/24
I'm not sure this one worked for me as well. It took awhile to get started, and after spending so much time explaining DER, I actually ended up wanting to know way more about PADE. Still, some telling stuff on a portion of the game that doesn't get enough attention. I hate individual fielding statistics, but I like the team ones.
wcarroll
5/24
Pitch perfect. There's some math here for those that want to sing along, some explanation for those of us that can't, and really nails the tone. If anything, he might overreach going to PADE as well as DER; it felt a bit rushed or tacked on when he touched on PADE, but it did make sense since Click was BP and is Rays. Mostly, I think he nailed the voice of it, explaining without feeling condescending. Very strong piece.
ckahrl
5/24
A tidy proposition, one told well that stuck with its intro and followed through to a neatly executed denouement. I like that Jeff was thoughtful in referencing the number of published (and in-industry, unpublished) metrics that are out there. I also like that he subtly challenges the johnny-come-lately assertions about how defense is the new "it" girl of sabermetrics, at least in that teams before the Rays were getting their moments of glory in no small part because of their improvements afield (how quickly some forgot the White Sox, for example). That said, like Jeff's initial contribution, the closing data, while interesting, was a bit limited in scope, where I'd love for him to tackle something more wide-ranging. Jeff's craftmanship as a writer and as an analyst is already impressive; what I want to see is his moving from the specific to the general to make his work more wide-ranging and important.
Oleoay
5/24
I thought this article was much better organized and easier to read than his initial entry. The PADE part felt tacked on but somehow was more relevant to the discussion of the Rays than the Defensive Efficiency. I would have prefeerred that one or the other statistical concepts were used, then a discussion on how the Rays used these concepts to improve their defensive efficiency and an actual "conclusion" paragraph to the article besides another chart. Thus, the ending doesn't mesh as well with the claim that Joe Maddon has "Maddon also apparently cracked the code for measuring team defense." Still I liked the article overall, just needs some tuning.
Oleoay
5/24
This article was one of the five that I gave a thumbs up to on my initial read through.
roughcarrigan
5/25
I just barely said yes to this. I thought Hissey's piece was slightly better but still gave this a thumbs up.
jcuddy
5/25
I was really drawn in by the opening, but the conclusion was muddled and slightly disappointing. Lost my vote by the end.
metty5
5/25
+1
wcarroll
5/25
How about instead of saying "+1" you click the plus button to rate the comment up? Pet peeve of mine ...
Shkspr
5/25
When I read Hissey's piece, one of the things I kept thinking was that when Bill James was writing these things thirty years ago, he was showing us the formulae at each step, which made me immediately want to grap some paper and a calculator and follow along. The meat of this article was what I wanted to see as a basics concept for that article. That said, both were strong enough to move on.
SkyKing162
5/25
If the piece is about fielding, start talking about fielding a bit sooner. And instead of explaining the older metrics in as much detail, I'd like the author to have skipped straight to the concept of removing subjectivity of errors and making sure to hold fielders who lacked range accountable for balls they can't get to.
Junts1
5/25
I read both fielding-metric articles, voted for both and actually thought this one was better: its a smoother read and transitions clearly from explaining the background, the intermediate, simple stat (def eff) before moving on to the better/more complex statistic (PADE) and emphasizing it: it's an excellent primer for understanding defensive measurement.
hotstatrat
5/25
This was the first entry on my docket. I've read through all of them now and am returning to make my comments and place my votes. Yes, votes. Although, I was tempted to make just one vote for the writer I believe is the clearly the best so far, I thought at this stage, I'll give support to the other margianlly worthy writers. Based on the comments, I don't think my favorite is in danger yet of being eliminated. I wasn't a fan of Jeff's qualifying entry - a fluffy piece of analysis, but found this one very well written - despite the rushed ending on PADE. For this, he has earned another chance.
Severe
5/25
One of the better so far for me -- and one of few to get my first round vote. He picked one point of reference (Joe Maddon's 9 run improvement) and touched back on it without too much fluff or rim-shot jokes. I don't know all these formulas by heart yet, and feel like I actually learned something.
gwguest
5/25
A second article ending with a table, but this one gave a good enough explanation before sending the readers on our way. The intro was a bit long and fluffy; otherwise very nice.
llewdor
5/25
Covered a relevant topic. Easy to read. Explained math using actual math. Provided raw data. There's literally nothing wrong with this article. Thumbs Up.
molnar
5/26
I liked this. It was well-organized, nicely balanced between setup and delivery. However, after reading about the three areas in which Maddon sought to improve, I did not realize until I finished the article that it was going to only be about defense! I especially liked the paragraph quoting Bill James, that allowed us to put the DER figures into context - just because .02 is a seemingly small number doesn't mean it's insignificant. This stood out in comparison to some of the other entries.
jtrichey
5/26
I thought this one was fantastic. Easy thumbs up, and good thing too, because I was on the fence about his qualifying piece. I liked Hissey's article on the same topic, but this one is better. Congrats! Also, I find it amusing how often Bill James name is popping up in all of these articles. James is probably one of the top 10 most influential people in my life, and not surprisingly, that seems to be the case with so many of us.
Oleoay
5/26
*puts his tongue in his cheek* All you need to do to be a sabremetrician is to invoke the Trinity of Bill James, His Son Sample Size, and include a table of Holy Scatterplots. *takes his tongue out of his cheek*
jdavlin
5/26
I really enjoyed how Jeff effortless draws the reader into the world of sabermetrics, in much the way Bill James did for me back when I was a baseball obsessed adolescent. Things do bog down a bit toward the end, but overall this is nearly flawless.
krissbeth
5/26
This article needs a conclusion. Ending on a stream of data isn't stylish and no data speaks for itself. The data could have been moved up in the article, if space constraints were too tight. Otherwise, good work.
caprio84
5/26
Best intro (in my opinion) of all the submissions. Told me something I didn't know and led me into the article. Well done.
DrDave
5/26
Started out great, then kind of petered out for me. I'll agree with Christina that the external references and historical story (including smarts pre-Moneyball) were well done. The ending was muddled and abrupt, though, and the importance (and complexity) of park adjustment glossed over too quickly. I would have liked to see the correlation between DEff and wins compared with other, more familiar stats, for shock value.