I hope everyone had a great Thanksgiving weekend. I spent the whole weekend with both sides of my family and had a great time. I went shopping on Friday and did some Albert Pujols-like damage to my holiday shopping list. But now back to the grind.

I explored some of the more common offensive metrics used by sabermetricians in my first post, and I’d like to talk about defensive metrics this time, namely the most popular and widely known: UZR. Dewan Plus-Minus is another defensive metric, but let’s stick to the one most of MMO already has some level of familiarity with.

UZR stands for Ultimate Zone Rating and measures the runs above or below average saved (or given up) by a player. It combines several factors of defense that all quantifies the skill into a runs above average number: Range Runs (RngR), Outfield Arm Runs (ARM), and Error Runs (ErrR) and Double-Play Runs (DPR).

The baseball field is split into 78 zones and UZR accounts for 64 of them. Note the chart on the right from retrosheet.org for more understanding on the zones. The numerical notation for each position is used to name the zone, and the zones in holes and gaps are numbered using the two position numbers. The gap in left-center is 78. The hole between first and second is 34. The letters stand for the depth of the zone.”S” is short, “D” is deep and “XD” is extreme deep (usually only notes the deepest zones of the ballparks). A ball hit to deep right-center is zone 89D. A bloop that lands just over shortstop lands in zone is 78S. A pop fly caught by a back-tracking second baseman will likely be in zone 4D or 34D (deep second or maybe deep second shaded a bit to the first base zone. The fourteen zones not accounted for in UZR include outfield foul territory (across the league, ballparks have very little foul ground in the outfield, not enough to consider. Yankee Stadium comes to mind.), infield pop-ups and zones for catchers and pitchers, as they don’t have UZRs.

It’s a little tricky calculating range runs, but I’ll do my best to make it as easy as possible using nice, round numbers. Buckle up. For each zone, a running total is kept of the hits in that zone, the outs made in that zone and the run value of the hits in that zone. For each player, the number of hits per zone and the number of outs per zone is recorded while the player is on the field. To establish a player’s range runs, first take the percentage of balls in that zone that result in outs by all players, called the average. If 1,000 hits and 1,500 outs were made in a certain zone, the out rate for that zone is 1,500 divided by 2,500 (total chances in that zone). That zone has a 60% out rate by the average fielder. Now, since UZR rate is expressed as a fraction of 1, you can determine the extra value of the balls caught by a specific fielder in that zone by subtracting the average out rate (60% or .60) from 1 to get a .40 value on all balls caught in that zone. Because the hit rate is 40%, so that’s the value given to an out made, or hit taken away. Get it? Don’t move on till you do, it doesn’t get any easier.

Let’s go back to that part about the running total kept for each player. Let’s suppose the player in question caught 150 balls in that zone over the year. So to get the extra value of his defense, multiply the 150 balls he caught by the value of balls he caught. He’s caught 60 extra balls than the average fielder in that zone (150 times .40).

I think determining the outs was the easy part, but determining the NEGATIVE value of the hits in that zone is more difficult. You’ll have to use how many outs were made by the average player at that position and the total number of outs made in that zone by any position. If you were using zone 78D (deep left-center), for instance, outs can be made in that zone by either the left fielder or the center fielder. If our player in question was hypothetically Angel Pagan, and we already determined 1,500 outs were made on 2,500 chances in that zone, how many were Pagan responsible for, and how many was Jason Bay responsible for? Well, if of those 1,500 outs, the centerfielder made 1,000 (67%) of them and the leftfielder made 500 (33%) of them, the centerfielder is similarly as responsible for 67% of the hits, 667 in this case (67% of 1,000 hits). If there were 75 hits in that zone while Pagan was playing center, he was responsible for 50 of them against 25 chalked up to Bay (or whoever the left fielder was at the time).

We already determined the value for the caught ball is .60, so Pagan gets a hit value of 45 (75 times .60). So he’s got a value of 45 hits against 60 outs. In that zone 78D, Pagan caught 15 more balls over the season than an average centerfielder. We determine the run value of Pagan’s defense in that zone by taking the swing in *average run value* of each hit compared to *run value of an out*recorded in that zone and multiplying it by 15.That’s tricky, I know. Let’s break it down. Let’s say the average expected run value of each hit was 0.4 runs, and the average run expectancy for an out made was -0.2. Therefore, multiply his 15 extra outs by 0.6 (the swing in run expectancy) and Pagan was 9 runs better than the average centerfielder in that zone. From there, run that same formula for every single zone in which any CF in baseball made at least one out. Add together those zone-specific range runs to get the player’s cumulative range runs to factor into UZR. You can also take specific zones to garner specific evaluations. For instance, if Pagan was +9 in 78D, +2.5 in 78, +1.5 in 78S and +2.0 in 78XD, all told, Pagan was +15 range runs above average going to his right.

To calculate the error runs, we don’t use zone-specific numbers, we use the chances across all zones. If we use the same total number of chances (the 2,500) and the total errors made in that zone across MLB we’ll get the MLB error rate. Say 100 errors were made in MLB by centerfielders for a rate of .04. If Pagan had 150 chances, as I gave you above, he SHOULD have made six errors. Let’s say he made four, for a gain of two errors (he made two fewer than he should have based on the average rate across MLB). If the run expectancy of reaching on an error is 0.5 runs, the swing is 0.7 expected runs (remember the -0.2 run expectancy for an out). So, 0.7 times the two errors gained is an error runs above average of 1.4 runs.

The same principle is used determining ARM and DPR, using the rates across all MLB and the rate of the player in question as compared to the run expectancy of the outcome. The four resulting numbers are combined to for UZR. Hooray? Not quite yet. UZR is a counting stat, just like home runs or RBIs. The more you play, the more you’ll have. To compare the defensive abilities of two players who have drastic differences in games played, UZR/150 is used, which is the number of runs above or below average a fielder is, per 150 defensive games. Now we can Hooray!

The goal of UZR/150 is to quantify runs into a total because putting a run value to defense allows one to compare with offensive metrics like wRAA (weighted runs above average, see previous post) to determine the true value of a player as a complete player, on both sides of the ball. According to Fangraphs, Joey Votto led the National League with 61.3 wRAA. Albert Pujols had 55.4. Votto also ranked higher than Pujols in UZR/150, though not by much. Votto’s UZR/150 was 2.2, edging Albert’s 1.1 by the slightest of margins. Combining UZR/150 and wRAA, it’s determined that Votto was worth exactly seven runs above average more than Pujols was in 2010 (Votto was 5.9 wRAA better and 1.1 UZR/150 better).

This and other stats make it pretty clear why Votto was a near-unanimous choice for MVP despite not leading the league in any of the traditional stats. While not leading in things like batting average, RBIs and home runs, the guy who won the MVP was first in the league in OBP, SLG%, wOBA and wRAA. The shift in thinking amongst the mainstream media who have MVP votes towards sabermetrics is a welcome change. This was also evident in King Felix winning the Cy Young Award despite 13 wins. He was clearly far and away such a better pitcher than anyone else in the league, almost in a league by himself.

After explaining offensive metrics like wOBA, which is inclusive of all outcomes of at-bats, and now UZR, I hope it’s becoming clearer why these stats are better than knowing simply the counting stats and totals. I, for one, am glad these stats already exist and are calculated already. Can you imagine the vein in your forehead if someone gave you a player’s stat line with individual numbers and asked you yourself to determine how many runs that player was worth offensively? Or someone asking you to determine Pagan’s UZR if you were given his total chances, put-outs and errors? I also hope it’s becoming clearer why sabermetrics generally quantifies statistics into runs and why that’s so important.