UZR gets to this number by combining "range runs" which are "the number of runs above or below average a fielder is, determined by how the fielder is able to get to balls hit in his vicinity," and "error runs," defined as "the number of runs above or below average a fielder is, determined by the number of errors he makes as compared to an average fielder at that position given the same distribution of balls in play." Added to standard UZRs are the UZR/150 numbers, or "the number of runs above or below average a fielder is, per 150 defensive games."
Developed by John Dewan in "The Fielding Bible" and marketed by his company, Baseball Info Solutions, this metric, conceived from BIS "video scouts" reviewing video of every play in every game over the course of a Major League season, awards or takes away credit (plus or minus) for making plays that other players at his position missed during the season or missing plays others made. The scores result from the detailed reports from the video that take into account the location of each batted ball, the speed of the ball, the type of hit and more.
Defensive Runs Saved (DRS)
DRS was introduced in Dewan's "The Fielding Bible Volume II" book and uses the Plus/Minus system, combined with "double play abilities, outfielder arms, bunt defense by corner infielders, pitcher stolen base defense, catcher stolen base defense and the catcher's ability to handle pitchers" and converts it all into one number of runs saved.
Or, as Dewan explains in the book, "Let's say there's a man on first with one out. The expected runs at that point are .528. The next play is a ground ball to the shortstop. He boots it for an error and we now have men on first and second with one out. The expected runs went from .528 to .919. That's an increase of .391 (.919 minus .528) runs. The play itself, the error, cost the team .391 runs. We don't have to follow it through and count the rest of the inning. We know what the value of the ending state is and can use it."
Probabilistic Model of Range (PMR)
This is the baby of David Pinto, a statistical analyst on BaseballMusings.com, and is similar to UZR. In describing the origins of PMR in 2005, Pinto wrote, "I calculate the probability of a ball being turned into an out based on six parameters: direction of hit (a vector), the type of hit (fly, ground, line drive, bunt), how hard the ball was hit (slow, medium, hard), the park, the handedness of the pitcher, the handedness of the batter."
For each ball in play, the program sums the probability of that ball being turned into an out, and that gives us the expected outs. Dividing that by balls in play yields expected defensive efficiency rating (DER). That is compared to the team's actual DER. A good defensive team should have a better DER than its expected DER."