Wins above replacement

Wins above replacement or wins above replacement player, commonly abbreviated to WAR or WARP, is a non-standardized sabermetric baseball statistic developed to sum up "a player's total contributions to his team". A player's WAR value is claimed to be the number of additional wins his team has achieved above the number of expected team wins if that player were traded for a replacement-level player: a player who may be added to the team for minimal cost and effort.
Individual WAR values are calculated from the number and success rate of on-field actions by a player, with higher values reflecting larger contributions to a team's success. WAR value also depends on what position a player plays, with more value going to key defensive positions like catcher and shortstop than positions with less defensive importance such as first base. A high WAR value built up by a player reflects successful performance, a large quantity of playing time, or both.

Overview

The basis for a WAR value is the estimated number of runs contributed by a player through offensive actions such as batting and base running, and runs denied to opposition teams by the player through defensive actions like fielding and pitching. Statistics such as weighted on-base average, ultimate zone rating, ultimate base running, and defense independent pitching statistics measure the effectiveness of a player at creating and saving runs for their team, on a per-plate appearance or per-inning basis. These statistics can be multiplied by the playing time of a player to give an estimate of the number of offensive and defensive runs contributed to their team.
Additional runs contributed to a team lead to additional wins, with 10 runs estimated to be equal to roughly one win. Therefore, a 1.0 WAR value for a player signifies a contribution of roughly 10 more runs than a replacement-level player, over a specified period of time. A replacement-level player is defined by FanGraphs as contributing 17.5 runs fewer than a player of league-average performance, over 600 plate appearances. Therefore, a 1.0 WAR player has contributed an estimated −7.5 runs relative to average over the same number of plate appearances, a 2.0 WAR player has contributed +2.5 runs, and a 5.0 WAR player has contributed +32.5 runs.
For an individual player, WAR values may be calculated for single seasons or parts of seasons, for several seasons, or across the whole career of the player. Collective WAR values for multiple players may also be estimated, for example to determine the contribution a team receives from its outfielders, its relief pitchers or from specific positions such as catcher. It is also possible to extrapolate a future WAR value from a player's past performance data.

Calculation

No clearly established formula exists for WAR. Sources that provide the statistic calculate it differently. These include Baseball Prospectus, Baseball-Reference, and FanGraphs. All of these sources publish the method they use to calculate WAR, and all use similar basic principles to do so. The version published by Baseball Prospectus is named WARP, that by Baseball-Reference is named bWAR or rWAR and that for Fangraphs is named fWAR. Compared to rWAR, the calculation of fWAR places greater emphasis on peripheral statistics.
WAR values are scaled equally for pitchers and batters; that is, pitchers and position players will have roughly the same WAR if their contribution to their team is deemed similar. However, the values are calculated differently for pitchers and position players: position players are evaluated using statistics for fielding, base running, and hitting, while pitchers are evaluated using statistics related to the opposing batters' hits, walks, and strikeouts in FanGraphs' version and runs allowed per 9 innings with a team defense adjustment for Baseball-Reference's version. Because the independent WAR frameworks are calculated differently, they do not have the same scale and cannot be used interchangeably in an analytical context.

Position players

FanGraphs

Similarly to bWAR, FanGraphs uses six components to calculate fWAR for position players. These components are all on the scale of runs, and are then converted to wins based on a runs-per-win value that changes each season based on the run environment. The formula used is

Batting runs is based on wRAA, adjusted using park and league factors.
As of 2024,, where XBR is Statcast's baserunning metric and wSB is Weighted Stolen Base Runs. For seasons prior to 2016, Ultimate Base Running and Weighted Grounded into Double Plays are used in place of XBR when available.
Fielding runs for non-catchers is based on either Outs Above Average converted to runs above average, Ultimate Zone Rating, or Total Zone, depending on era. For catchers, depending on era, either Statcast metrics, a combination of Stolen Base Runs and Catcher Framing, or TZ is used.
The positional adjustment is calculated as follows, where adj is a run value dependent on the player's position: +12.5 for a catcher, −12.5 for a first baseman, +2.5 for a second or third baseman, +7.5 for a shortstop, −7.5 for a left or right fielder, +2.5 for a center fielder, and −17.5 for a designated hitter.
The league adjustment is a minor adjustment added so each league's runs above average balances out to zero.
Finally, since all the previous run values are relative to league average, replacement runs are added to compare the player to a replacement level player.
Baseball Prospectus

Baseball Prospectus uses WARP. WARP relies on BP's proprietary set of statistics called "Deserved Runs", including DRA for pitchers and DRC for batters. BP claims that DRC is the superior metric for comparing player performance when the player changes teams, and also when accounting for changes to the physical game ball year to year. The WARP formula for position players is
BRAA = "Batting Runs Above Average is the number of runs a hitter produces relative to an average hitter, adjusted for park."
BRR = "Baserunning Runs measures a player's contributions on the basepaths based on activity during the run of play, on stolen base attempts, from tag-up situations, and other advancement opportunities. BRR is calculated as the sum of various baserunning components: Ground Advancement Runs, Stolen Base Runs, Air Advancement Runs, Hit Advancement Runs and Other Advancement Runs."
FRAA = "Fielding Runs Above Average...measure the defensive contributions made by players in the field."

Infielders, outfielders, and catchers are evaluated differently.
For outfielders, fielding impact is calculated by comparing their "assist opportunities" with the actual change in run expectancy on each play.
Evaluating catchers includes measuring SRAA and TRAA, both of which are also used for calculations for baserunners and pitchers. Catchers are also evaluated by their "framing" skills via CSAA as well as their skill at blocking errant pitches, via EPAA.

POS_ADJ = "The defensive value of the position a player plays, in total runs relative to the average player." BP does not make publicly available what values it uses for this adjustment.
REP_LEVEL = "The number of runs an average player would have produced relative to replacement level."
RPW = Runs Per Win.
The formula could then be written explicitly as

Pitchers

FanGraphs

Rather than focus on actual runs allowed, Fangraphs uses fielding independent pitching as their main component to calculate WAR as they feel it better reflects the contributions of the pitcher.
, the calculation of pitching fWAR is summarized as

lgFIPR9 and pFIPR9 are modified versions of league and player FIP, respectively. Unlike standard FIP, these values treat infield fly balls as strikeouts, are scaled with respect to RA9 instead of ERA, and are park adjusted. As of 2019, catcher framing is also considered for seasons with framing data. The difference between league and player FIP gives the pitcher's runs above average per 9 innings.
dRPW converts RAAP9 to wins per game above average. The exact value of dRPW varies between players since pitchers directly influence their run environment.
rep is added to give the pitcher credit for the number of wins expected from a replacement level pitcher.
LIM is a leverage index multiplier applied for relievers.
lgCorr is a minor correction to ensure WAR for all MLB pitchers sums to 430 each season.
Baseball Prospectus

Baseball Prospectus uses WARP. WARP relies on BP's proprietary set of statistics called "Deserved Runs", including DRA for pitchers and DRC for batters. DRA was found to accurately account for 72% of variance in pitcher performance, compared to 50% with Fielding Independent Pitching. The "environmental factors" beyond the pitcher's control which are evaluated include, for example, batter handedness, the identity of the umpire, and the temperature of the game at the time of the first pitch.

History

While it's not clear where the concept of "replacement level" originated, WAR has its roots in several early sabermetric statistics. In 1982, Bill James ranked players by "Offensive and Defensive Winning Percentage" and used a de facto replacement percentage to fill in data for part-time players. He used the term "replacement-level" explicitly the following year. James calculated that player with a.350 "winning percentage" was replacement-level. In his Historical Baseball Abstract, James writes

The line against which a player's value should be measured is the replacement level. The replacement level is a very real, very tangible place for a baseball team or a baseball player; drop under it and they release you. But nothing happens to you if you're a little better than average or a little worse than average.

By 1987, James started referring to "runs above replacement." James then converted the data to "wins", though without using the term "wins above replacement."
In the late 80s and early 90s, the publication Big Bad Baseball Annual ranked players based on WAR.
In 1992, Brock Hanke used the term "Wins Above Replacement" in the publication The Baseball Sabermetric.
In 2002, Clay Davenport introduced an early version of WARP. Keith Woolner created VORP, which focused exclusively on offensive value.
Mitchel Lichtman, inventor of the baseball defense statistic UZR, created SuperLWTS, which included baserunning and fielding on a play-by-play basis. SuperLWTS was in turn influenced by Pete Palmer and his systems Linear Weights and Total Player Rating.