Position Player WAR Calculations and Details

WAR for position players has six components:

• Batting Runs
• Baserunning Runs
• Runs added or lost due to Grounding into Double Plays in DP situations
• Fielding Runs
• Positional Adjustment Runs
• Replacement level Runs (based on playing time)

The first five measurements are all compared against league average, so a value of zero will equate to a league average player. Less than zero means worse than average, and greater than zero means better than average. These five correspond to the first half of our equation above (Player_runs - AvgPlayer_runs). The sixth factor is the second half of the equation (AvgPlayer_runs - ReplPlayer_runs).

Rbat, Batting Runs

For batting runs we use a linear weights system based on Tom Tango's wOBA (weighted on-base average) framework, but we add a number of improvements to our calculation of wRAA (weighted runs above average).

• Weights are based on the offense of a particular league season rather than all of major league baseball.
• Pitchers are excluded from the league wOBA calculation, so the run totals are not biased against players in seasons with DHs.
• We estimate CS totals for seasons in which we lack CS data
• From 2003 on, we differentiate between infield singles and outfield singles
• For all seasons, we differentiate between strikeouts and other outs. In early baseball, pre-1920 or so, this is especially vital because error rates were high and DP rates were low, so there was a lot of benefit to putting the ball in play.
• We include Reached on Errors for seasons that such data is available. For other years, we estimate the rate of ROEs and add that into our non-SO out values.
• Runs due to SB and CS are computed with wRAA, but we subtract them out from the batting total and add them into the baserunning total.

See How we compute wRAA for WAR for a full rundown of what we do.

Rbr, Baserunning Runs

Baserunning runs come from two places:

• Stolen Bases and Caught Stealing runs as calculated above for wRAA (see above for a link to an explanation of this), and
• Non-Basestealing baserunning which includes items like 1st to 3rd on singles, outs on the bases, tagging up on fly balls, scoring from third on a ground ball, etc...

This explanation describes the techniques used to estimate non-SB/CS baserunning contributions during the play-by-play era, 1931 to the present.

Obviously baserunners can advance while on first, second and third, but the mix of opportunities available to them is different for each of the three bases they may occupy. So for each player we track the number of occurrences for each of the following events for each base:

For Runners on 1st Base

1. For a single fielded by the RF or CF, the # of times the runner was held at second (and there was not a runner on second held at third).
2. For a single fielded by the RF or CF, the # of times the runner advanced to third.
3. For a single fielded by the RF or CF, the # of times the runner was out on the bases.
4. For a single fielded by the LF, the # of times the runner was held at second (and there was not a runner on second held at third).
5. For a single fielded by the LF, the # of times the runner advanced to third.
6. For a single fielded by the LF, the # of times the runner was out on the bases.
7. For a double fielded by any outfielder, the # of times the runner was held at third.
8. For a double fielded by any outfielder, the # of times the runner was scored.
9. For a double fielded by any outfielder, the # of times the runner was out on the bases.
10. For a fly ball with less than two out caught by any outfielder with no runner on second, the # of times the runner held at first.
11. For a fly ball with less than two out caught by any outfielder with no runner on second, the # of times the runner tagged and reached second.
12. For a fly ball with less than two out caught by any outfielder, the # of times the runner was out on the bases.
13. The # of advances on defensive indifference, passed balls, wild pitches, balks, and pickoff errors.
14. The # of outs recorded due to pickoffs (w/ no CS).

For Runners on 2nd Base

1. For a single fielded by any outfielder, the # of times the runner was held at third.
2. For a single fielded by any outfielder, the # of times the runner was scored.
3. For a single fielded by any outfielder, the # of times the runner was out on the bases.
4. For a fly ball with less than two out caught by any outfielder with no runner on third, the # of times the runner held at second.
5. For a fly ball with less than two out caught by any outfielder with no runner on third , the # of times the runner tagged and reached third.
6. For a fly ball with less than two out caught by any outfielder, the # of times the runner was out on the bases.
7. The # of advances to third on defensive indifference, passed balls, wild pitches, balks, and pickoff errors.
8. The # of outs recorded due to pickoffs (with no CS).
9. For infield ground balls with less than two out that are not an infield hit and with no runner on first, the number of times the runner advances to third.
10. For infield ground balls with less than two out that are not an infield hit and with no runner on first, the number of times the runner stays at second.
11. For infield ground balls with less than two out that are not an infield hit and with no runner on first, the number of times the runner is out.

For Runners on 3rd Base

1. For a fly ball with less than two out caught by any outfielder, the # of times the runner held at third.
2. For a fly ball with less than two out caught by any outfielder, the # of times the runner tagged and scored.
3. For a fly ball with less than two out caught by any outfielder, the # of times the runner was out on the bases.
4. The # of scores on passed balls, wild pitches, balks, and pickoff errors.
5. The # of outs recorded due to pickoffs (with no CS).
6. For infield ground balls with less than two out that are not an infield hit and with first or second base empty, the number of times the runner scores.
7. For infield ground balls with less than two out that are not an infield hit and with first or second base empty, the number of times the runner stays at third.
8. For infield ground balls with less than two out that are not an infield hit and with first or second base empty, the number of times the runner is out.

For each of the bases, we total these various events along with the total number of batters and or baserunning events where the player is at this base. This is the number of opportunities the player has for things like pickoffs and advancing on wild pitches.

Next, for the entire league, we find the total number of baserunning events of each type and the percent of the time that each occurs. So for example, we can see the percent of time a runner on first goes first to third on a single, first to second, or is thrown out.

Then we can compare the player to the league average for each of these categories to see how many bases they added or failed to add versus the league average, or how many extra or fewer outs they created on the basepaths versus the league average player.

From there we multiply the bases added above league average times a league-specific run value for each base added or not (generally around .20 runs/base) and subtract the number of outs added above league average times a season-specific run value for each out added (generally around .48 runs/out).

Therefore, a player that adds 20 bases via extra bases taken on batter singles or other means, and who also runs into 3 fewer outs than average, would add: 20 × .20 - (3 × (-0.48)) = 4.00 + 1.44 = 5.44 runs added.

This setup allows us to determine the effect of aggressive baserunning versus cautious baserunning and convert those actions into runs added or lost.

2011 Best: 8.0 runs by Ian Kinsler, 2011 Worst: -5.5 runs by Andre Ethier.

Reached on Error Runs

Reaching on an error may not seem like a skill we need to measure, but there is evidence that batters can have a large, non-random impact on the amount of time they reach base by error. Tom Ruane of RetroSheet did a large study showing exactly this.

In our original version of WAR this was a separate factor, but we've rolled this into wRAA for WAR, so ROE runs are now rolled into the batting runs.

Rdp, Grounded into Double Play Runs

GIDP avoidance is something that batters can have a significant impact on. Slow right-handed power hitters (e.g. Jim Rice) will ground into a ton of double plays, while someone like Carl Crawford won't.

Our run value for non-SO outs takes into account the league average # of DPs incurred by these number of outs. So everyone is assumed to ground into a certain number of DPs, but we know that this skill can allow some players to beat out the back end of the DP when other players wouldn't.

GIDP opportunities are any infield ground ball with a runner on first, less than two outs and at least one out is recorded on the play. The play must not be scored a hit as well.

The difference in runs scored between a "double play" and a "double play avoided" is, on average, .44 runs, but it can vary by the run environment of the league. The league GIDP rate is then calculated and:

R_gidp = .44 × ( GIDP_OPPS_player * GIDP_RATE_lg - GIDP_player)

2011 Best: 4.7 runs by Johnny Damon, 2011 Worst: -4.8 runs by Albert Pujols.

Rdef. Fielding Runs

Fielding measures obviously have a lot of controversy surrounding them. Previously, Baseball-Reference used Sean Smith's Total Zone Rating for all seasons. With our 2012 update, we have switched to using Baseball Info Solutions Defensive Runs Saved for seasons since its introduction in 2003, and TZR for previous seasons.

Fielding Runs: Defensive Runs Saved

Defensive Runs Saved (DRS) is the most sophisticated public system available. It includes 8 factors:

• Fielding Range Plus/Minus Runs Saved based on BIS-trained scorer observations and batted ball timing to determine the velocity of each batted ball.
• Outfield arm runs saved based on exact counts of baserunner advancements and kills and the velocity of the hit ball.
• Infielder double plays based on opportunities and rates they were turned based also on batted ball velocity.
• Good play-bad play values which include 28 positive play types. For example: HR-saving catches, backing up a play, blocking a pitch in the dirt, and 54 misplays like missing the cutoff man, failing to anticipate the wall and allowing extra bases, not covering a base, pulling a foot off the bag, etc...
• Bunt Fielding
• Catcher SB/CS data (which is tweaked by the pitchers caught)
• Pitcher SB/CS data (which is tweaked by the catchers behind the plate)
• Catcher handling of the pitching staff via things like pitch framing and pitch calling

See the Fielding Bible Volume III for a full run-down of the system.

2011 Best: Austin Jackson 29 runs, 2011 Worst: Logan Morrison -26 runs.

Fielding Runs: Total Zone Rating

Total Zone Rating is a fielding measure developed by Sean Smith and is used in WAR for all seasons prior to 2003. Total Zone Rating (TZR) is a non-observational fielding system that relies on various forms based on the level of data available ranging from basic fielding and pitching stats to play-by-play including batted ball types and hit location. As much data as is available is used for each season.

When play-by-play is available, TZR will use information like ground balls fielded by infielders and outfielders to estimate hits allowed by infielders. It uses baserunner advancement and out information to determine arm ratings for outfielders, double play acumen by infielders and arm ratings for catchers.

From 1953-2002, Runs Saved or Cost are calculated for:

• Fielder fielding range
• Outfield arms
• Turning the Double Play for infielders
• Catcher Throwing

For seasons we lack play-by-play data (pre-1953), we use information on opposition hitting, pitcher and batter handedness, fielding stats and more to estimate fielder opportunites and outs produced.

For seasons where observational data is not available (pre-2003), we believe TZR is the best system for estimating player defense.

Fielding Runs: Defensive Regression Analysis

Defensive Regression Analysis (DRA) is a fielding measure developed by Michael Humphreys and was introduced in his book Wizardry: Baseball's All-Time Greatest Fielders Revealed. Regression analysis is used to determine linear weights based on season-level data. These linear weights are applied to season-level fielding data such as putouts, assists, errors and pitcher handedness to estimate the number of plays a player made above or below average. The number of plays made are then converted into a run value, which estimates the number of runs saved.

DRA is used as the WAR fielding component for Negro League seasons. When comparing to the other fielding measures, such as DRS and TZR, DRA has a higher variance among the best and worst fielders. This leads to more extreme fielding runs estimates. Therefore we divide each player's runs saved by 1.75 to make this system compatible with other measurements.

Note: For leagues in which no fielding data is available, all players are given 0 fielding runs (league average).

The DRA calculation has been used in the SABR Defensive Index since 2013, which accounts for approximately 25% of the Rawlings Gold Glove and Platinum Glove Award selection process.

Rpos, Positional Adjustment Runs

If you take a quick look at the batting performance by defensive position, you'll quickly see that teams are willing to sacrifice offense at "defensive" positions (stats are prorated to 650 plate appearances).

When one quantifies these differences and also looks at the changes in fielding performance when players move to different positions, we can estimate the average differences between positions.

Current values (per 1350 (150*9) innings played) are:

1. C: +9 runs
2. SS: +7 runs
3. 2B: +3 runs
4. CF: +2.5 runs
5. 3B: +2 runs
6. RF: -7 runs
7. LF: -7 runs
8. 1B: -9.5 runs
9. DH: -15 runs
10. P: see Pitcher Positional Adjustment

Since fielding performances change over time, these values will vary as you travel back in time. For example, first basemen in the early 1900s through the 1940s were required to be better fielders than they are today. They were still the worst fielders on the infield, but the position had more of a defensive focus then than the corner outfield positions did.

To compute a player's Positional adjustment Runs, we add together for each non-pitching position: Position multiplier (from above) × innings played at position / 1,350 Innings. For players who are only pitchers this is Pitcher Positional Adjustment (from below) × (PA/4) / 150.

There is one final adjustment. The positional adjustment should be centered on zero if you sum across all of the players in the league. To force this to be the case, we do another step where we sum the league's positional runs and then allot the excess out to players based on playing time. This way the runs across the league effectively sum to zero.

NOTE (changed May 2017), we weight this adjustment by plate appearances, but only for non-DH's. Previous this was weighted by games played for each non-dh. Perhaps we should weight it by defensive innings, but this is a reasonable proxy for that and a million times easier to implement. As a matter of scale, when I made this change, Mike Trout added 3 position runs spread out over 2011-early 2017.

Here are the adjustments made year-by-year.

Pitcher Positional Adjustment. Now, pitchers also bat, so we need to calculate their Position Player WAR as well. Since pitcher fielding is included in Pitcher WAR, we do not need to consider it here. Pitchers are almost guaranteed to be below replacement hitters and likely no one uses a particular player as a pitcher because they are a good hitter. If that were the case, we might see teams trying for an advantage by routinely using Wilson Valdez as a pitcher or an NL team would always value Jon Lester less than an AL team. Rather, any pitcher hitting is gravy, so the pitcher position adjustment is set so that all pitchers taken together produce zero WAR as position players. As position players, we say a pitcher's games played equals PA/4.

To set the pitcher positional adjustment, we add together every pitcher's Batting Runs, Baserunning Runs,and GIDP Runs = Runs_sum_lg. We also add up their Plate Appearances, PA_sum_lg. The league-wide pitcher positional adjustment is then (-600 * Runs_sum_lg/ PA_sum_lg) which is runs per plate appearance. For a particular player, this number is then multiplied by their plate appearances for their individual positional adjustment.

The top and worst pitcher as position players in 2011 were: best: Carlos Zambrano 7.6 runs, worst: Ryan Dempster -4.8 runs.

note: Some pitchers like Red Ruffing, Bob Lemon and Jim Kaat (as a pinch runner) appeared in many games as a position player, pinch hitter or pinch runner. If we credit them these PA's as just a pitcher then we overvalue their offense, so to handle this we compute a percentage of time as a non-pitcher for all pitchers and they get credit under runs_replacement for this number of games and their runs_position as pitcher are reduced by one minus this factor. Since many of these occurred prior to full play-by-play data, we just use games played data for this adjustment.

 non-Pitcher factor
    = (Games Played - Games Pitched - .63 * Games as PH or PR)
      --------------------------------------------------------
                         Games Played

.63 is a factor so we value Games Played in the field differently than as a PH or PR which are likely just a single PA.

WAA, Wins Above Average and Runs to Wins Calculation

Our calculation of Runs to Wins is detailed on the runs to wins calculation page. To summarize, we are using PythagenPat along with the league average run environment and the player's contributions on offense and defense to adjust that run environment, and then plugging it into PythagenPat to get a win percentage, then computing wins above average from that.

The runs adjustments are based on Rbat, Rbr, Rdp, Rdefense, Rpos.

Replacement Level

Replacement level is something of a touchy subject with non-sabermetricians, and probably the least understood of the ideas here.

Currently, we set replacement level at .294 winning percentage (changed from .320 in March 2013) for the major leagues, which means there are 30*162*(.500-.294) = 1,000 Wins above replacement in the major leagues as a whole.

The wins, and therefore the runs, are further divided between pitchers and position players. We assign 41% to the pitchers and 59% to the position players. This corresponds to the salaries of free agent pitchers vs. hitters over the last four seasons.

On offense, this division and replacement level corresponds to 20.5 runs over the course of 600 plate appearances. So, if a team replaced a league-average starter with a replacement player we'd expect a 20 run difference in their run differential. We call this 20.5 runs the Replacement Level multiplier.

For pitchers, the replacement level starts with the replacement level pitcher's runs allowed per out, RpO_replacement. This value is the league average runs allowed per out multiplied by (20.5-1.8)/100). In this example, 20.5 is the league's replacement runs per 600 PAs and 1.8 is an empirical factor that makes the final result mostly closely align the sum of all player replacement runs to the desired league total.

For pitcher, runs_above_rep is then runs_above_avg + RpO_replacement * Outs pitched.

Now, for some leagues the competition was not that good, so the replacement runs multiplier will be smaller because the average player is worse relative to the replacement player. However, this is really only a case when looking at pre-World War I seasons and the World War II seasons. For example, we view the average player in the 1884 Union Association (the weakest major league by a wide margin) as a replacement level player, so the multiplier is zero for that league.

Also when the season is not 162 games, there are fewer wins to go around, so strike-shortened or 154 game seasons have fewer wins and a multiplier less than 20.5.

Finally, the leagues are not always equal in their quality levels as evidenced by things like inter-league play and player performances when shifting leagues. Taking these differences into account, assign slightly different multipliers to the leagues, but centered on 20.5 for 162-game seasons and 19 for 154-game seasons. One example of this is the post-war integration. The National League integrated far more quickly than the American League and was a higher quality league until the 1970s.

For Negro League seasons, we have set the replacement level equal to that of the American and National League for those seasons. For example, the 1920 AL & NL had a .294 replacement level: .500 - (228 + 279) / (16 * 154) = .294. Therefore, replacement level for the 1920 Negro National League is also set to .294.

Replacement level for the major leagues and the WAR assigned to a league by year and league.

+---------+-----+-----+------------------+-----------+
| Year    | NL  | AL  | Other            | Note      |
+---------+-----+-----+------------------+-----------+
|    2020 | 185 | 185 |                  | Covid-19  |
|    2019 | 475 | 525 |                  |           |
|    2018 | 475 | 525 |                  |           |
|    2017 | 475 | 525 |                  |           |
|    2016 | 475 | 525 |                  |           |
|    2015 | 475 | 525 |                  |           |
|    2014 | 475 | 525 |                  |           |
|    2013 | 475 | 525 |                  | HOU=>AL   |
|    2012 | 487 | 513 |                  |           |
|    2011 | 487 | 513 |                  |           |
|    2010 | 487 | 513 |                  |           |
|    2009 | 487 | 513 |                  |           |
|    2008 | 487 | 513 |                  |           |
|    2007 | 487 | 513 |                  |           |
|    2006 | 487 | 513 |                  |           |
|    2005 | 487 | 513 |                  |           |
|    2004 | 487 | 513 |                  |           |
|    2003 | 487 | 513 |                  |           |
|    2002 | 487 | 513 |                  |           |
|    2001 | 487 | 513 |                  |           |
|    2000 | 487 | 513 |                  |           |
|    1999 | 487 | 513 |                  |           |
|    1998 | 487 | 513 |                  | Expansion |
|    1997 | 420 | 513 |                  |           |
|    1996 | 420 | 513 |                  |           |
|    1995 | 373 | 456 |                  | Strike    |
|    1994 | 298 | 364 |                  | Strike    |
|    1993 | 420 | 513 |                  | Expansion |
|    1992 | 353 | 513 |                  |           |
|    1991 | 377 | 490 |                  |           |
|    1990 | 377 | 490 |                  |           |
|    1989 | 377 | 490 |                  |           |
|    1988 | 398 | 465 |                  |           |
|    1987 | 400 | 467 |                  |           |
|    1986 | 400 | 467 |                  |           |
|    1985 | 400 | 467 |                  |           |
|    1984 | 400 | 467 |                  |           |
|    1983 | 400 | 467 |                  |           |
|    1982 | 400 | 467 |                  |           |
|    1981 | 264 | 308 |                  | Strike    |
|    1980 | 399 | 466 |                  |           |
|    1979 | 376 | 489 |                  |           |
|    1978 | 377 | 490 |                  |           |
|    1977 | 377 | 490 |                  | Expansion |
|    1976 | 379 | 419 |                  |           |
|    1975 | 379 | 419 |                  |           |
|    1974 | 380 | 420 |                  |           |
|    1973 | 380 | 420 |                  |           |
|    1972 | 364 | 402 |                  | Strike    |
|    1971 | 399 | 399 |                  |           |
|    1970 | 400 | 400 |                  |           |
|    1969 | 400 | 400 |                  | Expansion |
|    1968 | 333 | 333 |                  |           |
|    1967 | 350 | 317 |                  |           |
|    1966 | 350 | 316 |                  |           |
|    1965 | 367 | 300 |                  |           |
|    1964 | 367 | 300 |                  |           |
|    1963 | 367 | 300 |                  |           |
|    1962 | 367 | 300 |                  | Expansion |
|    1961 | 288 | 288 |                  | Expansion |
|    1960 | 279 | 228 |                  |           |
|    1959 | 280 | 229 |                  |           |
|    1958 | 279 | 228 |                  |           |
|    1957 | 279 | 228 |                  |           |
|    1956 | 279 | 228 |                  |           |
|    1955 | 279 | 228 |                  |           |
|    1954 | 279 | 228 |                  |           |
|    1953 | 278 | 228 |                  |           |
|    1952 | 279 | 228 |                  |           |
|    1951 | 280 | 229 |                  |           |
|    1950 | 279 | 228 |                  |           |
|    1949 | 292 | 216 |                  |           |
|    1948 | 292 | 216 | NAL:30, NN2:54   |           |
|    1947 | 292 | 216 | NAL:30, NN2:63   |           |
|    1946 | 279 | 228 | NAL:20, NN2:69   | WWII      |
|    1945 | 235 | 213 | NAL:31, NN2:42   | WWII      |
|    1944 | 236 | 214 | NAL:21, NN2:43   | WWII      |
|    1943 | 236 | 214 | NAL:38, NN2:51   | WWII      |
|    1942 | 233 | 211 | NAL:28, NN2:48   | WWII      |
|    1941 | 254 | 254 | NAL:20, NN2:51   |           |
|    1940 | 252 | 252 | NAL:33, NN2:53   |           |
|    1939 | 239 | 264 | NAL:35, NN2:38   |           |
|    1938 | 236 | 261 | NAL:44, NN2:48   |           |
|    1937 | 239 | 264 | NAL:52, NN2:45   |           |
|    1936 | 240 | 266 | NN2:63           |           |
|    1935 | 226 | 277 | NN2:77           |           |
|    1934 | 224 | 274 | NN2:61           |           |
|    1933 | 226 | 276 | NN2:58           |           |
|    1932 | 228 | 278 | EWL:56, NSL:60   |           |
|    1931 | 228 | 278 | NNL:42           |           |
|    1930 | 228 | 279 | NNL:118          |           |
|    1929 | 227 | 277 | ANL:85, NNL:104  |           |
|    1928 | 227 | 278 | ECL:38, NNL:120  |           |
|    1927 | 228 | 279 | ECL:88, NNL:133  |           |
|    1926 | 227 | 277 | ECL:78, NNL:117  |           |
|    1925 | 227 | 277 | ECL:88, NNL:121  |           |
|    1924 | 227 | 277 | ECL:88, NNL:105  |           |
|    1923 | 228 | 279 | ECL:55, NNL:114  |           |
|    1922 | 228 | 279 | NNL:116          |           |
|    1921 | 227 | 278 | NNL:138          |           |
|    1920 | 228 | 279 | NNL:106          |           |
|    1919 | 206 | 252 |                  | WWI       |
|    1918 | 187 | 228 |                  | WWI       |
|    1917 | 227 | 277 |                  |           |
|    1916 | 227 | 278 |                  |           |
|    1915 | 227 | 277 | FL:122           |           |
|    1914 | 228 | 279 | FL:121           |           |
|    1913 | 224 | 274 |                  |           |
|    1912 | 226 | 276 |                  |           |
|    1911 | 226 | 276 |                  |           |
|    1910 | 228 | 278 |                  |           |
|    1909 | 239 | 264 |                  |           |
|    1908 | 241 | 266 |                  |           |
|    1907 | 236 | 261 |                  |           |
|    1906 | 237 | 262 |                  |           |
|    1905 | 239 | 264 |                  |           |
|    1904 | 239 | 264 |                  | 154g      |
|    1903 | 216 | 238 |                  |           |
|    1902 | 214 | 237 |                  |           |
|    1901 | 216 | 239 |                  |           |
|    1900 | 229 |     |                  |           |
|    1899 | 373 |     |                  |           |
|    1898 | 371 |     |                  |           |
|    1897 | 324 |     |                  |           |
|    1896 | 321 |     |                  |           |
|    1895 | 324 |     |                  |           |
|    1894 | 324 |     |                  |           |
|    1893 | 319 |     |                  |           |
|    1892 | 373 |     |                  |           |
|    1891 | 224 |     | AA:158           |           |
|    1890 | 219 |     | AA:153, PL:152   |           |
|    1889 | 214 |     | AA:142           |           |
|    1888 | 219 |     | AA:140           |           |
|    1887 | 203 |     | AA:139           |           |
|    1886 | 198 |     | AA:141           |           |
|    1885 | 183 |     | AA:115           |           |
|    1884 | 185 |     | AA:166, UA:0     |           |
|    1883 | 162 |     | AA:101           |           |
|    1882 | 138 |     | AA:61            |           |
|    1881 | 138 |     |                  |           |
|    1880 | 137 |     |                  |           |
|    1879 | 130 |     |                  |           |
|    1878 | 74  |     |                  |           |
|    1877 | 73  |     |                  |           |
|    1876 | 105 |     |                  |           |
|    1875 |     |     | NA:139           |           |
|    1874 |     |     | NA:96            |           |
|    1873 |     |     | NA:82            |           |
|    1872 |     |     | NA:73            |           |
|    1871 |     |     | NA:52            |           |
+---------+-----+-----+------------------+-----------+

Calculating the Replacement runs for a current player takes a bit of a conversion going from the league total WAR to a player-by-player basis. Even then we don't end up exactly on the button for the desired number, so we re-center on the desired number by assigning the difference to players based on their playing time.

WAR_rep, Replacement Level Wins

Our calculation of Runs to Wins is detailed on the runs to wins calculation page.

Fine-Tuning Replacement Level

After we make a first pass through the calculations, we determine how the league's current total WAR differs from the desired overall league WAR. We then add or subtract fractional replacement runs from each player's runs_replacement total based on their playing time, and recompute WAR_rep with this adjustment included.

WAR

WAR = WAR_rep + WAA