# 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
• 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, 1953 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.

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).

165 650 582 64 142 28 1 17 70 3 2 54 126 .245 .313 .389 .703 226 14 5 4 4 5 5 .282 95 100
152 650 575 75 155 32 1 22 85 3 1 63 121 .271 .345 .452 .797 259 15 5 0 5 7 5 .305 121 100
151 649 586 75 152 29 4 12 62 13 4 47 103 .260 .320 .389 .709 228 11 6 5 4 2 6 .294 97 100
155 649 583 69 147 31 2 14 71 7 3 52 111 .252 .317 .390 .707 227 14 5 3 5 4 7 .287 96 100
152 650 590 73 155 28 3 10 60 15 6 44 98 .263 .317 .380 .697 224 11 4 6 4 2 7 .298 94 100
152 649 585 75 149 29 3 17 70 14 5 52 124 .255 .320 .408 .728 238 10 5 2 4 3 6 .295 102 100
148 650 583 84 152 29 6 14 62 25 8 52 124 .261 .326 .410 .735 238 9 5 5 3 2 7 .307 104 100
152 650 578 78 155 31 3 20 75 11 4 59 123 .269 .341 .441 .782 254 11 6 1 4 5 5 .308 117 100
151 650 578 74 151 31 1 20 84 6 2 62 120 .263 .337 .427 .764 247 15 4 0 4 5 6 .298 112 100
283 649 556 29 78 12 0 3 31 3 1 21 213 .141 .175 .182 .357 101 6 1 69 0 0 5 .221 0 100

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.

year_IDruns_pruns_cruns_1bruns_2bruns_3bruns_ssruns_lfruns_cfruns_rfruns_dh
187112.895810.00000.00003.00005.000010.0000-10.0000-8.0000-10.0000-15.0000
187217.879710.00000.00003.00005.000010.0000-10.0000-8.0000-10.0000-15.0000
187338.964410.00000.00003.00005.000010.0000-10.0000-8.0000-10.0000-15.0000
187427.446210.00000.00003.00005.000010.0000-10.0000-8.0000-10.0000-15.0000
187517.755110.00000.00002.50005.000010.0000-10.0000-8.0000-10.0000-15.0000
187610.061310.00000.00002.50005.000010.0000-10.0000-8.0000-10.0000-15.0000
187727.715510.00000.00002.50005.000010.0000-10.0000-8.0000-9.5000-15.0000
187825.387810.00000.00002.50005.000010.0000-10.0000-8.0000-9.5000-15.0000
187921.601610.00000.00002.50005.000010.0000-9.5000-8.0000-9.5000-15.0000
188012.794010.00000.00003.00005.000010.0000-9.5000-8.0000-9.5000-15.0000
188124.333810.00000.00003.00005.000010.0000-9.5000-8.0000-9.5000-15.0000
188217.016810.00000.00003.00005.000010.0000-9.5000-8.0000-9.5000-15.0000
188325.937410.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
188419.924910.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
188527.578510.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
188625.934710.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
188743.054010.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
188838.555810.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
188948.185910.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
189044.751410.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
189145.110410.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
189239.664010.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
189342.033710.00000.00003.00005.000010.0000-10.0000-8.0000-9.0000-15.0000
189448.023910.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
189540.371310.00000.00003.00005.000010.0000-9.5000-8.0000-9.0000-15.0000
189645.589710.00000.00003.00005.000010.0000-10.0000-8.0000-9.0000-15.0000
189746.951410.0000-0.50002.50005.000010.0000-9.5000-7.5000-9.0000-15.0000
189836.284510.0000-1.50002.00005.000010.0000-9.0000-7.0000-9.0000-15.0000
189945.380410.0000-2.00001.50005.000010.0000-9.0000-6.5000-9.0000-15.0000
190044.970010.0000-2.50001.50005.000010.0000-8.5000-6.0000-8.5000-15.0000
190143.159510.0000-3.50001.00005.000010.0000-8.0000-5.0000-8.5000-15.0000
190242.656410.0000-4.00000.50005.000010.0000-8.0000-4.5000-8.0000-15.0000
190346.975510.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
190437.384410.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
190544.415510.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
190639.693010.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
190739.232710.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
190839.825510.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
190946.094110.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
191041.757710.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
191152.788910.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
191248.757110.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
191351.052410.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
191451.268010.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
191545.063010.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
191646.307510.0000-5.00000.00005.000010.0000-8.0000-4.0000-8.0000-15.0000
191740.80329.0000-5.50000.50005.000010.0000-8.0000-4.0000-8.0000-15.0000
191838.22478.5000-5.50001.50005.000010.0000-7.5000-4.0000-7.5000-15.0000
191944.33517.5000-6.00002.00005.000010.0000-7.5000-4.0000-7.5000-15.0000
192046.58027.0000-6.00003.00005.000010.0000-7.5000-4.0000-7.5000-15.0000
192155.86756.5000-6.50003.50005.000010.0000-7.0000-4.0000-7.5000-15.0000
192263.00475.5000-6.50004.50005.000010.0000-7.0000-4.0000-7.0000-15.0000
192356.66115.0000-7.00005.00005.000010.0000-7.0000-4.0000-7.0000-15.0000
192452.22125.0000-7.00005.00005.000010.0000-7.0000-4.0000-7.0000-15.0000
192562.14315.0000-7.00005.00005.000010.0000-7.0000-4.0000-7.0000-15.0000
192658.64195.0000-7.00005.00005.000010.0000-7.0000-4.0000-7.0000-15.0000
192759.19965.0000-7.00005.00004.500010.0000-7.0000-3.5000-7.0000-15.0000
192863.71175.0000-7.00005.00004.500010.0000-7.0000-3.5000-7.0000-15.0000
192967.24005.0000-7.00005.00004.000010.0000-7.0000-3.0000-7.0000-15.0000
193063.24115.0000-7.00005.00004.000010.0000-7.0000-3.0000-7.0000-15.0000
193161.56485.0000-7.00005.00003.500010.0000-7.0000-2.5000-7.0000-15.0000
193264.17945.0000-7.00005.00003.000010.0000-7.0000-2.5000-7.0000-15.0000
193358.23235.0000-7.00005.00003.000010.0000-7.0000-2.0000-7.0000-15.0000
193467.50465.0000-7.00005.00003.000010.0000-7.0000-2.0000-7.0000-15.0000
193561.06685.0000-7.00005.00003.000010.0000-7.0000-2.0000-7.0000-15.0000
193668.95045.0000-7.00005.00003.000010.0000-7.0000-2.0000-7.0000-15.0000
193765.97505.0000-7.00005.50002.500010.0000-7.0000-2.0000-7.0000-15.0000
193868.88545.0000-7.00005.50002.000010.0000-7.0000-1.5000-7.0000-15.0000
193965.81385.0000-7.00006.00001.500010.0000-7.0000-1.5000-7.0000-15.0000
194066.22855.0000-7.00006.00001.500010.0000-7.0000-1.5000-7.0000-15.0000
194165.58355.0000-7.00006.50001.000010.0000-7.0000-1.5000-7.0000-15.0000
194259.23375.0000-7.00006.50000.500010.0000-7.0000-1.0000-7.0000-15.0000
194350.05665.0000-7.00007.00000.000010.0000-7.0000-1.0000-7.0000-15.0000
194454.73715.0000-7.00007.00000.000010.0000-7.0000-1.0000-7.0000-15.0000
194553.06355.0000-7.00007.00000.000010.0000-7.0000-1.0000-7.0000-15.0000
194659.46675.0000-7.00007.00000.000010.0000-7.0000-1.0000-7.0000-15.0000
194764.56385.0000-7.00007.00000.000010.0000-7.0000-1.0000-7.0000-15.0000
194863.18005.0000-7.00007.00000.000010.0000-7.0000-1.0000-7.0000-15.0000
194965.52585.0000-7.00007.00000.000010.0000-7.0000-1.0000-7.0000-15.0000
195069.42065.0000-7.00007.00000.000010.0000-7.0000-1.0000-7.0000-15.0000
195159.25935.0000-7.00006.50000.00009.5000-7.0000-1.0000-7.0000-15.0000
195266.81825.0000-7.00006.50000.00009.5000-7.0000-1.0000-7.0000-15.0000
195364.63465.5000-7.00006.50000.50009.5000-7.0000-1.0000-7.0000-15.0000
195468.64066.0000-7.50006.00000.50009.0000-7.0000-1.0000-7.0000-15.0000
195559.43996.5000-7.50005.50000.50009.0000-7.0000-1.0000-7.0000-15.0000
195665.22737.0000-8.00005.50001.00009.0000-7.0000-1.0000-7.0000-15.0000
195764.38937.5000-8.50005.00001.50009.0000-7.0000-1.0000-7.0000-15.0000
195865.57258.0000-8.50005.00002.00009.0000-7.5000-1.0000-7.0000-15.0000
195968.67738.5000-9.00004.50002.50009.0000-7.5000-1.0000-7.0000-15.0000
196066.29828.5000-9.00004.50003.00009.0000-8.0000-1.0000-7.0000-15.0000
196169.57518.5000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
196272.45728.5000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
196366.31378.5000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
196469.53348.5000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
196569.72588.5000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
196664.64698.5000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
196765.79758.5000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
196860.63878.5000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
196970.06769.0000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
197071.71269.0000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
197163.60899.0000-9.00004.00003.00009.0000-8.0000-1.0000-7.0000-15.0000
197260.00579.0000-9.00004.00003.00009.0000-7.5000-1.0000-7.0000-15.0000
197367.03499.0000-9.00004.00003.00009.0000-7.5000-1.0000-7.0000-15.0000
197460.66469.0000-9.00004.00003.00009.0000-7.5000-1.0000-7.0000-15.0000
197565.89699.0000-9.00004.00003.00009.0000-7.5000-1.0000-7.0000-15.0000
197664.85009.0000-9.00004.00003.00009.0000-7.5000-1.0000-7.0000-15.0000
197771.25659.0000-9.50004.00003.00009.0000-7.5000-1.0000-7.0000-15.0000
197869.43109.0000-9.50004.00003.00009.0000-7.5000-1.0000-7.0000-15.0000
197972.21139.0000-9.50004.00002.50009.0000-7.5000-1.0000-7.0000-15.0000
198066.68889.0000-9.50004.00002.00009.0000-7.0000-1.0000-7.0000-15.0000
198165.76899.0000-9.50004.00002.00008.5000-7.0000-0.5000-7.0000-15.0000
198268.15009.0000-9.50004.00001.50008.5000-7.0000-0.5000-7.0000-15.0000
198371.44368.5000-9.50004.00001.00008.5000-7.00000.0000-7.0000-15.0000
198469.69248.5000-9.50004.00001.00008.5000-7.00000.0000-7.0000-15.0000
198567.65118.5000-9.50004.00001.00008.5000-7.00000.0000-7.0000-15.0000
198674.83688.5000-9.50004.00001.00008.5000-7.00000.0000-7.0000-15.0000
198783.97088.5000-9.50004.00001.00008.5000-7.00000.5000-7.0000-15.0000
198871.25158.5000-9.50004.00001.00008.5000-7.00000.5000-7.0000-15.0000
198966.66128.5000-9.50003.50001.00008.5000-7.00001.0000-7.0000-15.0000
199078.85718.5000-9.50003.00001.00008.5000-7.00001.5000-7.0000-15.0000
199170.94658.5000-9.50003.00001.00008.5000-7.00001.5000-7.0000-15.0000
199271.84318.5000-9.50003.00001.00008.5000-7.00002.0000-7.0000-15.0000
199380.02168.5000-9.50003.00001.00008.5000-7.00002.5000-7.0000-15.0000
199483.95748.5000-9.50003.00001.00008.5000-7.00002.5000-7.0000-15.0000
199579.07138.5000-9.50003.00001.50008.5000-7.00002.5000-7.0000-15.0000
199679.70108.5000-9.50003.00001.50008.5000-7.00002.5000-7.0000-15.0000
199787.69748.5000-9.50003.00001.50008.5000-7.00002.5000-7.0000-15.0000
199881.85228.5000-9.50003.00001.50008.0000-7.00002.5000-7.0000-15.0000
199988.65498.5000-9.50003.00001.50008.0000-7.00002.5000-7.0000-15.0000
200090.32458.5000-9.50003.00002.00008.0000-7.00002.5000-7.0000-15.0000
200186.84138.5000-9.50003.00002.00007.5000-7.00002.5000-7.0000-15.0000
200281.71738.5000-9.50003.00002.00007.5000-7.00002.5000-7.0000-15.0000
200383.30419.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
200487.27189.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
200582.81709.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
200692.94649.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
200788.24609.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
200884.42779.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
200984.70479.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
201080.55139.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
201175.91619.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
201284.71849.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
201378.06109.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
201481.08869.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
201582.36299.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
201686.26689.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000
201791.23189.0000-9.50003.00002.00007.0000-7.00002.5000-7.0000-15.0000

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, GIDP Runs, and Replacement Level 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.

#### 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.

## Replacement level for MLB and the WAR assigned to a league by year and league.

```| year_ID | WAR_NL | WAR_AL | WAR_other |
+---------+--------+--------+-----------+
|    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 |   expans  |
|    1997 |    420 |    513 |           |
|    1996 |    420 |    513 |           |
|    1995 |    373 |    456 |  strike   |
|    1994 |    298 |    364 |  strike   |
|    1993 |    420 |    513 |  expan    |
|    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 |   expan   |
|    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 |    expan  |
|    1968 |    333 |    333 |           |
|    1967 |    350 |    317 |           |
|    1966 |    350 |    316 |           |
|    1965 |    367 |    300 |           |
|    1964 |    367 |    300 |           |
|    1963 |    367 |    300 |           |
|    1962 |    367 |    300 |    expan  |
|    1961 |    288 |    288 |    expan  |
|    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 |           |
|    1947 |    292 |    216 |           |
|    1946 |    279 |    228 |   WWII    |
|    1945 |    235 |    213 |   WWII    |
|    1944 |    236 |    214 |   WWII    |
|    1943 |    236 |    214 |   WWII    |
|    1942 |    233 |    211 |   WWII    |
|    1941 |    254 |    254 |           |
|    1940 |    252 |    252 |           |
|    1939 |    239 |    264 |           |
|    1938 |    236 |    261 |           |
|    1937 |    239 |    264 |           |
|    1936 |    240 |    266 |           |
|    1935 |    226 |    277 |           |
|    1934 |    224 |    274 |           |
|    1933 |    226 |    276 |           |
|    1932 |    228 |    278 |           |
|    1931 |    228 |    278 |           |
|    1930 |    228 |    279 |           |
|    1929 |    227 |    277 |           |
|    1928 |    227 |    278 |           |
|    1927 |    228 |    279 |           |
|    1926 |    227 |    277 |           |
|    1925 |    227 |    277 |           |
|    1924 |    227 |    277 |           |
|    1923 |    228 |    279 |           |
|    1922 |    228 |    279 |           |
|    1921 |    227 |    278 |           |
|    1920 |    228 |    279 |           |
|    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 |        | AA,PL&UA  |
|    1891 |    224 |        |       158 |
|    1890 |    219 |        |       305 |
|    1889 |    214 |        |       142 |
|    1888 |    219 |        |       140 |
|    1887 |    203 |        |       139 |
|    1886 |    198 |        |       141 |
|    1885 |    183 |        |       115 |
|    1884 |    185 |        |UA:0,AA:166|
|    1883 |    162 |        |       101 |
|    1882 |    138 |        |        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