Calculating Wins Above Replacement

(WAR)

What is WAR?

WAR, an acronym for Wins Above Replacement, is an attempt by baseball statisticians (also known as sabermetricians) to reduce a baseball players worth to a team to a single number. The number tells us how many additional wins that a player can give a team over a replacement player that is available at no to low

cost and effort (usually a player on the bench or somewhere in the minor league farm system). For example, if a team won 80 games during the season and Player A has a WAR that calculates to 7.1, this suggests that the team won approximately seven more games (at 80) than they would have otherwise with a replacement level

player (only 73).Relatively new in concept, it combines hitting ability, baserunning, and defensive ability through a series of

formulas, then adjusts for specific position on the field (i.e. Catcher, First Baseman, Shortstop, etc.). To understand this concept, a brief overview of the game and sabermetrics in general is necessary.

The goal in baseball is to score the most runs, which happens ONLY when a player has safely reached base AND navigated the base paths back to his starting point at home plate.

Each team consists of nine defensive players who try to stop the runner from advancing to the next base. The teams switch sides after the defense records three outs, which can be achieved in a number of ways. Nine frames (18 half frames) are played and whoever is leading in runs at the end of these frames is declared the winner.

Though statistics have always been used since the invention of the game, prior to the advent of sabermetrics they did not account for the various ways that a player can reach base safely and many defensive statistics did not measure a players overall defensive worth on the field.

It is widely accepted that the term “sabermetrics” was coined by Bill James in reference to the Society for American Baseball Research (SABR) in which he is a member. James, known by the community as the father of modern sabermetrics, began publishing The Bill James Abstract in 1977 which started asking questions that no statistic that was widely used in baseball box scores (Average, RBI, Fielding Percentage) at the time could answer. For example, “Which pitchers and catchers allow the most bases to be stolen?”, “Why does batting average not include when a batter is walked to first base or hit by a pitch? Isn’t plate discipline part of being a good hitter?”, and “Shouldn’t a defensive player be evaluated on how many outs he is involved in recording rather than only on attempts at a cleanly fielded ball?” So James invented a new set of formulas and factors that adjust for these questions, including Runs Created for batters [RC=(Total Bases*(Hits + Walks))/(Plate Appearances)] and Range Factor for defensive plays [RF=(Assists + Putouts)/(Games Played)]

What are Sabermetrics?

Thus, sabermetrics became a mainstay in baseball record keeping and have been tweaked and refined since James initial publications and are sometimes used by the biggest sports coverage outlets, like ESPN and Fox Sports. Sabermetrics were also highlighted in the 2011 movie “Moneyball” based on the book by Michael Lewis, showing how the General Manager of the Oakland A’s pieced together a winning squad in 2002 based on these stats.

So what about W.A.R.?

Since the concept of WAR is fairly new, there are a few variations. The three most widely used formulas in the sabermetric community are taken directly from Baseball Reference, Fangraphs, and Baseball Prospectus (bWAR, fWAR, and WARP)

respectively. For this presentation, we will use the Fangraphs formula fWAR (found at http://www.fangraphs.com/library/misc/war/) since it is a bit more user friendly and because their website contains all

related information to sabermetrics. Because pitchers and position players WAR are calculated differently, we will be focusing on hitting and fielding statistics for simplicities sake.

fWAR combines a number of inputs and it’s formula for position players is as follows

fWAR = (Batting Runs + Base Running Runs + Fielding Runs + Positional Adjustment + League

Adjustment + Replacement Runs)/(Runs Per Win)We will break these inputs down piece by piece in the next few slides.

Commonly Used Statistics and Abbreviations

Below are a list of some of the most widely used statistics (and their abbreviations) in baseball and the ones that we will use for calculating our main statistic, W.A.R. (Note: These are only statistics for position

players, not pitchers)

1B – Single: hits on which the batter reaches first base safely without the contribution of a fielding error.2B – Double: hits on which the batter reaches second base safely without the contribution of a fielding error.3B – Triple: hits on which the batter reaches third base safely without the contribution of a fielding error.HR – Home Run: hits on which the batter successfully touches all four bases safely without the contribution of a fielding error.AB – At Bat: Plate appearances, not including bases on balls, being hit by pitch, sacrifices, interference, or obstruction.BA – Batting Average: hits divided by at bats (H/AB)BB – Base On Balls: (also called a "walk"): hitter not swinging at four pitches called out of the strike zone and awarded first baseH – Hits: times reached base because of a batted, fair ball without error by the defenseHBP – Hit By Pitch: times touched by a pitch and awarded first base as a resultIBB – Intentional Base On Balls: times awarded first base on balls (see BB above) deliberately thrown by the pitcher.UBB – Unintentional Base on Balls: times awarded first base on balls (see BB above) not deliberately thrown by the pitcher.PA – Plate Appearances: number of completed batting appearancesSF – Sacrifice Flies: Fly balls hit to the outfield which although caught for an out, allow a baserunner to advance

Calculating Batting RunsThe first step in calculating fWAR is to find a players wOBA (pronounced Whoa-ba) which stands for Weighted On Base Average. The formula is as follows:

wOBA = (0.690×uBB + 0.722×HBP + 0.888×1B + 1.271×2B + 1.616×3B +2.101×HR) / (AB + BB – IBB + SF + HBP)

These weights change yearly and Fangraphs has a table that provides all weights for a specific season from 1871 to present here. This chart also provides scales and league wide wOBA to calculate wRAA (Weighted Runs Above Average) which in

turn calculates to Batting Runs, our first input in our fWAR formula.

wRAA = ((wOBA – league wOBA) / wOBA scale) × PA

wRAA calculates the number of offensive runs a player provides over his replacement. This is obviously important because runs wins games. wRAA is not adjusted for league or park factors so we use this formula.

Batting Runs = wRAA + (lgR/PA – (PF*lgR/PA))*PA + (lgR/PA – (AL or NL non-pitcher wRC/PA))*PA

Joey Votto’s 2013 Season in

Batting Runs Let’s work an example using Joey Votto and his 2013 numbers to calculate

batting runs, found herewOBA = (0.690*uBB + 0.722*HBP + 0.888*1B + 1.271*2B + 1.616*3B + 2.101*HR) / (AB + BB – IBB + SF + HBP)

0.400 = (0.690*116 + 0.722*4 + 0.888*120 + 1.271*30 + 1.616*3 + 2.101*24) / (581 + 135 – 19 + 6 + 4)

We then take his wOBA, substract the league wOBA and divide by the wOBA scale to convert it to wRAA:wRAA = ((wOBA – league wOBA) / wOBA scale) *PA

48.9 = ((.400-.314)/1.277)*726

We then adjust for league and park factors to come to Batting Runs

Batting Runs = wRAA + (lgR/PA – (PF*lgR/RA))*PA + (lgR/PA – (AL or NL non-pitcher wRC/PA))*PA

46.3 = 48.9 + (.110 – (1.01*.110))*726 + (.110 – (9783/86959))*726

Calculating Base Running RunsCalculating baseruns is relatively simple as well. It is the sum of Ultimate Base

Running (UBR) and Weighted Stolen Base Runs (wSB). UBR cannot be calculated by hand since it uses video tracking data from another

source, Baseball Info Solutions (link here) but the numbers are here

We can, however, calculate wSB by this formula

wSB = (SB * runSB) + (CS * runCS) – (lgwSB * (1B + BB + HBP – IBB))

SB and CS are abbrevations for Stolen Bases and Caught Stealing, respectively. runSB and runCS are linear weights and are estimates. Over all seasons 0.2=runSB

and runCS changes yearly.

To obtain lgwSB (League Weighted SB) use this formula for any particular season:lgwSB = (lgSB * runSB + lgCS * runCS) / (lg1B + lgBB + lgHBP – lgIBB)

Using the same example, Joey Vottos 2013 season with a lgwSB of 0.0035 for the 2013 season

-0.8 = (6*0.2) + (3*-0.384) – 0.0035*(120 + 135 + 4 -19)

Therefore, looking up his UBR at 0.2 and adding to his wSB of -.08 we arrive at -.06

Fielding Runs and Positional AdjustmentFielding runs are found under the defensive ratings table on Joey Vottos 2013 statistics and the explanation for calculation

is here and here. This is another statistic that uses video tracking to calculate. Joey Votto had a 2.2 Fielding Runs Above Average in 2013. Because playing shortstop is relatively more difficult than first base, for example, we must also adjust for position. Fangraphs uses the formula for position adjustment:

Positional Adjustment = ((Innings Played/9) / 162) * position specific run value

and uses these numbers based on difficulty of position

Catcher: +12.5 runs (all are per 162 defensive games)First Base: -12.5 runsSecond Base: +2.5 runsThird Base: +2.5 runsShortstop: +7.5 runsLeft Field: -7.5 runsCenter Field: +2.5 runsRight Field: -7.5 runsDesignated Hitter: -17.5 runs

Joey Votto plays first base so his formula for the 2013 is thus:-12.3 = ((1430.66/9) / 162) * -12.5

League Adjustment: American League (AL) or National League (NL)

Since baseball is split into two different divisions with slightly different rules in each regarding hitting, we must adjust for the player in the league by the following formula

League Adjustment = ((-1)*(lgBatting Runs + lgBase Running Runs + lgFielding Runs + lgPositional Adjustment) / lgPA)*PA

Joey Votto plays for the Cincinatti Reds who are in the National League. Here are the NL wide numbers for 2013 and Votto’s League adjustment number

0.8 = (-1)*((-653.1 – 11.7 +74.0 +491.4)/92116)*726

Calculating Replacement Runs We now have the numbers we need for our player example, Joey Votto, but what about the other component, our replacement player and his run contribution? Here is Fangraphs explanation of how they calculate Replacement Level Runs:

“To do this, we start with 570 WAR (or 57% of the total 1,000 WAR) and multiply it by the number of games in the season divided by 2,430 because position players make up 570 WAR per 2,430 games. We then multiply that by Runs Per Win divided by League PA to convert WAR into runs per PA and then we multiple that by the player’s PA to determine their (the players) share of runs.”

The formula is as follows:

Replacement Level Runs = (570 * (MLB Games/2,430)) * (Runs Per Win/lgPA) * PA

Therefore, based on 2013 season numbers we can calculate this as such

20.7 = (570 * (2,431/2,430)) * (9.264/184873) * 726

Joey Vottos statistic here says that he contributed +20.7 runs over his replacement player

Piecing Together The FormulaWe have all the numbers necessary except for our divisible number, Runs Per Win. Runs Per Win (R/W) is the number of

extra runs a team needs to win a game, league averaged. This number varies yearly as well and is found at http://www.fangraphs.com/guts.aspx?type=cn.

League average for the year was 9.264

Lets combine our numbers into the formula to calculate Vottos WAR statistic

fWAR = (Batting Runs + Base Running Runs + Fielding Runs + Positional Adjustment + League Adjustment + Replacement Runs)/(Runs Per Win)

fWAR = (46.3 - 0.6 + 2.2 - 12.3 + 0.8 + 20.7)/(9.264)fWAR = 6.160

=6.2

Though not an exact science yet, we can thus conclude, by the fairest measure we have, that Joey Votto contributed approximately 6 more wins to his teams overall 2013 record than a replacement level player would have during that same season.

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

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8Joey Votto Career WAR

Season

WAR

Since we now know how to calculate WAR, here are Joey Vottos WAR numbers since the start of his major league career. 2007 and 2014 seasons do not accurately reflect his numbers since

2007 only included 89 PA and 2014 was an injury season.

2007 2008 2009 2010 2011 2012 2013 20140

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8 Joey Vottos Career Average Rate Of Change

Vottos WAR Plot Points

Average Rate of Change at +0.52

Season

WAR

Omitting Vottos starting season and his injury season, 2007 and 2014 respectively, we can use the Slope as an average Rate of Change formula to obtain a line that reflects how his worth to the Cincinnati Reds has changed over the course of 5 seasons.

= = .52 Average Rate of Change

We can conclude that Votto would have had a WAR of 6.62 in 2014, however we are also talking about performance based numbers and a game that must take into account hundreds of variables. If healthy though, it may have come to pass.

We can also use Babe Ruth’s career war to graph his performance over his career. Between the years 1914 and 1918, Ruth played in less than 100 games. 1925, and injury season and 1935, the last year of his career are also unreflective of Ruth’s overall performance.

1910 1915 1920 1925 1930 1935 19400

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1.40.9

1.5

5.2

9.4

13.313.9

6.5

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12.5

3.6

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10.6

7.8

10.5 10.7

8.7

6.7

5.2

0.2

Babe Ruth Career WAR

Season

WAR

1916 1918 1920 1922 1924 1926 1928 1930 1932 1934 19360

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16Babe Ruth Average Rate Of Change 1918-1923

Ruths WAR Plot Points

Average Rate of Change 1918-1923

Season

WAR

Using the same Slope as Rate Of Change formula, I omitted his season where Ruth played in less than 100 games (except for 1925 for comparison). I used the formula to calculate his years between 1918 (first season with over 100 games) and 1923

(his highest ever WAR contribution).

= = 1.96 Average Rate of Change

Therefore, Ruth improved by an average of approximately 2 games over a replacement player between 1918-1923. That’s unheard of in MLB by todays standards.

We can also see that far surpassed expectations between 1919-1921 but fell way below average in 1922. It may have been attributed to having nearly 200 less PA during 1922.

1916 1918 1920 1922 1924 1926 1928 1930 1932 1934 19360

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16Babe Ruth Career WAR

Ruths WAR Plot Points

Average Rate of Change 1918-1923

Average Rate of Change 1923-1934

Season

WAR

Here is the same graph and same formula but projecting the downward trend from his highest WAR of 15 in 1923 to near the end of his career in 1934.

= = -0.89 Average Rate of Change

We can see that between 1924 and 1928, this solution nearly comes close to the actual numbers, especially 1926 and 1928. Again, because these are performance based numbers, lots of variables are involved, however the formula does accurately

reflect certain seasons and numbers.

Side by side comparison of the two charts

1916 1918 1920 1922 1924 1926 1928 1930 1932 1934 19360

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Babe Ruth Career Average Rate Of Change

Ruths WAR Plot Points

Average Rate of Change 1918-1923

Average Rate of Change 1923-1934

Season

WAR

2007 2008 2009 2010 2011 2012 2013 20140

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8Joey Vottos Career Average Rate Of Change

Vottos WAR Plot Points

Average Rate of Change at +0.52

Season

WAR

Further Reading and Works Cited

James, Bill The New Bill James Historical Baseball Abstract Free Press, 2003

Lewis, Michael Moneyball: The Art of Winning an Unfair Game W.W. Norton and Company, 2004

Tango, Tom, Mitchel Lichtman and Andrew Dolphin The Book: Playing The Percentages in Baseball CreateSpace Independent Publishing Platform, 2014

Editor WAR for Position Players www.fangraphs.com

www.baseball-reference.com

www.baseballprospectus.com

http://www2.baseballinfosolutions.com/