As I mentioned in the first part of this Wins Above Replacement series, I will be walking through the calculation of a specific player's WAR in this post. I'll use David DeJesus as an example. I will mirror the format of the aforementioned WAR post in order to make the following calculations easy to follow.
First, we need to convert DeJesus' wOBA to wRAA. To do that, we simply use the following formula:
DeJesus' wRAA, unadjusted for the ballparks that he played in, was 8.6 runs.* Therefore, DeJesus' batting was worth 8.6 runs.
DeJesus played all three outfield positions in 2012. He primarily played in right field, (86 total games, 74 starts), with somewhat fewer starts in center field, (50 games, 36 starts), and a few starts in left field, (three games, two starts). DeJesus' UZR in right was -4.5, his UZR in center was -1.3, and his UZR in left was 0.9; we just add the three UZRs to get the runs DeJesus was worth in the field. Therefore, DeJesus' fielding was worth -4.9 runs.
Defensive Position (or Positional Adjustment)
DeJesus played the equivalent of 85 full games in right field, 37 games in center field, and two games in left field. His positional adjustment is calculated below:
Therefore, DeJesus' positional adjustment was worth -3.5 runs.
Replacement Level Adjustment
We need to credit DeJesus for how frequently he played in 2012. The replacement level calculation is as follows:
Therefore, DeJesus' replacement level adjustment was worth 19.4 runs.
We now add the runs above average from each of the four components that we evaluated: Batting, Fielding, Defensive Position, and Replacement Level Adjustment. Below is a chart that summarizes DeJesus' component run values calculated by both us and FanGraphs:
As you can see, DeJesus was worth 19.6 runs above average in 2012. After converting these runs to wins using the 10 runs to one win ratio, we find that DeJesus was worth 2.0 WAR.
In actuality, FanGraphs has DeJesus at 1.7 WAR. As we can see in the chart above, the key differences in the numbers stem from the fact that: (i) FanGraphs gave DeJesus 6.0 runs in his batting component, which is lower because it likely accounts for the better than average ballpark conditions that DeJesus played in, (ii) FanGraphs now calculates a baserunning component of WAR in which DeJesus put up a negative score, and (iii) FanGraphs used a slightly different runs-to-win ratio. Once we account for baserunning (-.1 WAR), and the ballpark adjustment (-.2 WAR), we get 1.7 WAR.
Despite some inconsistencies, our hand calculated WAR of 2.0 was very close to DeJesus' actual WAR of 1.7. With the inclusion of the baserunning component*, the addition of a ballpark adjustment on the batting component, and a more accurate runs to win ratio, we could perfectly calculate WAR.
* An aside: When I have some time, I'll try and figure out how to calculate the ballpark adjustment and the baserunning score. If I do, I'll update these posts. In the meantime, if anyone else knows where to find an explanation for either of these, then please mention it below.