Here’s the brief explanation of how to calculate WAR.
First we start with all of the things that a player can do on offense: hit singles, double, triples, home runs, walks, steal bases, strike out, get thrown out stealing, etc, etc. Including some marginal things like reaching base on error. For each one of these events we know (historically) how many runs a team has scored, on average, from the occurrence of an event of that type to the end of the inning. This gives you the run value of each event. (In practice it makes sense to use the run value from recent seasons, what was going on in 1912 doesn’t really matter.) For example – and going from memory here, so numbers may be off – the run value of a home run is about 1.4, a single is about 0.3, a walk is a little bit less than a single. That is, in recent years, from the time that a player hit a home run (including the home run itself) to the end of the inning, on average a team will score 1.4 runs. So the first step in the calculation is to take all of the things that a player does on offense, all the singles that he hit, all the doubles that he hit, and so on, and multiply them by their run values. (Of course the run values for bad things – like striking out – will be negative.) Add that all up.
The next step is to convert the player’s offensive run value into wins. This is pretty straight forward. As a rule of thumb, if the number of runs you score, plus the number of runs you prevent, equals 10, you should expect your team to win an additional game. (There will be some year-to-year variation on this, but 10 is a nice rule of thumb.) So divide the player’s run value by 10 to get his expected offensive win value.
Then we need to look at defense. There are a few ways to do this, but we’ll keep it simple. (Differences in WAR that you find on Fangraphs or Baseball-Reference usually come down to the fact that they calculate defense differently.) Cut the field up into zones – basically think of them as concentric circles radiating out from where fielders usually stand. The further the circle is from the player’s usual position, the more credit that they get for making a play in that zone. (Roughly speaking at least; how one fielder interacts with another’s chances changes this some, and shifts throw things off a bit, but it’ll do for our summary.) If a ball is hit right toward you, since almost anyone could catch that ball, you don’t get much credit for making that play. But you lose points if you boot the ball. On the other hand, if you have to go a really long way to make a play, you’ll get lots of credit for making the play, and lose basically nothing if you don’t manage to make it (since we shouldn’t have expected you to make it anyway).
Each ball that gets hit has an expected run value attached to it – a screaming line drive down the first base line has a high run value, since it’s probably a double or a triple, whereas a soft grounder right to the shortstop has a very low run value because it’s almost always an out, and even when it’s not it’s just a single. So take the run value of each ball that gets hit, note whether the player in question made a play on it or not, and then multiple the run value of the ball by the weight attached to the zone that it was hit to. So if a center fielder makes a diving catch deep in the hole, you take the run value of a double and multiply it by something close to 1, since it almost certainly would have been a double if the play wasn’t made, and since it was a hard play to make, we shouldn’t just expect any old center fielder to make it. Or say that a shortstop fails to make a play on a ball hit up the middle. Take the run value of a single (0.3), since the play wasn’t made, reverse the sign (-0.3), and then multiply that value by the weight of the zone, say 0.5, since we’d only expect a shortstop to make that play half the time. Failing to make that play then costs the shortstop 0.15 runs. Once you’ve done all that, add up the results, and again divide by 10 to get the player’s expected defensive wins.
Now, WAR doesn’t measure wins, it measures wins _above replacement_, so we need to know what replacement level is. Essentially, replacement level is supposed to be the amount of performance – runs scored and prevented – that you could get from freely available talent. The kind of guy that almost every team has stashed in AAA. We’re not talking top prospects here, just career minor leaguers who can get called up in a pinch. Figure out how many runs such a player should be expected to score and how many he should be expected to prevent. Convert that to wins (divide by 10). Then subtract that amount from our target player’s win totals.
Finally, we need a positional adjustment. Good hitting shortstops and catchers are hard to find. Good hitting first basemen are easier to find. We need to adjust for that. So take the number of games a player plays at each position and multiple that by the per-game adjustment for each of those positions. A full-time center fielder gets a bonus of about 0.3 WAR from this, a full-time first baseman gets a penalty of about 0.8 WAR. Catchers (IIRC) get the highest bonus, DHs get the largest penalty. Once you’ve made the positional adjustment you have the player’s WAR.
The run values and the weights are all determined historically, it’s important to see that they are just records of what has happened in the past. Replacement level is also an empirical matter, as there are plenty of replacement level players. Guys who are on the border of being MLB bench players but also spend a lot of time in the minors. Replacement level is just what those guys, on average, can give you. There’s nothing theoretical about these components of WAR, they’re all determined by what happened on the baseball diamond.
It is important, however, to recognize that statistics are tools, and, as with any tool, that are some jobs that WAR is good for and others that it isn’t. Don’t try to turn a screw with a hammer, and don’t try to pound a nail with a screwdriver. What WAR tells you is how many extra games an arbitrary team would be expected to win or to lose if they added the player in question. This is a useful thing to know for some purposes and not for others. It makes sense, for instance, when talking about the hall of fame, or when trying to evaluate a player’s career as a whole. But it’s not so helpful if you are a team’s general manager and are trying to decide whether to sign a player. In that case you don’t care about wins above a generic AAAA player, you care about wins above the actual guy whose job this player would take. But the general strategy can still be useful. If you are considering signing a third baseman, but you’ve already got a good one, then you can keep the win calculations and adjust replacement level up to reflect the good incumbent. Alternatively if you’ve got a black hole at third base and no help on the farm team, you can keep the outlines of the calculation but drop replacement level down.
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