Pitch Type Linear Weights Explained

Yesterday David Appleman announced a new section at FanGraphs showing the Linear Weights Run Value for each pitcher and pitch type. He asked me to write a short explanation of how these values are calculated.

The run value of any event is the change in the expected number of runs scored over the rest of the inning from before and after the event happened. The expected number of runs scored is the average number scored from a given out and base-occupancy state. Let’s take Tuesday’s Oakland at Tampa Bay game as an example. At the top of the 1st inning with zero on and zero out the average team scores 0.55 runs. Orlando Cabrera hit a single off of Jamie Shields. Now with a runner on first and none out the average team scores 0.95 runs. So the run value of the single was 0.55-0.95=0.4.

You can do the same thing taking each pitch as as event rather than the outcome of each at-bat. To do this you need to know the run expectancy from each count, in other words, the average run value of all events from at-bats which pass through a given count. For example the run expectancy of a 3-0 count is 0.2, on average at-bats from that count are worth about half of a single.

Now we can run through Cabrera’s at-bat with Shields as an example of valuing each pitch in an at-bat.

0-0: Run Value 0.00

Pitch 1: Fastball for a ball

1-0: Run Value 0.03

So the value of that first pitch was 0.03 runs. On average the A’s will score 0.03 more runs than before Shields threw the pitch.

1-0: Run Value 0.03

Pitch 2: Fastball for a called strike

1-1: Run Value -0.02

The run value of that fastball was -0.02-0.03 =-0.05.

1-1: Run Value -0.02

Pitch 3: Fastball for a called strike

1-2: Run Value -0.08

The run value of that fastball was -0.06. You can see here that the run value of a strike (or any other event) is count-dependent.

1-2: Run Value -0.08

Pitch 4: Fastball for a ball

2-2: Run Value -0.04

The run value of that fastball was 0.04.

2-2: Run Value -0.04

Pitch 5: Change fouled off

2-2: Run Value -0.04

Since the count did not change the run expectancy did not, so the run value of this changeup was 0.

2-2: Run Value -0.04

Pitch 6: Fastball hit for a single

Runner on first no outs: Run Value 0.4

The run value of this fastball is the change in run expectancy, 0.4-(-0.04) = 0.44.

Shields threw five fastballs valued at 0.03, -0.05,-0.06,0.04 and 0.44. He threw one changeup that had a value of 0.00. These values are the change in run expectancy in the game, so a negative number is a good for the pitcher (fewer runs scored). On the player pages the numbers are flipped so a positive number indicates a good pitch, the number of runs saved by those pitches.

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Dave Allen's other baseball work can be found at Baseball Analysts.

21 Responses to “Pitch Type Linear Weights Explained”

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  1. NadavT says:

    Thanks for the clear explanation!

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  2. Chris says:

    Will these be posted on the leader board so we can see who had the best pitch a certain year?

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    • Dan says:

      A leaderboard would be phenomenal. Sorting by pitch type plus a board of all pitches put together would be such a great tool. Because as of now, these pitch type linear weight stats are great, but a little hard to comprehend, as we have no frame of reference. A leaderboard would let us know what a true “great pitch”, for comparison’s sake.

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  3. Matt Harms says:

    This just opens up a whole new batch of questions now that we’re finally starting to get data on this stuff. This further reinforces how important defense is. And it will be interesting to see how that whole “pitch to contact” theory shakes out with this new data. IE, if your pitching coach wants you to “pitch to contact” (ahem, Dave Duncan), then which of your pitches is statistically best to reduce the run outcome of the play.

    Man, this stuff is fantastic. Thanks again guys, FanGraphs is just blowing my mind right now.

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  4. JIK says:

    Possibly dumb question:

    Does this take batted-balls into account?

    In other words, say it’s a 2-2 count with nobody on base: would a fastball that results in a swinging strike 3, be of equal weight/value as a fastball that results in a flyball out?

    What if there were fewer than 2 outs, and a runner on third?

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    • A fly out and a strikeout in these instances are worth essentially the same amount.

      I’m weighting sac flies differently, but over a full season it’s not going to make much of a difference.

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  5. Matt Harms says:

    One more question, though… is there anyway to show back-to-back pitch combinations? Because data like this–if taken to the extreme–could be a good justification for someone like Verlander to only throw his fastball, or for Santana to only throw his change-up. But these pitches don’t occur in a vaccuum, they’re part of a logical pitching progression where one pitch sets up the next.

    Ultimately, is the data really that statistically relevant?

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  6. Fett42 says:

    Wouldn’t a pitcher’s linear weights value for each of his pitches be expected to change if he mixed up his percentage or pattern of throwing each time regardless of how good the pitch is? Or am I understanding this wrong?

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  7. Xeifrank says:

    I also would ask the same question as Matt does in post #6. Logic tells me that previous pitches influence current pitch. If you threw a fastball every single pitch, it would start to get hit hard. But if you mix your pitches up, the fastball is more effective.

    Secondly, is this really “linear weights”, or more along the lines of Win Probability Added (WPA)? Seems like the latter to me. When I think of linear weights, I think of a y=mx+b.

    Lastly, is it correct to give such heavy weight to the last pitch of the at-bat? That pitch that gives up a hit or makes an out? If so, wouldn’t you want to use only fielding independent resultants (hr, bb, k)?

    Very intersting stuff, but needs more discussion.

    vr, Xei

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    • Well, the whole thing uses linear weights. I guess the calculation is a bit different because it takes the difference in those weights, but at least in my mind it’s really the same thing. It is not like WPA because it doesn’t take anything into account except for the ball/strike count. A strike in an 0-1 count is always worth the same. A ball in an 0-1 count is also always worth the same. And so on and so on.

      The last pitch of the at-bat is not necessarily that heavily weighted because it depends on the count. I’m perfectly happy taking away more than 1.4 runs for a home run if he’s up 0-2 if he throws a really crummy fastball and if he’s down 3-0 and throws a fastball for a home run, I’m taking away less than 1.4 for that particular pitch.

      We’re keeping track of the fielding independent results too, so if we want to see what the results would look like later on with singles, doubles, and triples removed, that is also a possibility.

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      • Sky says:

        You’re using run expectancy based base-out state, right? So a strikeout to start an inning has a different pitch value than a strikeout with two runners on and two outs, right? That’s what Dave Allen’s article implies, at least. If not, are you just using typical context-neutral linear weights for all events and a common pitch-count chart for every situation?

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      • We’re using context neutral linear weights for each event and also a common pitch-count chart for every situation.

        The base runners/out situation is never taken into consideration.

        I think where the confusion arises is that the weights I’m using are derived by using run expectancy matrix on play-by-play data from 2005-2008 and then taking the average of worth of what a plate appearance through an 0-1 count would be, etc…

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  8. Curtis says:

    You guys continue to surpass even my loftiest of expectations.

    Is there a plan to do the same for hitters? I’d love to see the tale of the tape on Albert Pujols vs. fastballs or Ryan Howard vs. sliders.

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  9. Gdawg says:

    This stuff is great, but I have a question about a possible new feature.

    With all this information on how a pitcher does for each pitch, are there plans to incorporate this type of data for hitters? This way, you could see who hits fastballs best vs. who the best curveball hitter is or w/e you might be interested in. Or also some specific plate discipline data on specific pitches would be cool too (like swing and miss%, contact%, and all that good stuff).

    Anyways, thanks for the explanation and keep up the great work guys.

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  10. Jay says:

    Hey guys i’m understanding you until the single in the at bat.

    If i’m correct, you are using this chart: http://baseballanalysts.com/archives/2008/02/writing_about_t.php

    Or do you have an updated one you are using?

    Because based on the above chart, the run value of a single would be 0.5 and not 0.4.

    Also, i’m guessing the numbers are regardless of base state? So a single is worth 0.4 regardless if it’s with the bases loaded or no one on?

    Great stuff as always.

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  11. Dave Allen says:


    Good point. The value of a single with no outs and no one on is 0.4, but the average value of a single (over all base and out states) is about 0.5.

    So the value of the pitch given the out and base state was 0.44, as I showed in the demonstration. But in point of fact, what I do in my articles, and what I think David has done here, is ignore the base and out state and take the average value of the single (or an out, home run, etc). So the value he would attach to the pitch for the Shields’ fastball would be 0.54 as you suggest.

    I hope that clears it up.

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  12. Jay says:

    Thanks Dave. That makes sense and just wanted to make sure I was understanding it correctly.

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  13. bhinnant says:

    I find this impractical except on a macro theoretical basis. To reach a true linear weighting, you would need to factor in the specifics of a pitcher, the batter being faced, and the defensive ability of the team, plus men on base, plus outs. Otherwise, you equate Rivera in the ninth against Eckstein with no one on and 2 outs as the same as Bruce Chen against Pujols in any inning with bases loaded and no outs. I suspect the real results are higher than the theoretical projections, but you’d need some serious programming to make it work. On the other hand, at one time only wins was a significant stat, so you have to start somewhere.

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  14. Ethan says:

    Would a pitcher, pitching optimally, have the same linear weights value for each pitch type (because they’d be mixing their pitches to make it so)? In that case, this doesn’t really tell us how good in an objective sense any given pitch was (for that we’d probably use PitchFX data), but rather it tells us how good the pitcher is overall and how well he mixes his pitches.

    Or, am I misunderstanding completely?

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  15. Linds0r says:

    Pitch 5 should change the value of the subsequent pitches, since the linear weight should change not only on count, but also on pitch number. I would image the run expectancy of the first pitch of the at bat is vastly different than the 10th. True? If so, the run expectancy must also take into consideration the number of pitches thrown in the at bat. The same should be said of the first pitch to a full count and the fifth pitch to a full count.

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  16. Amy says:

    Reader’s New Year will be greater with this idea!

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