The Size of the Strike Zone by Count
I recently became fascinated with the strike zone and its effective boundaries. The strike zone laid out in baseball’s rule book is simple; it extends a total of 17 inches across the width of home plate, between the hitter’s knee and midsection and covering the entire depth of the plate. The strike zone as it actually gets called by umpires is complex. It shifts quite significantly depending on the handedness of the hitter for one.
That’s not the whole story though, not even close. The dexterity of the hitter isn’t the only significant variable in how the strike zone is called. Compare the following two strike zone heat maps, fitted from 2012 data on called pitches to right-handed hitters.
The strike zone in 3-0 counts (sample size (n): 3,665 called pitches) has an area of 3.73 square feet (sq. ft.) while the strike zone in 0-2 counts (n: 12,339) shrinks down to 2.39 sq. ft., a size about 64% as big. Interestingly, the zone does not appear to constrict itself equally, but rather by getting shorter (vertically) slightly more than it gets skinnier (horizontally). Look at this possibly overcrowded contour overlay.
I super imposed both the 50% threshold and the 75% threshold to show just how much it shrinks. Coincidentally, the 75% level in 3-0 counts is close to, but notably still larger than, the 50% level in 0-2 counts. That means that a pitch that would be a 50/50 toss-up when thrown in a 0-2 count would be called a strike about 85% of the time if the count had been 3-0 instead.
Strike zones in 3-0 counts are called over a larger area than in 0-2 counts. That is neither new nor surprising information. What I wanted to do was get more specific though and so I broke apart the 2012 data sample by both hitter side and count, fitted strike zones to each of the 24 categories and then computed the area of the zone, in square feet, created by the 50% threshold.
| Count | LH SZ Size | RH SZ Size |
|---|---|---|
| 0-0 | 3.40 | 3.49 |
| 0-1 | 2.75 | 2.89 |
| 0-2 | 2.38 | 2.39 |
| 1-0 | 3.43 | 3.52 |
| 1-1 | 3.00 | 3.07 |
| 1-2 | 2.53 | 2.43 |
| 2-0 | 3.55 | 3.63 |
| 2-1 | 3.10 | 3.21 |
| 2-2 | 2.92 | 2.88 |
| 3-0 | 3.70 | 3.73 |
| 3-1 | 3.22 | 3.36 |
| 3-2 | 2.87 | 2.97 |
| x-2 | 2.54 | 2.44 |
| 3-x | 3.33 | 3.45 |
| Any | 3.27 | 3.37 |
There were two points I immediately took away from the list. First is that the right-handed strike zone is almost uniformly larger than the left-handed one. There were only two counts where left-handed hitters faced a larger 50% strike zone: 1-2 and 2-2. However, with 0-2 counts were nearly identical, an interesting but unexplained grouping emerged; counts with two strikes saw a much tighter zone called on right-handed hitters than to their lefty counterparts.
Secondly, I expected that the zone in 0-0 counts would be fairly neutral compared to the other counts. Instead, it was firmly in the pitcher’s favor. Ordering the counts by their average size places 3-0 at the largest, no surprise, followed by 2-, 1-0 and then 0-0. Counts at 3-1 actually showed a smaller called zone than first pitches.
Matthew Carruth is a software engineer who has been fascinated with baseball statistics since age five. When not dissecting baseball, he is watching hockey or playing soccer.
Fantastic stuff, thanks!
Awesome visual. It might be good to add p-values for the differences in size using a ChiSq
I hate that this is even a thing. I wish there was more pressure on the Umpires to improve their performance instead of cherishing their inconsistency as a charming tradition.
THIS!!!! When oh WHEN are we going to automate this “feature” of umpiring “ability”.
But human factors!!!
I wish they would just let Questec call balls and strikes.
Amen, bro’. I can’t think of any other job outside of politics where this much incompetence is tolerated.
NBA official.
Me either.
I think I agree with automating the strike/ball calls, but think how much it would change the game–at least initially. Batters and pitchers are already conditioned to these changing strike zones as well as the LH vs. RH strike zones and changes from where the batter stands…etc. I would predict that some players would benefit much more than others by a true strike zone–and the results of the change would be interesting and unpredictable.
There’ll be an adjustment period to be sure, but I bet both pitchers and hitters would love having a consistent strike zone.
I imagine that it would result is many more walks and strikeouts, particularly as batters adjust, and a aesthetically less pleasing product overall as fewer PAs are decided by balls in play.
It should have an interesting impact on pitch values, since the de facto strike zone is wider and squatter than the de jure one. I think we’d see the curveball gain effectiveness while the slider lost some, while fastball velocity might gain even more value as it becomes harder to shave the outside of the zone with middling velocity.
MANY more walks and strikeouts? I don’t see why that would be the long term outcome; after hitters and pitchers have adjusted it should feel like watching the same game.
Well, I suppose that it would depend on your subjective definition of “many,” but yes. It’s not just a matter of batter and pitchers adjusting, it’s a matter of the very obvious evidence above that the umpire fudge the zone quite a bit to extend 3-0/0-2 counts. With that gone, more Ks and BB are inevitable.
I think this data suggests that a consistent strike zone would force pitchers to throw more hittable pitches in 3-0 counts, which would limit walks. They would also be able to throw less hittable pitches in 0-2 counts, so I agree that that should lead to more strikeouts.
Robo. Ump. Now!
1)This could be the result of the league desiring hitters to put the ball in play, so the umpires are instructed accordingly.
2)Most umpires are right handed and see the left pitches better.
3)These numbers could also be skewed a bit because good pitchers have more control than bad pitchers. Good pitchers are more likely to get the benefit of the doubt and are less likely to see 3-1 counts.
4)Also bad catchers at pitch framing are skewing the results as well because the top Aces of baseball rarely pitch to bad pitch framers (look it up yourself). Accordingly, bad pitchers get most of the bad calls from bad pitch framing.
All the more reason to allow computers, and not umpires, call the strikezone. A strike is a strike is a strike. Pitch framing, poor umps, vengeful umps….. none of it should alter the strikezone, which has a set definition.
Tommy Lasorda: “If you’re going to do that, why don’t we just get robots and let them play the game? If you don’t need umpires out there, and you can put robots out there, then why do we need ballplayers?”
Um…because the umpires are not playing the game. A great ump is one you don’t even notice. A robot should be very good at this. A ballplayer plays the game. Playing and officiating are completely different things.
Because people don’t pay to watch umpires make calls, they play to see players play the game. Umpire mistakes take away from the fairness of the game, player mistakes do not.
This is really fantastic stuff. Any chance you could post a LH-vs.-RH breakdown by count for the 75% thresholds, too?
I think it’d be fascinating to see a strike zone chart by pitch type, as well (I know, it’s easy to say this when someone else is crunching the numbers). I feel like there are tons and tons of curveballs that drop into the top edge of the strike zone that never get called by umps. But maybe that’s a corollary to the vertical shrinkage of the zone in x-1 and x-2 counts (assuming that my intuition is correct and that curveballs tend to be thrown more with 1 or 2 strikes than with zero strikes on hitters).
You can do it yoursel for each umpire:
http://www.baseballheatmaps.com/graph/umpire.php
Let’s take a step back and recognize that you have highly biased samples. You forget that these pitches only get counted if the batter chooses to not swing. Batters will swing at a lot more pitches on the corners in an 0-2 count than they will in a 3-0 count. The pitches they swing at are more frequently the easy-to-recognize strikes. The tricky pitches will more frequently go by without a swing, and thus make it into your sample. It seems quite likely that this sample bias explains the “changing” strike zone rather than a conscious decision by the umpires. Rather, in 0-2 counts umpires have a group of pitches that are much more challenging to categorize and we should expect a much higher error rate.
“we should expect a much higher error rate.”… YES!
Which is why we should not be asking human beings to do a job that computers/machines can do much more accurately and much more consistently.
I don’t buy it. I think these heat maps demonstrate that for a given location, the same pitch is more likely to be called a strike in a 3-0 count than an 0-2 count. It’s not “tricky” pitches on the corner vs pitches down the middle, it’s corner vs. corner.
Interesting theory, but I don’t buy it either. This data just supports what everyone pretty much knows already from watching the games. Just watching a game, you see pitches all the time at 0-2 that you are sure would have been a strike at 0-0 or 3-0, which are called balls.
They may be in the same location, but in a 3-0 count, those pitches are a lot more likely to be fastballs, and in an 0-2 count more likely to be breaking balls.
Sorry SG, but your argument makes no sense at all. It would imply that batters down 0-2 only swing at borderline pitches and always let center zone strikes go by.
No, it means that if a batter is letting a pitch go by 0-2, it’s more likely to be one that doesn’t look like a strike, which could be a breaking ball on the corner. Some batters take 3-0 almost routinely, and those pitches are often fastballs.
Those 3-0 fastballs should be easy to call, which doesn’t explain why the zone is so large.
So why has it been that the error has consistently and to an extreme degree always gone in the same direction?? If it is a matter of mere difficulty, we would expect to see an expansion of the zone just as often as we see a diminution, but that is in fact never what we see or have seen.
Exactly correct. Sample selection bias all the way.
To me this demonstrates that umpires are affected by their own expectations of pitch location and maybe even by the batter.
Own expectations: Counts where you would expect the pitcher to throw a strike, the zone grows. Potentially illustrating the umpire look in the zone- expecting the pitch to be there.
Batter impact: In 0-2 counts, umpires would probably not expect a pitch to be anywhere near the middle of the plate- combine that with the hitter not swinging and the umpire may be less inclined to call boarder line pitches.
What this clearly and unmistakably demonstrates is that humans are unreliable when it comes to accurately and repeatedly calling balls and strikes.
Plain and simple.
I’m not convinced that umpires can’t be trained (or threatened into) doing a better job. MLB just doesn’t seem to care.
I don’t want to get rid of umpires, or threaten them, or retrain them. I just want technology to do the part that human beings are not well designed to do! Human beings are not good at repetitive accuracy… especially tracking small high speed objects and precisely locating them in a theoretical three dimensional space.
I am in no way damning umpires. I think they do as good a job as human beings can do at calling balls and strikes.
I think they’d be even better if technology took over balls and strikes, and freed the umps to more closely watch other aspects of the game.
How do we use technology to locate a ball in a theoretical three dimensional space?
Also, baseball is played all over the world and many levels. Instigating technology like this in MLB would alter the game too much from the way it is played everywhere else.
Baseball works just fine, as long as umpires can remain consistent during the course of a game. Reward umpires that can do that, and get rid of the ones who can not.
legendaryan…
ask the people that do the systems for professional cricket… or tennis.
And the “non-major leagues” in those sports (without the technology) continue to get along just fine.
Not implementing it everywhere is no excuse for not implementing it at the highest level when that capability exists… as it surely does.
And why weed out those umps that aren’t as good… when you can replace that piece of the task for all umpires with something that is essentially “perfect”… ALL the time, EVERY time ?
@Dave S
I don’t care about Umpires at all; if they show any resistance to an increased role for technology to assist them in calling the game correctly I would rather they be fired. Officials are ideally invisible; I’m often amazed at how little I see NFL refs during play. Every once in a while I’ll notice that a ref is set up in the middle of the field amongst the players, but I almost never notice them on replays. NFL refs go largely unnoticed until they need to make a call, which is exactly how it should be.
Those heat maps should be exhibit 1 for ending the misuse of human beings for calling balls and strikes.
Matthew, I cannot thank you enough for posting them.
Bookmarked.
I am curious if the handedness of the pitcher shows any unusual variations.
Fantastic research, really enjoyed the article.
For some research background into this topic, you can start here:
http://www.hardballtimes.com/main/article/the-compassionate-umpire/
bookmarked that one too Mr. Studeman, thanks!
I came in here to post that, too late!
I wonder if there is a bias, on the umpire’s side, to be more likely to call a ball in a 3-1 count for instance as opposed to an 0-2 count?
What I am trying to say is there a tendency by umpires to call a ball or strike, during any particular selected count, simply because it “is” that count?
Someone may have already looked at this but it would be interesting to know if a called ball was more likely to happen after an 0-2 count versus a 3-1 count? How much does human emotion and expectation enter into the equation in those situations that might bias the call?
Interesting work Matt!
3-1 is in the table.
What if we let the home plate ump wear some sort of glasses that projected a strike zone box in his field of vision? Seems like it would be a helpful guideline.
Or just have him wear an earpiece that says ball or strike based on a computer, but he still gets to do his silly routine. Don’t even tell people you’re doing it. Everyone wins.
Wow. This is a great idea. I’ll admit that I enjoy the aesthetic of an umpire calling a batter on a strike out, and I think we can all admit those tense moments in big games would be a little less tense with those umpires behind the plate. This idea is a win-win.
Argh. Without* those umpires behind the plate.
Tango once proposed grouping of plate counts as:
3-0, 3-1, 3-2 // 2-0, 1-0, 0-0 // 2-1, 1-1, 0-1// 2-2, 1-2, 0-2
(“3-ball counts first, and then you clump by the strike counts”)
Or alternatively, he suggested grouping by balls minus strikes. I think both of these methods might help the data in your table stand out more.
Has anyone done a breakdown like this comparing established pitchers (Rivera, Halladay) to young pitchers?
Watching a rookie on your favorite team seemingly get squeezed for being a rookie is incredibly frustrating. Then watching the situation worsen if the pitcher reacts, and the umpire perceives it as being ‘shown up’ and makes the strike zone even smaller, forcing the kid to throw meatballs, which causes him to get rocked, which makes oh god I’m getting to angry to talk about this just please automate the strike zone.
That’s it. You’re tossed.
I seems like the biggest factor in determining size of the strike zone is number of strikes. More balls expand it slightly but more strikes shrink it dramatically.
I think that one day people will look back on this era of baseball as the “fuzzy strike zone” era… the way we look back at the “deadball” era.
The two things I take from this:
1. Conventional wisdom dictates that the batter “expand the strike zone” in a 2-strike count to avoid a backwards K. This suggests quite the opposite.
2. Anecdotally, it seems that LHB post better numbers than RHB. This suggests that some of that may be due to the fact that they have smaller strike zones.
The batter may still expand the strikezone in that he is more prone to swing at a borderline pitch for fear that it may strike him out if he watches it go by, but umps seem to be giving the batter quite the advantage in an 0-2 count.
I have a psychological explanation for this, but mine is different than some that others have suggested. I think umpires, in the case of the borderline pitch, rather than make the “expected” call, they are making the “uncontroversial call.”
If it is 0-2 and a pitch comes in on the border of the strikezone, the ump has two choices. He can make the strong, controversial call of strike or the safe, non-controversial call of ball. If he calls strike, he’s questioning the batter’s judgment of the pitch, he’s ending the at bat, and possibly making an impact of the outcome of the game. He will likely earn some response from the batter and the batter’s manager. Or, he could just call a ball. The pitcher may shake his head, but the outcome of the game will probably be almost the same. The pitcher still has three more chances to get that third strike. In fact, if he hits the same spot, the umpire, still guilty about not giving him that call, ring up the batter on essentially the same pitch. This way, we have the same result as if he had called a strike, but this time he gave the batter a chance, giving him less reason to complain.
My conclusion: umpires are cowards.
Yep, there is a fundamental bias towards letting the at-bat continue, thereby decreasing the chance the umpire ends the at-bat. I believe this is all the more so when that at-bat matters more (hence why pitchers get routinely rung up on 0-2, and why Drew Storen shouldn’t try to paint the corners when looking for strike 3 in the 9th inning of Game 5).
POUR OUT A LITTLE LIQUOR
ROLL UP THE WINDOW, BLAZE UP SOME INDO
GET TO’ DOWN FOR MY NIGGAZ IN THE PEN, YO
Computers please. And for yardage in football.
How does batter height play into this? The stance should vary to some degree by batter and by stance. Not that I’ve noticed batters crouching more in 0-2 counts, but if they did it would change the zone.
A computer-called strike zone would be a disaster for the pitchers. An unchanging strike zone with no variance depending on the count would greatly favor hitters as they learn the exact limits of their personal swings. Part of baseball is the uncertainty as a hitter where the exact strike zone is at the plate. It’s not codified in the rules like that but has been played since time immemorial in that manner.
I could not disagree with that more completely.
Computer strike zone would be completely adaptable. I believe there would be some initial issues, until you got the strike zone designed to balance pitchers and batters… but once you locked in the proper size of it, you’re golden. Then every batter would get the same exact strike zone, all day, everyday. And so would the pitchers.
As a side comment, I also, would like to see the strike zone be exactly the same for all batters, no matter how tall or short they are.
Tall batters shouldn’t have to cover more strike zone simply because they are tall. Short batters shouldn’t have to cover a smaller zone, simply because they are short. It should be democratic. One guaranteed uniform strike zone for all batters and pitchers.
But that’s getting well ahead of things…
FG, hire this robot to write articles, it will save you loads on writers!
I would love to see an automatic strike zone for reasons already listed, but also for two more:
mlb would have to fess up exactly where the zone ends upstairs, and I want to see how long it takes before mlb starts to change the zone to get more hitting or more pitching.
I think those would both be very good things. And I wouldn’t have a problem with MLB altering the zone to increase offense or defense… as it sees fit. At least it would be known and transparent to everyone. Thats at least fair.
But again, this is getting ahead of things. Would just be happy with a uniform strike zone for now.
How long before conspiracies popped up that Albert Pujols or Giancarlo Stanton or whoever is the superstar of the day gets a tweaked strike zone because he is a superstar?
Immediately
Is there anyone who actually thinks that isn’t already happening?
You cannot be so naive as to think that star players already don’t get preferential treatment in every sport.
One thing to be careful of with this is the total number of called pitches in a given count. In an 0-2 count a batter is much more likely to swing at a pitch anywhere close to the strike zone, thus reducing the number of pitches in the zone that the umpire can call a strike. When using the locally weighted regression technique, as you have, this can cause for an over-fitted zone, and an exaggeration of the decrease in strike-zone size. While some of this is definitely due to the umpire, at least some of it is a result of the location of the pitches that they have an opportunity to call.
I suspect this may be a factor as well, but no idea as to the magnitude.
Actually, doesn’t pitch f/x track location swing or no swing?
Great work. Any chance you could give us some tips on how to do this sort of analysis ourselves? (most importantly, where and how to download the data)
I’d be interested to see how this lays out by pitch type, or break. How much of this variation is due to the count and how much by the break of the ball?
I also wonder where the decide the plane of the strike zone to be. Since the zone actually has a depth, it sort of bothers me to always see it represented as a plane surface. I wonder if umpires change the plane on breaking balls, and maybe we could find a different plane that mirrors the called zone size of fastballs. Ive umpired for years, and this is a great article to me.
How would the strike zone be automatically adjusted from batter to batter?
Thats a pretty trivial problem.
If the cameras can pick up a the rotation, speed, and position of an object in 3D space, do you really think they can’t pick up a guys knees?
Can’t wait for the day that there’s an automated strike zone so that it becomes a new opportunity for cheating. Old cheating: vaseline on the baseball. New cheating: hack the automated strike zone to be “off” by a fraction of an inch.
Also can’t wait for a day that a game is called on account of computer crash.
That said, this is legitimately groundbreaking research here, and some fascinating insights into how much the limits, bias or competence of the umpires may skew some results.
At some point I imagine we can get to some overlays of what the strike zone is by count, what a pitcher throws, what the hitter swings at, what the ump called and what the ‘true’ strike zone is, to get some unmistakeable trend lines in individual games. I.e. we should be able to use this game to look at individual at-bats and determine which ones were so far ‘off’ that it eventually radically altered the rest of the game. We could invalidate individual at-bats, perhaps throw out entire games or series, based on having some hard data that the umpiring was game-changingly terrible.
“Also can’t wait for a day that a game is called on account of computer crash.”
We have imbedded computer systems running pretty much every critical piece of equipment in the world. Its trivial at this point to design simple systems like this and have them run reliably.
You ever see a fetal heart rate monitor crash? I haven’t either.
Couldn’t this be (partially) explained by pitch type?
Maybe umpires have a smaller strike zone for off-speed pitches (more likely to be thrown in 0-2 counts) than for fastballs (more likely to be thrown in 3-0 counts, and probably 0-0 counts as well).
cyberwulf,
Some of the effect does seem to come from pitch type (I have not broken it down visually though). We tend to see fastballs called strikes much more often than other pitches in the same area and in the same given count. This variance is sucked up into the across-count changes. While this won’t change the actual result (strike zone size across count), it could bring into question the causal effect of count on umpire decisions.
From what I have found, it is a combination of the count, pitch type, and a number of other things.
PS-I would like for Matt to do the visual breakdown for sure!
how bout breaking them down by league…id love to see statistical analysis either proving or disproving the theory of the national league’s larger strike zone.
Matthew,
1. You should really be using an interaction of the horizontal and vertical locations in a single smooth. The strike zone is not symmetric as it is shown here. I ran into problems before when I initially proposed GAM instead of loess a while back: http://princeofslides.blogspot.com/2010/12/rethinking-loess-for-binomial-response.html
2. Which visualization package are you using in R to make the heat maps? It looks like “image”, but I find “filled.contour” to be prettier (Dave Allen and others have used this in the past).
3. In one of your older posts at Lookout Landing, it looks like you were having the same problem I had: http://princeofslides.blogspot.com/2011/03/having-problem-with-r-2122-64-bit-and.html. But here it looks like you fixed it (using mgcv now, I assume?).
4. Why not refer people back to previous work by Walsh (THT), Dave Allen (BA), J-Doug (BtB), Weinstock (THT), Mills (PoS), Scorecasting on this? How is this piece adding to that?
http://www.hardballtimes.com/main/article/which-umpire-has-the-largest-strikezone/
http://baseballanalysts.com/archives/2009/04/the_effect_of_t.php
http://www.hardballtimes.com/main/article/the-compassionate-umpire/
http://princeofslides.blogspot.com/2012/03/strike-zone-changes.html
http://www.beyondtheboxscore.com/2010/12/24/1892898/benefit-of-the-doubt-odd-patterns-in-umpire-compensation
I think this is all very nice and great for another audience. But give credit where credit is due?
1. I’ll look at that. Thanks.
2. I don’t like filled.contour because I find the attached key to be distracting, unneeded and takes space away from the main graph.
3. I’m not recalling what you’re referring to exactly.
4. Because I didn’t/haven’t read those.
Sorry, the problem in #3 is the vertical stretching of the oval shape. This is not present here.
Addendum to 4)
I did find some of the basic R code from a post of yours at some point and clearly the look of the images are similar to yours since out of all the R-based heat maps I’ve seen, I liked your visual presentation best. I should have mentioned that, my apologies.
Regarding 1) I don’t see where you correct it in the link you provide? I see you raised the concern, but there’s no discussion of an outcome from that concern.
With 1), do you mean the vertical stretching or the interaction? The vertical stretching was a glitch between a new version of R and the “gam” package, which was fixed by using the “mgcv” package (which works better anyway and cross-validates your smoothing parameter). If you mean the interaction term, I have no gone over this at all on my website, but find that it is more representative of what is going on in the zone (though I don’t know that it would have significant effects on the size of the zone).
Much of my code comes from Dave Allen, but I was mostly talking about the specific problem and methodology. Code is code and that is called open source for a reason. I think everyone (including this) has had additional things to say that are interesting. Just always like to see everything come together and build on the last.
I meant the interaction, as in what exactly you changed in the code to use that instead.
However, I’ve actually not been using the mgcv package, wasn’t aware of it. I’m going to test that out now and see how it looks.
The interaction would simply use:
fit1 <- mgcv(strike_call ~ s(px, py, k=50), data=…)
It gets a bit more into the nitty gritty of the strike calls from what I've seen (evaluates up-and-out, up-and-in, etc.).
It won't change the relative impacts across counts much, nor will it impact the size of the zone much (well, actually the cross-validated smoothing parameter does). However, it might be more accurate from an R^2 point of view, as well as a "correct call" POV for each umpire.