Last week, Russell Carleton wrote a thought-provoking article for Baseball Prospectus about the automatic ball-strike system, which will be creeping into the major league level during spring training in just a few months. What I found really fascinating was the particular distinction Carleton drew between the current zone and the robot one. “I think that there is a human element that we need to consider when talking about the automated strike zone,” Carleton wrote. “It’s just not that human element. It’s the one no one wants to talk about.” The element he was referring to was probability.
Assuming it’s functioning properly, the robot zone is perfectly black and white. Every pitch either touches the strike zone or doesn’t and that’s that. On the other hand, humans are imperfect, so the zone they call features plenty of gray. Pick any spot in or near the strike zone, and you can look up the probability that it will be called a ball or a strike. In the moment, for any one batter and pitcher, that’s completely unfair; a robot would know with 100% certainty whether the pitch should have been called a strike or a ball, whereas roughly 7% of the time, the human umpire will make the wrong call, screwing somebody over in the process. But over the course of a long season, things tend to balance out, and you can construct some reasonable arguments in favor of the current, unintentionally probabilistic approach.
If you’re familiar with the work of Umpire Scorecards, you’re likely used to the idea of a probability-based strike zone already. Umpire Scorecards grades umpires not simply by how well they adhere to the rulebook zone, but by how much better or worse than average they are at adhering to it. In order to make that judgement, it’s necessary to consider sorts of factors that might affect the call of an average umpire: location, speed, break, handedness, count, and so on. “The reality is that there’s the ‘definitely a strike’ zone,” Carleton wrote last week. “There’s the ‘definitely not a strike’ zone. And there’s the fuzzy zone. There are different rules in the fuzzy zone. Taking away the fuzzy zone and forcing it into the yes/no zone is going to have some very unpredictable consequences.” Take the count as an example. As you surely know, umpires see their zones tighten up with two strikes and loosen up with three balls. If that tendency disappeared, walk and strikeout rates would likely go up. Do we want that?
Because an ever-increasing number of umpires rose through the ranks under a system that rewards them for adhering to the Statcast zone, accuracy has been rising and rising. Another way to phrase it is that humans have been successfully trained to perform more and more like robots. We’ve already seen some of the consequences Carleton mentioned. Accuracy has increased faster for pitches inside the zone than outside the zone, which has resulted in more called strikes and depressed offense. Another effect is that umpires have been calling more strikes at the bottom of the zone – or if you prefer, catchers have been stealing more strikes at the bottom of the zone. Today, we’re particularly interested in the top and bottom, because when I was reading Carleton’s article, one thing kept popping into my mind. Here’s a diagram of the strike zone pulled straight from the MLB rulebook. Whoever posed for this thing has some serious cheekbones. Seriously, this dude is absolutely smoldering:
The rulebook zone starts at the midpoint between the shoulders and the top of the pants, which is why each time a new batter comes to the plate, the umpire stops the game, pulls out their trusty tape measure, and calculates that exact spot. Wait, sorry, the umpire doesn’t do that. As a result, the top and bottom of the zone are blurrier than the sides. Players on the extremes of the height spectrum often bear the brunt of that. If you look at the players who led the league in called strikes above the zone in 2024, you’ll find that five of the top eight – Sal Frelick, Corbin Carroll, Seiya Suzuki, Josh Smith, and Jose Altuve – stand 5-foot-10 or shorter. Likewise, the umpire never squats down to make sure they register the exact height of the hollow beneath the kneecap, so if you look for players who got the the most called strikes below the zone, you’ll find that four of the top 11 – Michael Toglia, Oneil Cruz, Elly De La Cruz, and Aaron Judge – stand 6-foot-5 or taller. It’s not as dramatic a percentage as the short players at the bottom of the zone, but the trend is clear and it’s understandable. The torso midpoint and the knee hollow are just guidelines based on dubious anatomical landmarks – it might help to think of them the way a hitting coach thinks of instructional cues: You don’t actually want the batter to hit a low line drive to the opposite field every single time, but focusing on that goal can help them keep their swing right – and they’re every bit as fuzzy as the calls of the umpires tasked with abiding by them.
The ABS zone eschews body parts. It knows nothing of knees and shoulders, and if a batter were to sag their pants extremely low, it wouldn’t care that the midpoint between their top and the shoulders had just shifted down dramatically, reducing the size of the strike zone. (To be clear, a human umpire wouldn’t adjust the strike zone based on saggy pants either, but according to the letter of the law, they should.) ABS determines the top and bottom of the zone by using a percentage of the batter’s height, which is why hundreds of minor leaguers suddenly shrank last fall. The top of the zone is 53.5% of the batter’s height, while the bottom is 27%. If you’re keeping score at home, that means that the total height of the strike zone is 26.5% of the batter’s height. If that strikes you as a small percentage, you’re not wrong. I ran some quick measurements on our rulebook strike zone friend in the diagram above. His strike zone represents a whopping 41% of his crouched height. As it turns out, that’s because the proportions of the diagram are a bit off. If you measure everything based on the width of the strike zone in the diagram, 17 inches, you’ll discover that our friendly guy only stands 4-foot-5. Once again, this is the actual diagram that describes the strike zone in the official Major League Baseball rulebook! The height of the zone in the diagram works out to 22 inches. In order for it to be accurate according to the ABS zone – in which the height of the zone represents 26.5% of the batter’s total height – the batter would need to be 6-foot-9. When he stood up out of his crouch, our tiny batter would somehow need to find an extra an extra 27 inches of height!
I understand that umpires are being judged based on the Statcast zone, and that they’re also working off decades of experience. It’s not as if they’re pulling this diagram out of their pockets as a refresher between pitches. And maybe the foreshortening here is just a little bit dramatic. But also, uh, it may be time to update the officially sanctioned illustration of the zone that they see in their rulebooks.
All of this led me to one question: How much bigger is the strike zone for a tall player than a short player? Because ABS uses simple percentages based on the batter’s height, we can determine that exactly. Here’s the thing about the strike zone, though. The effective size of the strike zone is a lot bigger than its actual size. If one electron on the baseball’s outer edge passes through the zone, then the pitch counts as a strike. The zone that pitchers aim for and batters protect isn’t just 17 inches wide. It’s 17 inches wide plus the diameter of a baseball on either side. Regulation balls are between 2.865 to 2.944 inches in diameter, and we’re going to make our calculations using the bigger size, simply because, once again, we care about the effective zone that the batter actually has to protect. In all, that means the zone is just a hair under 22.889 inches wide for everyone.
The same goes for the height of the zone. Because this is the variable part, let’s just start with an average, 6-foot-2 major leaguer. The top of the zone will be 53.5% of their 74-inch height, which is to say 39.590 inches. Add the height of the ball and that brings us to 42.534 inches. For reference, a standard kitchen counter is 36 inches tall, so put a bobblehead on your counter and you’ve got the top of the zone for an average player. The bottom of the zone is 27% of their height, and once we factor in the diameter of the baseball, that works out to 17.036 inches off the ground. The average newborn baby is 19 to 20 inches tall, so for reference, head to the nursery of your local hospital, borrow the shortest baby you can find, and politely ask them to stand up. That’s the bottom of the average player’s zone.
To get the total area of the zone, we’re back in geometry class: Simply multiply the base times the height. Well, actually, that’s not quite true in this case. We need to remove some area around the corners because of the roundness of the baseball. Let me show you what I mean. Here’s the top-left corner of the zone:
There are three baseballs here. The one on the bottom and the one on the right are just barely touching the rulebook strike zone, so they’re definitely strikes. But what about the one on the top left? The edges of the ball, both on the bottom and on the right side, are within the parameters of the strike zone, but because it doesn’t have corners, the ball isn’t actually touching the zone. I don’t know how the Hawk-Eye system works, but I have to assume that it’s prepared for such a scenario. Right? Maybe? Even a perfect rulebook strike zone needs to have curved corners to account for this. I can’t tell you the exact area that we need to subtract from each corner of the zone because I have forgotten approximately 100% of the trigonometry I’ve ever learned. However, I used Photoshop to cheat and get an approximate measurement. I simply threw a whole bunch of baseballs on the same diagram, all of them touching the exact corner of the zone, and then measured the area in pink relative to the size of the ball.
[Update: Reader Joe Wilkey pointed out in the comments that the solution to this corner conundrum is actually very simple geometry. For each corner, you take the area of a square whose sides are the same diameter as the baseball (8.670 inches), then you subtract from it a quarter of the area of a circle whose radius is the diameter of a baseball (6.809 inches). The diagram below should help explain how that works. That means that we’ll subtract 1.860 inches per corner, or 7.442 inches in total. The following numbers have been updated to account for that figure.]
With that last puzzle piece in place, we can calculate the exact size of each player’s strike zone. The formula looks like this:
Area of Strike Zone = (((Width of Plate + (Width of Baseball x 2)) x (53.5% of Height – 27% of Height + (Width of Baseball x 2))) – (4 x ((Width of Baseball x Width of Baseball) – (pi x Width of Baseball x Width of Baseball ÷ 4)))
If all those parentheses make you want to die, we can hop into algebra and simplify the formula so it looks like this:
Area of Strike Zone = (22.9 x (26.5% of Height + 5.9)) – 7.4
Now that our formula is settled, let’s see how much of the strike zone different players actually have to cover.
Strike Zone Area Based on Height
Let’s go to everyone’s favorite odd couple. Aaron Judge’s strike zone is 3.45 inches taller than Jose Altuve’s, and its total area is a whopping 78.9 square inches larger. To put that in context, a marbled composition notebook, the kind you used to use in school, has a total area of 70.7 inches. That’s a pretty significant extra amount to cover, and don’t even get me started on the difference between Sean Hjelle’s zone and Shakira’s. If the 5-foot-4 Wee Willie Keeler were to come back and play as a zombie batter today, his strike zone would be almost perfectly square. For anyone shorter, the zone would be wider than it is tall.
Maybe even more interesting are the columns for the top and bottom. Judge’s zone starts seven inches above Altuve’s, but it ends just 3.5 inches below it. That’s just a result of using a percentage as the determining factor. It makes all the sense in the world to do so, but it’s likely the reason that list of players who get lots of unjust called strikes at the top of the zone is more densely packed with short players. The knees of short and tall players are much closer in height than their shoulders. When taking the height of the batter into account, umpires should be adjusting more at the top of the zone than the bottom, but clearly, that’s not so easy to do.
As for whether or not all of this is fair – bigger players having so much more zone to worry about than smaller players – my answer is a firm maybe. In absolute terms, Oneil Cruz has a much bigger strike zone to cover than Corbin Carroll, which is patently unfair. However, proportionally speaking, he doesn’t have to reach any higher or lower than Carroll does to get to the top or the bottom of the zone. The angles are exactly the same. Moreover, if we keep analyzing things proportionally, it’s clear that the strike zone is much narrower for him. Because Cruz’s larger height leaves him with longer arms and a longer torso to lean with, Carroll has to reach for an outside pitch in a way that Cruz doesn’t. The stills below are both taken from hard-hit balls on pitches that hit the outside corner.
Carroll’s whole swing is affected by the need to reach out for the ball, but look how much more upright Cruz is on the left. Even on the outside corner, the pitch is in his wheelhouse and he’s able to pull it approximately 9,000 feet. I’d guess that more than offsets the extra 54.6 inches of zone that Cruz has to cover. Even if we use an ABS system to implement a perfect strike zone, we still can’t make it perfectly fair.
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