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.

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.

For the 21st consecutive season, the ZiPS projection system is unleashing a full set of prognostications. For more information on the ZiPS projections, please consult this year’s introduction and MLB’s glossary entry. The team order is selected by lot, and the next team up is the San Francisco Giants.

Batters

Well, the Giants have solved at least one problem: finding another Buster Posey. Not in the form of Joey Bart, as was the original intention for a few years, but rather in Patrick Bailey. Now, Bailey isn’t quite peak Posey, an unreasonable expectation to have of anyone, but he has become a legitimate star behind the plate. Bailey also doesn’t exhibit the same distribution of talent as Posey did, as Bailey is arguably the most valuable defensive player in baseball with just enough bat to make that drool-worthy. To make a reference that’s even too old for me, Bailey’s a bit like a reboot of The Six Million Dollar Man in which they had the technology to build the cyber-Platonic ideal of Austin Hedges.

Bailey isn’t the only high spot in the lineup. Matt Chapman, who it seems the projections were not too high on in 2024 after all, should have at least a few good years left in him, and the Giants are generally at least average-ish elsewhere. ZiPS is higher than the other systems on Tyler Fitzgerald, and both the computer and I are hoping to see what Jung Hoo Lee can do after injuries cost him the opportunity to make good on what was shaping up to be a middling-at-best debut in the US. Read the rest of this entry »

It’s a bit of a cliché that all-time great basketball players like to add an element to their game every offseason. You come back from summer vacation and Tim Duncan has a new post move or LeBron’s shooting three-pointers now. This truism informs something I like to ask baseball players during breakout seasons: Do you have an eye on the next thing you want to learn? Sometimes you get some banality about being more consistent, or just an outright “no,” but on occasion a pitcher will reveal a hitherto hidden desire to learn a palmball, so it’s worth asking.

Nobody has embodied this drive for self-improvement like Shohei Ohtani. The man who already does everything showed up at the start of 2024 and decided to turn his plus running speed from a curiosity into a weapon. Shotime had previously topped out in the 20-steal range, and usually with pretty ugly success rates. In 2022, he needed 20 attempts to swipe just 11 bags; that year, he also stole the George Springer Trophy for Most Mystifyingly Bad Basestealer for a Fast Guy. Read the rest of this entry »

Nick Yorke went from the Boston Red Sox to the Pittsburgh Pirates at this past summer’s trade deadline in exchange for Quinn Priester. Some months earlier he’d gone back to the approach that made him a first-round pick in 2020, and from there a productive hitter in his first full professional season. The adjustment was needed. While Yorke remained a promising prospect in 2022 — a campaign compromised by injuries — and again in 2023, his productivity was less than what was expected, and certainly less than what he’d hoped for.

The changes Yorke made this year proved a panacea. After getting off to a so-so start in cold-weather Portland, Maine, he swung a hot bat after being promoted to Triple-A Worcester, and from there at Indianapolis following the trade. Over 344 plate appearances at the highest level of the minors, the 22-year-old infielder/outfielder slashed .333/.420/.498 with 25 doubles, eight home runs, and a 143 wRC+. Moreover, he stuck out at a lower rate than he did in a season-plus at the Double-A level. Upon getting called up in mid-September, Yorke went 8-for-37 with a pair of home runs and an 82 wRC+ in 42 plate appearances across 11 major league games.

Yorke sat down at Pittsburgh’s PNC Park during the final week of the regular season to discuss his successful turnaround this year.

———

David Laurila: We first talked hitting in April 2021 as you were beginning your first season of pro ball. How would you compare now to then?

Nick Yorke: “I would say pretty different while being the same at the same time. I felt — especially that first year when I was 19 — that I was doing really well approach-wise. I was driving the ball the other way. I feel like I kind of got away from that the past couple of years.”

Laurila: How and why did you get away from your old approach? Read the rest of this entry »

For the 21st consecutive season, the ZiPS projection system is unleashing a full set of prognostications. For more information on the ZiPS projections, please consult this year’s introduction and MLB’s glossary entry. The team order is selected by lot, and the next team up is the Los Angeles Angeles.

Batters

For the first month of the offseason, the Angels have been one of the most active teams, acquiring Jorge Soler and the apparently-still-in-baseball Scott Kingery in trades, claiming Ryan Noda off waivers, and signing Travis d’Arnaud, Yusei Kikuchi, Kyle Hendricks, and Kevin Newman in free agency. Doing this tightens up the team’s secondary talent and adds to its depth.

The larger question is what the Angels actually intend to do with these moves. These are the types of things that should have been done back in the days when they had a healthy Mike Trout or were getting 8-10 wins a year from Shohei Ohtani. From 2018 to 2023, all the Angels had to do to contend was build a 75-win team around Trout and Ohtani, something they never succeeded at doing. Now, it looks like they have that 75-win team, except Ohtani isn’t around anymore and Trout is aging and injury prone. (ZiPS is projecting Trout to have around 300 plate appearances in 2025.) Read the rest of this entry »

Today, at FanGraphs dot com, we’re turning over a new leaf. The last two times Aroldis Chapman changed teams — when he signed with the Pirates last January and when he was traded from Kansas City to Texas seven months prior — Jay Jaffe and I both referenced the Tattoo Infection Incident of 2022. It’s memorable and useful as a shorthand for the ignoble end to Chapman’s tenure with the Yankees — though both of his stints in New York were to a greater or lesser extent ignoble throughout.

More than that, Lindsey Adler’s story on the situation introduced a novel clause to the sportswriting canon, a literary construction so vivid it clearly fascinated both Jay and myself for months after the fact. But no more. I’m going to write an Aroldis Chapman story without quoting the phrase, “veritable moat of pus.”

Oh crap, I said the phrase that pays. What a pity; with that said, I’ll surely have another opportunity to write a clean transaction story about the veteran left-hander when he changes teams again. Because if Chapman is still able to command a one-year, $10.75 million contract from the Red Sox, it seems major league teams are determined to keep giving chances to a player who ought to have exhausted the sport’s patience by now. Read the rest of this entry »

Let’s be honest: Headlines aside, trying to dub one pitch the “best” in baseball is a silly way of thinking about things. There are so many pitches a year that anointing exactly one the best doesn’t make much sense. Emmanuel Clase threw hundreds of unhittable cutters this year. Blake Snell’s curveball, when correctly weighted, might as well be made of smoke. Paul Skenes and Jhoan Duran both throw 100-mph offspeed pitches. How can you separate one of these from the rest?

One easy way? Ask one of our pitch models. PitchingBot gives every single pitch three grades. There’s a pure stuff grade, a pure command grade, and a holistic overall score. Those work basically how you’d expect. Stuff is just the raw characteristics of the pitch, ignoring location and count. Command accounts for count and location. The overall grade isn’t a straight combination of the two; it uses all the same inputs, but instead of separately considering pitch shape and location, it grades the combination.

If, for example, you wanted to see the nastiest pitch of the year, you’d look at each individual pitch’s stuff grade. You’d want something with a ton of movement, good velocity, and probably some kind of funky release point to make the other attributes play up. It almost certainly won’t be a fastball, because there’s no way you can match the pure bat-missing prowess of a breaking pitch that way. You’d be looking for something like this:

That Kevin Gausman splitter is just the ticket. It’s not his most consistent pitch – splitters are tough that way. The movement profile is all over the place depending on his exact grip, which leads to the occasional floating ball that hitters can obliterate. But that variance works in his favor sometimes, too, like on that pitch to Giancarlo Stanton. That splitter fell 37 inches, six more than his average one, because he killed the spin on it absolutely perfectly. Read the rest of this entry »

After 17 seasons as a professional baseball player – very nearly half his life – Kyle Higashioka has signed his first major league free agent contract. And the timing couldn’t have been better. Higashioka entered a thin catching market coming off the most productive offensive season of his career, and he cashed in to the tune of a slightly back-loaded two-year, $12.5 million deal with the Rangers. The deal also has a $7 million mutual option for 2027 with a $1 million buyout, which means Higashioka is guaranteed to make $13.5 million.

One very disappointing year removed from a World Series championship, the Rangers are hoping that the 34-year-old’s consistency can help them bounce back into contention. Higashioka has now strung together three consecutive seasons in which he’s played at least 83 games and put up at least 1.3 WAR. Texas would love to see him make it four. Read the rest of this entry »

As I occasionally mention, I worked in finance before I started writing about baseball. One of my early bosses told me something that pretty much everyone in the industry has heard at one time or another. I had just presented a fancy trade that took advantage of about seven different financial instruments to eke out a small profit with minimal risk. He took a long look at my page of notes, scrunched up his nose, and gave me a tip that has stuck with me ever since: “Hedges are for gardens.”

That’s not something you’ll learn in a book. Financial theory is all about reducing variance and then doing the resulting low-risk trade you’ve built over and over. They call them hedge funds for a reason, after all: hedging against loss is a lot of the point. But the secret those books won’t tell you is that this behavior has a logical limit. If I showed you a risk-free way to make a dollar, theory would tell you to replicate that exact trade a billion times. If I showed you a riskier way to make five dollars, theory would tell you to reject it in favor of the first trade and make up the foregone four dollars in volume.

But in the real world, that’s not how things work. As it turns out, you can’t replicate things infinitely. Plenty of the decisions I’d made that reduced variance also reduced expected return per unit of the trade. You can think of it in simplified terms: I’d taken something that would make me four dollars, plus or minus five dollars, and turned it into something that made me two dollars for sure. Two is less than four. If I could select the guaranteed two dollar option twice, that would be clearly better than the risky four dollar option, but my boss pointed out that just doing twice as much isn’t always easy, or even feasible. The better trade, he told me, was the one that didn’t sacrifice quite so much expected value in the name of hedging.

What does this have to do with baseball? More than you’d think. Accepting lower returns in exchange for lower risk is a time-honored tradition across all sports. Whether it’s the running game in football, mid-range jumpers in basketball, or setting up deep and playing defensively in soccer, old school tactics were heavy on risk mitigation. Baseball has tons of these: shortening up to put the ball in play, pitching to contact, sacrifice bunting, letting your starter go seven regardless of how he’s pitching that day. Those strategies are all about minimizing variance around your central outcomes rather than trying for the highest effective value. Read the rest of this entry »

The Chicago Cubs got their offseason into gear Monday morning, with the reported signing of veteran left-handed pitcher Matthew Boyd to a two-year contract worth $14.5 million per year, plus incentives. Hey, Boyd is a name people know, and he had that one really good year a while back, didn’t he? There’s got to be a reason the Cubs are handing out a multi-year deal for almost $30 million to a pitcher who made eight starts in 2024, hasn’t broken 80 innings in a season since 2019, and turns 34 before the start of spring training.

It makes sense, but you have to work a little to see it. Read the rest of this entry »