Last week, Michael Rosen wrote about Jack Flaherty’s delayed free agency market. Michael advanced a number of theories about why Flaherty hadn’t yet signed a deal, and what that might mean about his fastball, teams’ perceptions of his fastball, and the trajectory of his career broadly speaking. I found that piece really interesting – and I also started thinking about what Flaherty not having signed yet means in a larger sense.

You don’t have to look any further than last year to get an idea of what could happen to Flaherty. Blake Snell and Jordan Montgomery both waited a long time before settling for short-term deals. The year before that, Carlos Correa’s multiple failed physicals kept him on the market until the very end. In 2022, Correa, Kenley Jansen, and Trevor Story all found themselves looking for employment well into March.

All of those players came into the offseason expecting a major contract, and all of them ended up getting less than anticipated, bringing to mind some classic FanGraphs articles from Travis Sawchik, back in the halcyon days of 2018. Those articles drew on a study by Max Rieper that separated free agents into pre- and post-New Year’s signings and found a large discount for the latter group. Read the rest of this entry »

Earlier this week, I threw some numbers together on the value of productive outs. I focused on Corbin Carroll, and rightly so: His electric skill set is a perfect entry point for explaining how hitters can add (or subtract) value relative to average even when making an out. Putting the ball in play? We love it. Avoiding double plays? We love that too. The Diamondbacks are a team full of speedsters, and Carroll’s productive outs gave their baserunners a chance to show off their wheels.

A quick refresher: I calculated the difference in run scoring expectation between the average out and a specific type of out (strikeout, air out, non-GIDP groundout, double play) for each base/out state. Then I had a computer program tag each out made in 2024 with that difference. For example, the average out made with a runner on second and no outs cost teams 0.35 runs of scoring expectation in 2024. Groundouts in that situation only cost 0.25 runs, a difference of 0.1 runs.

Thus, on every groundout that occurred with a runner on second and no out, I had the computer note ‘plus 0.1’ for the “productive out” value. A strikeout in that situation, on the other hand, lowered scoring expectancy by 0.43 runs, a difference from average of -.09 runs. So the computer noted ‘minus 0.09’ for every strikeout with a runner on second and no out. Do this for every combination of base/out state and out type, add it all up, and you can work out the total value of a player’s productive outs. Read the rest of this entry »

Not every out is created equal. Take this fly out from Corbin Carroll, for example:

A lot of things can happen when you make an out with the bases loaded. You could strike out, leaving every runner in place. You could hit into a double play, an inning-ending one in this case. You could ground out some other way, or hit an infield fly. But Carroll’s here was the most valuable imaginable; with one out, he advanced every single runner, including the runner who scored from third.

Mathematically speaking, you can think of it this way. The average out that took place with the bases loaded and one out lowered the team’s run expectancy by a massive 0.61 runs in 2024. That’s because tons of these outs were either strikeouts (bad, runner on third doesn’t score) or double plays (bad, inning ends). But Carroll’s fly out was far better than that. It actually increased the run expectancy by a hair; driving the lead runner home and moving the trail runners up a base is exquisitely valuable.

That’s not the only way this could have gone. Consider a similar situation, a groundout from Aaron Judge:

Like Carroll, Judge batted with a runner on third and fewer than two outs. In this situation, the average out is bad, lowering run expectancy by 0.514 runs. But Judge’s was obviously worse. It cost the Yankees all the expected runs they had left in the inning, naturally, which added up to just a bit more than 1.15. Read the rest of this entry »

In 2008, the first year of PitchF/X pitch tracking, 13.9% of all pitches across the major leagues were sliders. Ah, those were the days – flat, crushable fastballs as far as the eye could see. More or less every year since then, sliders have proliferated. Don’t believe me? Take a look at the graph:

Are you surprised? Of course not. You’ve seen Blake Snell pitch – and Lance McCullers Jr., Sean Manaea, five of your team’s best relievers, and pretty much anyone in the past half decade. Pitchers are flocking to sliders whenever they can get away with throwing one. It used to be a two-strike offering, then an ahead-in-the-count offering, and now many pitchers would rather throw sliders than fastballs when they desperately need to find the zone. Look at that inexorable march higher.

Only, maybe it’s not so inexorable anymore. Between 2015 and 2023, the average increase in slider rate was 0.9 percentage points year-over-year. The lowest increase was half a percentage point; each of the last three years saw increases of a percentage point or more. But from 2023 to 2024, slider rate stagnated. In 2023, 22.2% of all pitches were sliders. In 2024, that number only climbed to 22.3%, the lowest increase since the upward trend started a decade ago.

That’s hardly evidence of the demise of the slider. For one thing, the number is still going up. For another thing, it’s one year. Finally, 2024 marked the highest rate of sliders thrown in major league history. If I showed you the above graph and told you “look, sliders aren’t cool anymore,” you’d be understandably unmoved.

Not to worry, though. It might be January 9, but I won’t try to pass that off as genuine baseball analysis even in the depths of winter. I’ve got a tiny bit more than that. Raw slider rate is a misleading way of considering how pitcher behavior is changing. There are two ways to increase the league-wide slider rate. First, pitchers could adjust their arsenals to use more sliders and fewer other pitches. Second, the population could change – new, slider-dominant pitchers could replace other hurlers who throw the pitch less frequently.

For example, Adam Wainwright retired after the 2023 season. He threw 1,785 pitches that year, and only five were sliders. Plenty of the innings Wainwright filled for the Cardinals went to Andre Pallante, who graduated from the bullpen to the rotation and made 20 starts in 2024. Pallante actually threw fewer sliders proportionally in 2024 than he did in 2023 – but his pitch count ballooned from 1,139 to 1,978. Similarly, Michael McGreevy made his big league debut in 2024 and threw 311 pitches, 19% of which were sliders.

The numbers can lie to you. Pallante, the only one of our three pitchers to appear in both years, lowered his slider rate. But in 2023, Pallante and Wainwright combined for a 7% slider rate. In 2024, Pallante and McGreevy combined for a 17.1% slider rate. That sounds like a huge change in behavior – but it’s actually just a change in population composition.

The story we all think about isn’t Wainwright retiring and handing his innings to McGreevy and Pallante. It’s Brayan Bello going from 17.5% sliders to 28% sliders while pitching a similar innings load – something that also happened in 2024, just so we’re clear.

To measure how existing pitchers are changing their slider usage, we shouldn’t look at the overall rate. We should instead look at the change in each pitcher’s rate. That’s a truer reflection of the question I’m asking, or at least I think it is. And that answer differs from the chart I showed you up at the top of this article.

There were 315 pitchers who threw at least 50 innings in 2023 and 2024, and threw at least one slider in each of those two years. Of those 315 pitchers, 142 increased their slider usage, 24 kept their usage the same, and 149 decreased the rate at which they threw sliders. The story was similar from 2022 to 2023. There were 216 pitchers who fit the criteria in those years; 90 increased their slider usage, 19 kept theirs the same, and 107 decreased the rate at which they used the pitch. From 2021 to 2022, the effect went the other way; 122 pitchers threw sliders more frequently in 2022 than they did in 2021, 22 kept their usage the same, and 74 decreased their usage.

Put that way, the change is quite striking. The slider craze kicked off in earnest in 2017. From 2016-2017, 114 pitchers increased their slider usage and 89 decreased theirs. That rough split persisted in 2017-2018 and 2018-2019. Everything around the 2020 season is a little weird thanks to the abbreviated schedule, but the basic gist – more pitchers increasing slider usage than decreasing slider usage – was true in every pair of years from 2014-2015 through 2021-2022.

That sounds more like a trend than the overall rate of sliders thrown. Graphically, it looks like this:

Let’s put that in plain English. From 2015, the start of the spike in slider usage, through 2022, there were far more pitchers increasing their slider frequency than decreasing it. On average across those years, 1.3 pitchers threw more sliders for every one pitcher who threw fewer. In the past two years, that trend has reversed; more pitchers are reducing their reliance on sliders than increasing it. The population is going to continue to change – they don’t make a lot of Adam Wainwrights these days – but on a per-pitcher basis, the relentless increase in slider usage has halted.

I tried a few other ways of looking at this phenomenon. I held pitcher workloads constant from year one and applied year two slider rates to each pitcher (pitchers who only threw in year one obviously keep their rate unchanged). The same trend held – the last two years have seen a sharp divergence from the boom times of 2015-2022. I looked at the percentage of starters who started using a slider more than some other pitch in their arsenal and compared it to the ones who de-emphasized it; same deal. I also should note that I’ve grouped sweepers and slurves among the sliders for this article, so this reversal is not about pitchers ditching traditional sliders to get in on the sweeper craze.

No matter how you slice it, we’ve seemingly entered a new phase of pitch design. For a while, most pitchers took a hard look at what they were throwing and decided they needed more sliders. Now, though, it appears that we’ve reached an equilibrium point. Some pitchers still want more. Some think they’re throwing enough, or even a hair too many. Now splitters are on the rise, and hybrid cutters are starting to eat into sliders’ market share.

It’s far too early to say that sliders are on the decline. Factually speaking, they’re not. But to me, at least, it’s clear that the last two years are different than the years before them when it comes to the most ubiquitous out pitch in baseball. Sure, everyone has a slider now – but in the same way that four-seam fastballs were inevitable right until sinkers made a comeback, the slider is no longer expanding its dominance among secondary pitches. An exciting conclusion? I’m not sure. But it’s certainly backed by the evidence.

As of the time I’m writing this article, roughly half of our Top 50 free agents have signed new contracts this offseason. That sounds like a great time to take a look at how the market has developed, both for individual players and overall positional archetypes. For example, starting pitchers have been all the rage so far, or so it seems. But does that match up with the data?

I sliced the data up into three groups to get a handle on this: starters, relievers, and position players. I then calculated how far off both I and the crowdsourced predictions were when it came to average annual value and total dollars handed out. You can see here that I came out very slightly ahead of the pack of readers by these metrics, at least so far:

To be fair, none of us have done particularly well. The last two years I’ve run this experiment, I missed by around $1 million in average annual value, and the crowd missed by between $1 and $2 million. Likewise, I’ve missed by roughly $10 million in average annual value per contract, with the crowd around $18 million. This year, the contracts have been longer than I expected, and richer than you readers expected, though you did a much better job on a relative basis when it came to predicting total dollar outlay. We were all low on every category, though, across the board.
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Sports Info Solutions has been tracking every pitch thrown in Major League Baseball since 2002, and since the beginning, those pitches have been hitting the strike zone less and less frequently. You can check the tumbling year-over-year numbers over on our pitch-level data leaderboard, but if you want to spare yourself a click, I pulled them into the graph below. It paints a damning picture of the command of today’s stuff-over-stamina, throw-it-hard-before-your-elbow-explodes pitchers. Don’t go near this graph if you’re on roller skates:

If you ever feel the need to shake your fist at young pitchers and mutter about loud music and fastball command, this is the graph for you. SIS has documented the percentage of pitches that hit the strike zone dropping from the low 50s to the low 40s over the last 20 years. Combine that with the game’s ever-increasing focus on velocity and stuff, and you’ve got a nice, tidy narrative: today’s pitchers are too focused on throwing hard to know where the hell they’re throwing the ball. However, the truth is a bit more complicated. It’s important to keep in mind that the SIS numbers come from real life human beings who analyze video to track pitches, while the friendly robot that powers Statcast has its definition of the strike zone set in digital stone. Read the rest of this entry »

Right after FanGraphs published my piece on the Kirby Index, the metric’s namesake lost his touch. George Kirby’s trademark command — so reliable that I felt comfortable naming a statistic after him — fell off a cliff. While the walk rate remained under control, the home run rate spiked; he allowed seven home runs in May, all on pitches where he missed his target by a significant margin.

Watching the namesake of my new metric turn mediocre immediately following publication was among the many humbling experiences of publishing this story. Nevertheless, I wanted to revisit the piece. For one, it’s December. And writing the story led me down a fascinating rabbit hole: While I learned that the Kirby Index has its flaws, I also learned a ton about contemporary efforts to quantify pitcher command.

But first, what is the Kirby Index? I found that release angles, in concert with release height and width, almost perfectly predicted the location of a pitch. If these two variables told you almost everything about the location of a pitch, then a measurement of their variation for individual pitchers could theoretically provide novel information about pitcher command.

This got a few people mad on Twitter, including baseball’s eminent physicist Alan Nathan and Greg Rybarczyk, the creator of the “Hit Tracker” and a former member of the Red Sox front office. These two — particularly Rybarczyk — took issue with my use of machine learning to make these predictions, arguing that my use of machine learning suggested I didn’t understand the actual mechanics of why a pitch goes where it goes.

“You’re spot on, Alan,” wrote Rybarczyk. “The amazement that trajectory and launch parameters are strongly associated with where the ball ends up can only come from people who see tracking data as columns of digits rather than measurements of reality that reflect the underlying physics.”

While the tone was a bit much, Rybarczyk had a point. My “amazement” would have been tempered with a more thorough understanding of how Statcast calculates the location where a pitch crosses home plate. After publication, I learned that the nine-parameter fit explains why pitch location could be so powerfully predicted by release angles.

The location of a pitch is derived from the initial velocity, initial release point, and initial acceleration of the pitch in three dimensions. (These are the nine parameters.) Release angles are calculated using initial velocity and initial release point. Because the location of the pitch and the release angle are both derived from the 9P fit, it makes sense that they’d be almost perfectly correlated.

This led to a reasonable critique: If release angles are location information in a different form, why not just apply the same technique of measuring variation on the pitch locations themselves? This is a fair question. But using locations would have undermined the conclusion of that Kirby Index piece — that biomechanical data like release angles could improve the precision of command measurements.

Teams, with their access to KinaTrax data, could create their own version of the Kirby Index, not with implied release angles derived from the nine-parameter fit, but with the position of wrists and arms captured at the moment of release. The Kirby Index piece wasn’t just about creating a new way to measure command; I wanted it to point toward one specific way that the new data revolution in baseball would unfold.

But enough about that. It’s time for the leaderboards. I removed all pitchers with fewer than 500 fastballs. Here are the top 20 in the Kirby Index for the 2024 season:

And here are the bottom 20:

A few takeaways for me: First, I am so grateful Kirby got it together and finished in the top three. Death, taxes, and George Kirby throwing fastballs where he wants. Second, the top and bottom of the leaderboards are satisfying. Cody Bradford throws 89 and lives off his elite command, and Joe Boyle — well, there’s a reason the A’s threw him in as a piece in the Jeffrey Springs trade despite his otherworldly stuff. Third, there are guys on the laggard list — Seth Lugo and Miles Mikolas, in particular — who look out of place.

Mikolas lingered around the bottom of the leaderboards all year, which I found curious. Mikolas, after all, averages just 93 mph on his four-seam fastball; one would imagine such a guy would need to have elite command to remain a viable major league starter, and that league-worst command effectively would be a death sentence. Confusing this further, Mikolas avoided walks better than almost anyone.

Why Mikolas ranked so poorly in the Kirby Index while walking so few hitters could probably be the subject of its own article, but for the purposes of this story, it’s probably enough to say that the Kirby Index misses some things.

An example: Mikolas ranked second among all pitchers in arm angle variation on four-seam fastballs, suggesting that Mikolas is intentionally altering his arm angle from pitch to pitch, likely depending on whether the hitter is left-handed or right-handed. This is just one reason why someone might rank low in the Kirby Index. Another, as I mentioned in the original article, is that a pitcher like Lugo might be aiming at so many different targets that it fools a metric like the Kirby Index.

So: The Kirby Index was a fun exercise, but there are some flaws. What are the alternatives to measuring pitcher command?

Location+

Location+ is the industry standard. The FanGraphs Sabermetric library (an incredible resource, it must be said) does a great job of describing that metric, so I’d encourage you to click this hyperlink for the full description. The short version: Run values are assigned to each location and each pitch type based on the count. Each pitch is graded on the stuff-neutral locations.

Implied location value

Nobody seems particularly satisfied with Location+, including the creators of Location+ themselves. Because each count state and each pitch type uses its own run value map to distribute run value grades, it takes a super long time for the statistic to stabilize, upward of hundreds of pitches. It also isn’t particularly sticky from year to year.

The newest version of Location+, which will debut sometime in the near future, will use a similar logic to PitchProfiler’s command model. Essentially, PitchProfiler calculates a Stuff+ and a Pitching+ for each pitcher, which are set on a run value scale. By subtracting the Stuff+ run value from the Pitching+ run value, the model backs into the value a pitcher gets from their command alone.

Blobs

Whether it’s measuring the standard deviation of release angle proxies or the actual locations of the pitches themselves, this method can be defined as the “blob” method, assessing the cluster tightness of the chosen variable.

Max Bay, now a senior quantitative analyst with the Dodgers, advanced the Kirby Index method by measuring release angle “confidence ellipses,” allowing for a more elegant unification of the vertical and horizontal release angle components.

Miss distance

The central concern with the Kirby Index and all the blob methods, as I stated at the time, is the single target assumption. Ideally, instead of looking at how closely all pitchers are clustered around a single point, each pitch would be evaluated based on how close it finished to the actual target.

But targets are hard to come by. SportsVision started tracking these targets in the mid-2010s, as Eno Sarris outlined in his piece on the state of command research in 2018. These days, Driveline Baseball measures this working alongside Inside Edge. Inside Edge deploys human beings to manually tag the target location for every single pitch. With these data in hand, Driveline can do a couple of things. First, they created a Command+ model, modifying the mean miss distances by accounting for the difficulty of the target and the shape of a pitch.

Using intended zone data, Driveline also shows pitchers where exactly they should aim to account for their miss tendencies. I’m told they will be producing this methodology in a public post soon.

Catcher Targets (Computer Vision)

In a perfect world, computers would replace human beings — wait, let me try that sentence again. It is expensive and time-intensive to manually track targets through video, and so for good reason, miss target data belong to those who are willing to pay the price. Computer vision techniques present the potential to produce the data cheaply and (therefore) democratically.

Carlos Marcano and Dylan Drummey introduced their BaseballCV project in September. (Drummey was hired by the Cubs shortly thereafter.) Joseph Dattoli, the director of player development at the University of Missouri, offered a contribution to the project by demonstrating how computer vision could be used to tag catcher targets. The only limitation, Joseph pointed out, is the computing power required to comb through video of every single pitch.

There are some potential problems with any command measurement dependent on target tracking. Targets aren’t always real targets, more like cues for the pitcher to throw toward that general direction. But Joseph gets around this concern by tracking the catcher’s glove as well as his center of mass, which is less susceptible to these sorts of dekes. Still, there’s a way to go before this method scales into a form where daily leaderboards are accessible.

The Powers method

Absent a raft of public information about actual pitcher targets, there instead can be an effort to simulate them. In their 2023 presentation, “Pitch trajectory density estimation for predicting future outcomes,” Rice professor Scott Powers and his co-author Vicente Iglesias proposed a method to account for the random variation in pitch trajectories, in the process offering a framework for simulating something like a target. (I will likely butcher his methods if I try to summarize them, so I’d encourage you to watch the full presentation if you’re interested.)

The Powers method was modified by Stephen Sutton-Brown at Baseball Prospectus, who used Blake Snell as an example of the way these targeting models can be applied at scale to assess individual pitchers. First, Sutton-Brown fit a model that created a global target for each pitch type, adjusting for the count and handedness of each batter. Then, for each pitcher, this global target was tweaked to account for that pitcher’s tendencies. Using these simulated targets, he calculated their average miss distance, allowing for a separation of the run value of a pitcher’s targets from the run value of their command ability.

“Nothing”

On Twitter, I asked Lance Brozdowski what he saw as the gold standard command metric. He answered “Nothing,” which sums up the problem well. This is a challenging question, and all the existing methods have their flaws.

There are ways that the Kirby Index could be improved, but as far as I can tell, the best way forward for public command metrics is some sort of combination of the final two methods, with active monitoring of the computer vision advancements to see if consistent targets can be established.

But one would imagine the story is completely different on the team side. By marrying the KinaTrax data with miss distance information, these methods could potentially be combined to make some sort of super metric, one that I imagine gets pretty close to measuring the true command ability of major league pitchers. (In a video from Wednesday, Brozdowski reported on some of the potential of these data for measuring and improving command, as well as their limitations.) The public might not be quite there, but as far as I can tell, we’re not that far off.

Editor’s Note: This story has been updated to include Vicente Iglesias as a co-author on the 2023 presentation, “Pitch trajectory density estimation for predicting future outcomes.”

The Winter Meetings always feature trades, but two stood above the fray last week. First, the Guardians traded Andrés Giménez to the Blue Jays in a two-part transaction that briefly left Cleveland with three lefty-hitting first basemen. Then the White Sox traded Garrett Crochet to the Red Sox for four prospects. The best of that group, Kyle Teel, happens to play catcher, the same position as Chicago’s top prospect Edgar Quero. They even have the same future value grade of 50, which is the cutoff for top 100 prospects.

The Guardians made an extra trade to avoid doubling up on similar archetypes, sending Spencer Horwitz to the Pirates for three young pitchers, but the White Sox just kept both catchers. I heard a lot of murmured questioning of that decision as I walked around the Dallas hotel that briefly hosted the center of the baseball universe. But I think both teams were acting rationally, and that worrying about Teel and Quero overlapping is silly. I can’t prove it for you – but I did come up with some data that will hopefully sway your opinion.

Cleveland’s case was straightforward. Steamer projects Horwitz as a 2.5 WAR/600 PA player. It projects Kyle Manzardo as a 1.8 WAR/600 PA player. Josh Naylor? Steamer has him down for 2.4 WAR/600 PA. Three players for two positions — first and DH. (Yes, Horwitz has played second base, too, but he really shouldn’t be a second baseman, and I don’t think the Guardians would’ve used him there.) One of them would ride the bench despite being an above-average contributor, a poor decision for a team that’s trying to maximize its resources. Something had to give.

On the other hand, there are the White Sox. They, too, traded a young star, and the best player they got back plays a position where they already had a similar option. Teel was our 42th-ranked prospect on our updated Top 100 list in 2024, a polished all-around catcher who we expect to reach the majors at some point in the next two years. Quero was our 40th-ranked prospect, and you’re never going to believe this, but he’s a polished all-around catcher who we expect to reach the majors at some point in the next two years.
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I don’t know the name for this phenomenon, but I’m guessing everyone has experienced it at some point. You hear something enough times, and you start to repeat it without really thinking critically about it. My example: the breakeven stolen base rate. I’ve heard that term so many times over the years, often in connection with whether teams were stealing too much or not enough, that I incorporated it into my thought processes like it was my own.

But then someone asked me why the optimal stolen base success rate was around 70%, and I realized that I’d been wrong. It was a bolt-of-inspiration kind of moment – you only need to hear the counter-argument once to re-assess your old, uncritically assumed thought. Why should teams keep stealing so long as they’re successful more than 70% (ish) of the time? I couldn’t explain it to myself using math.

The other side of the coin, the notion that teams should be successful at far better than the breakeven rate in the aggregate, is incredibly easy to understand. There’s a difference between marginal return and total return. Consider a business where you’re making investments. Your first investment makes $10. Your next one makes $8, and then $6, and so on. You could keep investing until your business breaks even – until you make a negative $10 investment to offset that first one, more or less ($10+$8+$6+$4+$2+$0-$2-$4-$6-$8-$10). But that’s a clearly bad decision. You should stop when your marginal return stops being positive – when an investment returns you $0, you can just stop going and pocket the $30 ($10+$8+$6+$4+$2+$0). Read the rest of this entry »

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.