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Jon Roegele is a columnist at Beyond the Box Score. He currently lives in Vancouver, British Columbia. You can follow him on Twitter at @MLBPlayerAnalys.

The strike zone is always a hot topic within the baseball community, and the attention on called strikes has only intensified since the public introduction of the PITCHf/x tracking system in 2007.

Thanks to this technology, the position at which every pitch crosses the plane of home plate is reported by means of a fitting algorithm and extrapolation. While the reported measurement is not perfect, due to potential error from a number of sources (including fitting error and camera characterization error), the reported data gives us a very good idea of pitch locations and whether pitches were deemed balls or strikes.

Numerous studies on the strike zone have been published over the past several years. Some of these studies have shown that the strike zone is called differently based on factors such as batter handedness and the count. This phenomenon fascinates me, so I decided to examine the called strike zone across a range of factors to investigate which ones appear to consistently affect its size and/or shape.

To assess the strike zone at the most granular level, I divided the plane at the front of home plate into 1×1 inch squares. I then binned every called pitch into its appropriate square by its location and calculated the percentage of pitches in each grid location that were called strikes. If more than half of the pitches in a cell were called strikes, then that cell was included in the strike zone. With this framework in place, I then adjusted one variable at a time through a range of values to examine its potential effect on the size and shape of the called strike zone. With one-square-inch grid locations, it is simple to sum the cells and report the called strike zone in terms of its area in square inches.

In order to have more confidence in the effect of a particular factor, I examined a number of years independently. The factors that I report herein are those that consistently altered the strike zone in a particular manner throughout the observed seasons. They are addressed in the order of significance of their effect.

I should also point out that while I am adjusting one variable at a time, in reality changing many of these variables implicitly suggests that another related underlying variable would not be randomly distributed given the fixed range of the variable under study. I will attempt to point out these circumstances where applicable.

All images are shown from the catcher’s/umpire’s perspective.

Factors Affecting Strike Zone Location

Batter Handedness
The fact that the called strike zone differs for left-handed hitters and right-handed hitters was first reported by John Walsh and reaffirmed by Mike Fast. While the zone for righties is quite symmetric about the center of home plate, the zone is shifted by roughly two inches away from left-handed batters. I recently looked at these so called “lefty strikes” and how well left-handed hitters perform on pitches in this area. The shift can clearly be seen through a sample side-by-side comparison of one season, which is typical across the period of the study.

Called strike zone for LHH and RHH, 2012

Another observation that holds true across the seasons studied is that the strike zone for hitters on each side of the plate favors low-and-away pitches to down-and-in pitches, as can be seen in the juxtaposed images.

Batter Height
Of all the factors investigated in this study, this is the only one that should have a direct impact on both the location and size of the zone, given the relatively larger vertical component of the strike zone for taller batters. As logic would suggest, the strike zone does move up as the height of the hitter increases. Here is a sample of the strike zone shifted north for taller hitters.

Called strike zone for LHH <= 70in tall and LHH >= 75in tall, 2011

Between the shortest and tallest hitters, it looks like the bottom of the strike zone shifts up about one inch while the top expands by about two inches. This makes sense to me—the taller hitter’s knees will of course be higher, leading to the shift at the bottom, and his knee to shoulder-belt midpoint distance would be longer, driving an even more elevated top of the zone.

League
The strike zone called in the National League tends to be slightly lower than that of the American League. More specifically, NL umpires call more pitches for strikes in the row of the grid between 1.67 feet (20”) and 1.75 feet (21”) off the ground than their AL counterparts. Here is a sample of an AL and NL zone from the same season.

Called strike zone for RHH in AL and RHH in NL, 2010

Of course, one explanation for this could be that the average AL batter was taller than the average NL batter. As a quick test for this, I looked at the zone location for all hitters in each league that were 72 inches tall (6’2”). The roughly one-inch-lower zone was still visible with the height controlled for in this fashion.

Factors Affecting Strike Zone Size

Count
The effect that the count has on the called strike zone has also been reported previously. For the sake of sample sizes, in this study I examined the effect that the number of strikes in the count has on the zone, ignoring the number of balls. While the number of balls does also alter the zone (in particular on 3-0 counts), the number of strikes makes an enormous difference.

Consider the size differences between zones called with no strikes and two strikes:

 Year 0 strikes 1 strike 2 strikes Max Delta 2009 452 406 374 78 2010 454 404 382 72 2011 467 417 385 82 2012 477 425 400 77

Average size of strike zone in square inches by strike count for LHH, RHH

Note the dramatic decline of the strike zone size as the number of strikes climbs. The area of the strike zone decreases on the order of 77 square inches with two strikes as compared to a strike-free count.

That’s a lot—it’s like taking away a 9×9 box from the strike zone! This is akin to what you might hear if you have an environmental assessment of an older house—the report will tell you that your house is leaking as if you have a 4’x4’ hole in one of your exterior walls. Of course, the truth in that case, as in this one, is that the area is accumulated incrementally over a number of places, in this case squeezing in from all directions around the perimeter of the called zone.

Called strike zone for LHH with 0 strikes and LHH with 2 strikes, 2011

Admittedly, the image with two strikes is not pretty. There are a few factors at play here, including the fact that the image is ball count-indifferent, that more breaking balls and offspeed pitches would be thrown with two strikes, and also that the sample size is smaller given that many plate appearances never reach a two-strike count.

While it is purely conjecture, one potential explanation here for the difference in sizes could be the reluctance of umpires to be put on the spot by ringing up a hitter on a called third strike pitch that they might perceive as borderline. Early in the count, they may be more likely to rule such pitches as strikes given the lower immediate impact of their decision on the game.

The types of pitches thrown in these strike counts would also tend to be different. With no strikes, one might expect mostly fastballs, given that this is still the most common first pitch of a plate appearance, and also the pitch most pitchers will go to when behind in the count. Keep this in mind as we progress though the other factors.

Outs
The strike zone relaxes to encompass a greater area the closer the inning gets to completion.

 Year 0 out 1 out 2 out Max Delta 2009 425 429 463 38 2010 421 436 463 42 2011 432 448 468 36 2012 446 454 482 36

Average size of strike zone in square inches by outs for LHH, RHH

With two out, the area of the called strike zone is roughly 38 inches larger than it is when there is nobody out. I find this factor particularly interesting, both because I can’t see any significant underlying variable that would be confounding this result and because I can’t think of a good theory to explain why it looks the way it does.

Called strike zone for LHH with 0 out and LHH with 2 out, 2009

From the images across the seasons, it would appear that with two out, the zone expands slightly more off the edges of the plate than it does up and down.

Pitch Type
Pitch type has already surfaced as a variable that is coupled to some extent with the strike count in the plate appearance. It will also be intertwined with pitch velocity, which will be covered shortly.

While they’re very good now, raw Gameday pitch type classifications have been a work in progress throughout the PITCHf/x era, in particular in the earlier seasons. An example of this is the handling of pitches that are now labeled as two-seam fastballs (FT), which were not introduced as a pitch class until the 2010 season. Classifying pitches is not an easy task, so these results always should always be viewed with this in mind.

With that said, we can see how the pitch type plays a role in the size of the strike zone. While I did compare all pitch types, I will show only those that have likely been the most consistently classified across the seasons in question, especially because they show the largest discrepancy in strike zone size.

For this category, the pitch sample sizes for the non-fastballs are a little smaller than most other ranges measured in this study, so while I believe the general trend here is valid, you may choose to take the measured deltas with a grain of salt.

 Year FF SL CU CH Max Delta 2009 438 416 417 416 22 2010 437 423 412 419 25 2011 452 438 425 421 31 2012 466 442 435 436 31

Average size of strike zone in square inches by pitch type for LHH, RHH

Fastballs clearly expand the zone relative to all breaking and offspeed pitches. In particular, four-seam fastball-based strike zones were found to be about 25 square inches larger on average than those for changeups and curveballs. For the most part, this finding lines up with expectations based on the count effect above.

Batter Height
Batter height also affects the size of the strike zone.

 Year <= 70in 71-72in 73-74in >= 75in Max Delta 2009 421 432 442 442 21 2010 425 424 441 445 20 2011 444 441 446 460 19 2012 441 455 459 470 29

Average size of strike zone in square inches by batter height for LHH, RHH

The strike zone size peaks for hitters over 75 inches (6’3”) tall, with these batters having to deal with a zone that is on average 21 square inches larger than hitters up to 70 inches (5’10”) tall.

The data also relies on reported player height data, which may or may not be accurate, as some players “round up” their heights slightly.

Velocity
The velocity of the pitch also affects the size of the strike zone. By binning pitches by speed, we can see a clear shrinking of the strike zone at both velocity extremes:

 Year < 84mph 84-88mph 88-92mph > 92mph Max Delta 2009 425 435 442 426 17 2010 421 439 442 428 21 2011 438 450 454 440 16 2012 442 452 469 451 27

Average size of strike zone in square inches by pitch velocity for LHH, RHH

The size of the strike zone consistently peaks in the 88-92 mph range and reaches its minimum on pitches below 84mph. The difference in zone area is about 20 square inches. It is interesting to note that the area called on fastballs above 92 mph actually reverts to a smaller zone again. There are several potential reasons I can think of that could play a part in this difference.

The hardest pitches are likely some of the most difficult for the catcher to receive, especially if the pitch does not hit the initial target. I would expect that having to move your glove and catch a pitch at this velocity while maintaining a “quiet” receiving posture is quite challenging, which may influence the umpire indirectly toward believing a borderline pitch was outside the zone. In addition, many flamethrowers struggle with command in general but make up for that shortcoming with elite velocity. It is not hard to imagine that a pitcher who routinely misses targets by relatively large margins would not get the benefit of the doubt compared to a control artist.

There is, of course, the underlying factor here of velocity implying pitch type to some extent. This result lines up well with the previous observations regarding pitch type, with fastballs showing a larger strike zone than other, slower offerings. Pitch type is certainly intermixed with these results, given that the pitches studied were thrown not only with different velocity but with different movement as well.

Base State
Having runners on base also causes the size of the strike zone to shift. Again, to keep sample sizes reasonable, I grouped all cases with a runner on a particular base together, rather than splitting them into the subsets.

 Year Empty 1XX X2X XX3 Max Delta 2009 442 428 435 426 16 2010 434 424 432 428 10 2011 454 440 440 430 24 2012 462 448 449 440 22

Average size of strike zone in square inches by base state for LHH, RHH

Here we see that when a runner reaches base, the strike zone tightens up relative to its size with the bases empty. With a runner on third, the zone contracted by about 17 square inches on average over the period of the study.

This one is also interesting. I suppose this is consistent with the strike zone being smallest with two strikes, in so much as the pitcher is given the benefit of the doubt initially, but is forced to “earn” his way out of a troublesome inning with runners on just as he must “earn” that last strike for a strikeout. This is just conjecture again, but it’s an interesting exercise to imagine what is at play here.

League
The National League zone is not only very slightly lower than the American League zone, it’s also very slightly larger.

 Year AL NL Max Delta 2009 428 437 9 2010 425 443 18 2011 445 448 3 2012 454 461 7

Average size of strike zone in square inches by league for LHH, RHH

The difference is not staggering, but it appeared in every year of the study. The set of umpires calling pitches are distinct between the two leagues.

Batter Handedness
We already saw that the strike zone for left-handed hitters is shifted off the outside edge of the plate, but some of the difficulties that shift presents to lefties may be evened out by the fact that the zone for right-handed hitters is actually slightly larger.

 Year LHH RHH Max Delta 2009 432 438 6 2010 434 439 5 2011 446 451 5 2012 453 460 7

Average size of strike zone in square inches by batter handedness

Since the player pools for these groups are so distinct (aside from switch-hitters), I tried controlling for batter height for this particular factor to make sure that the difference is not simply caused by right-handed batters being taller on average than their left-handed counterparts. The larger RHH zone was still visible by height bin, indicating that a true gap, albeit small, does appear to be present.

One area that can be highlighted to illuminate this difference between the sizes of the called strike zones for left-handed hitters and right-handed hitters is pitch type. Pitches classified as cut fastballs and sliders were called with disproportionately larger strike zones for RHH than LHH. The zone for cutters (only in years since 2010) averaged 24 square inches larger for RHH, while the difference was roughly 18 square inches for sliders.

Temperature
I found a very minor fluctuation in the size of the strike zone due to temperature.

 Year < 65F 65-75F 75-85F > 85F Max Delta 2009 428 433 441 427 14 2010 431 435 440 437 9 2011 439 449 450 444 11 2012 456 458 455 449 9

Average size of strike zone in square inches by temperature for LHH, RHH

Showing the same pattern as with velocity, the two extremes show the smallest strike zones, with games played between 75-85 Fahrenheit typically having the largest zone. The called strike zone in the warmest weather is typically about seven square inches smaller than games played at a more pleasant 75-85 F temperature.

Other Factors
I investigated other factors that did not turn up any effect that was consistent across the period of study. These mainly consisted of temporal factors like month, time of day, and inning. Certainly month and to a lesser extent time of day would be somewhat coupled with temperature, but apparently not so much that they consistently followed the same trend. I also investigated pitcher handedness, but its effect was not visible across all the seasons in question.

Putting it All Together
This is not a perfect study, but by examining the complete set of results above, can we make any overarching statements about what affects the strike zone? We can make some educated guesses, I believe, although to be certain would require some follow-up work to ensure that all other factors are controlled while varying the one under scrutiny. To do this would require the entire PITCHf/x dataset across six years to be combined, in order to create reasonable sample sizes for this method of strike zone measurement.

Batter handedness affects the location of the zone, with the zone for left-handed hitters shifted toward the outside edge of the plate. The zone adjusts to the height of the hitter in a manner that would be expected based on its rulebook definition. The NL strike zone appears to be slightly lower than the AL version, although I could imagine uncontrolled factors outside the scope of this study (such as run-scoring environment) proving to be the root explanation for this observation.

I do not believe that from what I have presented above we can separate the individual effects of the count, pitch type, and velocity factors on the called strike zone. The fact that the strike count shows the largest delta leads me to think that it plays a part in the zone size regardless of the pitch type and velocity. The fact that the zone for pitches with a velocity over 92 mph is smaller than that for pitches slightly slower, yet four-seam fastballs as a group form the largest strike zone, suggests to me that the fastest of fastballs yield a smaller strike zone than more typical fastballs. And the fact that all breaking and offspeed pitches and offerings slower than 84 mph are called with a smaller zone seems to indicate a real effect here, where I would propose that it is likely more the movement of these pitches than their velocity that is making the difference.

To my mind, the fact that the strike zone appears to expand as much as it does with two outs in the inning is the most intriguing of these individual results. Someone may already have discovered this, but I was not aware of this effect before undertaking this study.

Expanding Strike Zone
You may have noticed a trend in the above tables across all of the individual factors: the strike zone appears to have covered a progressively larger area over the seasons in the study. This is absolutely the case.

 Year Strike Zone 2009 435 2010 436 2011 448 2012 456 2013 (up to end of June) 460

Average size of strike zone in square inches for LHH, RHH

For the overall picture of the strike zone, I’ve included an interim result for the 2013 season for all games up to the end of June. The strike zone’s size is on the rise in recent years, with the last year and a half operating with a zone about five percent larger than the one in effect just a few seasons earlier. This expansion may be part of the reason for the increase in called strikes reported by James Gentile. Extra called strikes would be one catalyst for the fairly substantial rise in strikeout rates in today’s game.

Since scoring has been on the decline over the years in the study, I wondered if the zone might appear to have grown simply due to a higher percentage of pitches being delivered with the bases empty (as we’ve determined that the zone tends to swell in these games states). For this reason, I checked the particular case of bases empty, nobody out, with a 0-0 count to batters between 70-72 inches tall. The effect was still very striking, with the average zone for LHH and RHH increasing in square inches by year from 430 in 2009, to 438 in 2010, to 450 in 2011, and up to 459 in 2012.

Consider the picture of the strike zone as it stood in 2009 compared to 2012.

Called strike zone for LHH in 2009 and LHH in 2012

Called strike zone for RHH in 2009 and RHH in 2012

If you look closely, you can see that the strike zone may have been squeezed in ever so slightly in 2012 compared to 2009, but it has expanded at the bottom quite significantly, by up to two inches. In fact, if you look at the images of all pitches by year from 2009 forward, which you are free to do at the bottom of this article, the trend of the bottom of the zone creeping lower is apparent for both left-handed and right-handed batters. Another way to view this change is by measuring the area of the called strike zone below 1.75 feet (21”) off the ground throughout these seasons.

 Year LHH RHH 2009 1 0 2010 8 5 2011 12 10 2012 22 17 2013 (up to end of June) 30 26

Average size of strike zone less than 21” from the ground in square inches for LHH, RHH

This finding is also very interesting. Is this a result of a heightened awareness of pitch framing? In other words, have catchers around the league become so much better on average at receiving borderline low pitches that they have turned these former balls into strikes? Are pitchers throwing more to this area than ever before, such that umpires are being conditioned to deem more of these pitches acceptable? Have umpires been instructed to call the lower strike? This topic bears more research. One possibility that we can rule out here is that the American League has been slowly adopting the National League’s lower zone, as the pattern of the bottom of the zone falling has been visible in both leagues over this time.

There is ongoing debate about whether the current strikeout rate and its skyward trajectory are good for the game. While I haven’t proven that the larger zone directly leads to more strikeouts, if it is decided that strikeouts need to be curtailed, the way the strike zone is called, particularly at the knees, might be one avenue through which the league could attempt to alter the situation.

Next Steps
There is more that I could do with this framework in place. Deriving an equation for the strike zone is one interest. I have done some initial experiments by taking the grid location where the pitch arrives and looking at the percentage of pitches in that location that are called strikes based on the most significant of the individual factors discussed in this article. Even from that work, I can see that Mike Fast developed a very elegant formula for the strike zone that is both simple and very accurate in his groundbreaking study linked earlier. By running some regressions, I am seeking to determine which of the factors above truly have the most impact on the called zone when taken concurrently. There appear to be diminishing returns for adding in extra factors, and at some point the effect is so small as to no longer be statistically significant. There is a trade-off between accuracy and simplicity.

With a definition of the strike zone based on some subset of these factors, looking at individual hitters, pitchers, and possibly catchers and even umpires could be interesting. For example, how much could Z-Swing rate and O-Swing rate change for hitters if a more accurate zone is utilized? Similarly, which pitchers hit these less obvious areas of the zone most frequently? If we calculated net expected balls called strikes minus expected strikes called balls for catchers, would it change our opinion slightly about anyone’s receiving skill level? I would not expect any league-wide impact from this sort of exploration, but perhaps it could highlight certain individuals who may be underrated or overrated if the zones typically used for these measures are less accurate than might be possible.

I would also like to delve deeper into the dropping bottom of the strike zone across the league over the PITCHf/x era to see if it might be possible to determine a root cause for this behavior.

Strike Zone Images
While I realize not everyone will be interested, for those who are, the images of the strike zones as determined by this study are available below. Simply select the desired factor and range, and the strike zone image will refresh accordingly.

As usual, these images are shown from the catcher’s/umpire’s perspective. Note that the “box” drawn in the image is just a two-foot square that is included as a point of reference to allow for an easier comparison between images.

According to the rulebook, the strike zone extends from one edge to the other of the 17” home plate in the horizontal axis. The top and bottom of the strike zone are based on the location of the hitter’s body. The top of the zone is defined by the midpoint between the top of the batter’s shoulders and top of his uniform pants, and the bottom of the zone is defined as the hollow beneath the kneecap.

Note: You will need to enable Javascript in your browser in order for the drop down list to function as intended.

Credit and thanks to Baseball Heat Maps for the PITCHf/x database and Baseball-Reference for player height data upon which this analysis was based.

This is a free article. If you enjoyed it, consider subscribing to Baseball Prospectus. Subscriptions support ongoing public baseball research and analysis in an increasingly proprietary environment.

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onetimeforall
7/24
Great stuff. Not only does this contribute to the increase in strikeouts, but the decrease in walks. This year's league-wide walk rate is 7.9% (was 8.0% last year), after hovering between 8.5% and 8.9% from 2001-2009.
SixToolPlayer
7/24
One factor that may explain the expansion of the strike zone overall (and the apparent increase in conformity to the rulebook zone) is that PITCHf/x and/or other tracking systems might be shaping better awareness of umpire tendencies to MLB and the umpires themselves.
jroegele
7/24
Interesting idea Dan. You're right in that it seems like at the left/right edges of the strike zone, they may be being pulled in ever so slightly, which would be more toward the rulebook strike zone (even in the case of the inside edge of the LHH zone).

What we can't see here of course is where "hollow beneath the kneecaps" are of the batters, to know whether expanding slightly down is conforming more or breaking away from the rulebook definition. I would guess in the direction that you're suggesting too, that the slow changes are in fact to get into better accordance with the rulebook definition given the availability of this data for the league and umpires to review.
JDanger
7/24
Bookmarked!
sahmed
7/24
One note on a great article: The separate umpire crews for both leagues were merged about a decade ago. Perhaps the NL's slightly larger zone is due to pitchers-as-batters getting a larger strike zone?
jroegele
7/24
Thanks for pointing that out Sean!

This is a good theory for the effect of the league on the zone, as I do recall a study lately where it was found that the difference between the number of called strikes vs. expected strikes was greatest with pitchers at the plate compared to all other positions.
pft1957
7/25
This is awesome stuff. Love to see something like this done for individual umps, top 10 biggest and smallest zones and its effect on offense.
walrus0909
7/25
I know Mavs owner Mark Cuban says his team does this with NBA refs. If I had to guess, there are several MLB teams that do exactly what you suggest, and maybe adjust their strategy accordingly.

And Jon, I know the sample size is probably too small, but I would still love to see the strike zone for knuckleballs.
ericmvan
7/28
Next thing to look at: the impact of the identity of the hitter, and of the pitcher.

For each, I'd look at three factors: experience, overall quality, and strike zone command. Do veteran hitters have a smaller zone than rookies? Among hitters with three previous years of experience, do good hitters (based on the last three years) have a smaller zone than bad ones? And do hitters who draw a lot of walks have a smaller zone than those who don't? (You can also look at this by including strikeout totals, and I'd use K% - 2 * BB% as a measure of overall strike zone command.) The last two questions are obviously related, so you might want to also create four groups, including hitters who aren't very good overall but draw a lot of walks, and hitters who are good but don't walk much.

And of course you'd look at the pitchers the same way.
Shankly
8/01
Amazing study. Thanks.
lichtman
10/22
Great stuff!

I wonder if some of the changes in strike zone size are not really changes in size, such as with the count. Let me explain.

Even though you are using 1x1 inch grids, there is some wiggle room within each square, plus there are calibration and reliability issues with the pitch f/x system itself.

If in certain counts or other situations a pitcher is throwing more to the edge of the strike zone (like with slower pitches too, and even faster pitches, since they tend to be controlled less), is it possible that the pitches on the edge of the zone are more "outward" within those borderline 1x1 squares, making it appear as if the strike zone were smaller, even though it isn't?

In other words, what if a "perfect umpire" were calling the strike zone, such that it is always the same regardless of the situation (batter, count, outs, etc.). And what if pitcher A was trying to throw pitches right in the middle of the zone, and pitcher B was trying to throw pitches only at the edges. I contend that with pitcher B your method would produce a smaller strike zone because on those 1x1 squares on the edges of the zone, with pitcher B there would be more pitches located on the outer edges of those squares and our perfect umpire would call those balls, whereas with pitcher A, there would be more pitches actually located on the inner part of those squares and more likely to be called strikes.

Your method would assume that all pitches in those borderline squares should be called a ball or strike the same percentage of time for both pitchers.

Basically, I think that the scatter plot of pitches is a big factor in terms of the measured (using your method, which is probably the best you can do) size of the zone.