A certain young Cincinnati pitcher named Aroldis Chapman has generated some buzz over the past week by throwing the fastest pitch ever thrown in the history of the world. Or did he?

It turns out that’s not an easy question to answer either simply or accurately, even though we have a lot more and better data than they did in the days when Walter Johnson’s fastball was measured against, variously, a gravity drop interval recorder in a laboratory and a speeding motorcycle on Aberdeen Proving Grounds.

I’ll try to explain how one would go about answering that question using pitch tracking data collected over the last few years. Most notably that involves the PITCHf/x camera system installed by Sportvision beginning in the 2006 playoffs and progressively installed throughout major league ballparks during the 2007 season. PITCHf/x systems have been in place in all major league ballparks since the start of the 2008 season, and well over 95% of the pitches thrown since then have been tracked by the system. PITCHf/x data is notable because it is used in Major League Baseball’s online Gameday application, and it is available freely on the web for use by analysts. There are other pitch tracking systems using different technology, such as the TrackMan doppler radar system, but the data from these systems has not been made public.

Of course, there is also radar gun data, which I don’t mean to dismiss. However, I’m not going to cover that in this article since the data has not been made public in any systematic fashion and I am not an expert on the use of radar guns.

There are two important things to understand about pitch speed data. The first is that a baseball thrown through air slows down during its travel due to the force of drag. PITCHf/x data tells us that fastballs lose about nine percent of their speed from the pitcher’s hand to home plate. This means that where the speed measurement is made on the trajectory of the pitch is very important.

### The drag force and measurement distances

Pitchers release the ball about 54 or 55 feet from the point of home plate. Radar guns attempt to measure the peak speed right out of the pitcher’s hand, although there is evidence that they don’t really pick up the speed until a few feet later. That evidence comes by way of comparison to PITCHf/x.

To explain why requires a little detour into how PITCHf/x works. If you want to skip this part, you can pick back up at the next paragraph. PITCHf/x measures the pitch trajectory using two video cameras, each capturing images at 30 frames per second. It tracks the ball across the area of green grass between the mound and the home plate area. These 15 or 20 images of the baseball are used to compute a smooth trajectory for the baseball, assuming constant acceleration. Such a trajectory can be fully described by nine parameters: an initial position, an initial velocity, and the acceleration, each in three dimensions. From these parameters we can calculate the position of the baseball at any point along the trajectory. For example, we could calculate where the baseball was when it was 10 feet from home plate or 50 feet from home plate. These nine parameters are reported on the MLB Gameday website for every pitch.

The values reported on Gameday are from a point on the trajectory that is 50 feet from home plate. This means that the speed that Gameday reports from PITCHf/x is not the speed at which the pitch left the pitcher’s hand. In fact, it is a little slower than the release speed, by about 0.6 to 0.8 mph, depending on the speed of the pitch and the exact release point of the pitcher.

Sportvision and MLBAM experimented with reporting values at different distances from home plate before settling on the 50-foot reporting distance for the 2008 season. During the 2006 playoffs and 2007 season, PITCHf/x pitch speeds were reported at distances of 40, 45, 50 and 55 feet from home plate. PITCHf/x data from 2008, 2009 and 2010 is all from a 50-foot reporting distance (except for a brief period of time in August 2009 when 45 feet was used).

Ultimately, Sportvision and MLB Advanced Media (MLBAM) settled on the 50-foot distance for calculating pitch speeds because they believed this matched up best with the radar gun measurements that the baseball community was already familiar with. When they had reported from 55 feet, they were criticized for inflating the speeds. I have not done a detailed comparison between PITCHf/x data from 2007 and the BIS pitch speeds for 2007 available on Fangraphs, which presumably would have come from stadium radar guns. However, my impression from working with both data sets is that the 50-foot PITCHf/x speeds do generally agree pretty well with the BIS speed data. Making the same comparison after 2007 is problematic because stadium and TV speed displays began increasingly to use PITCHf/x data rather than radar gun data.

So, we see that the distance at which the speed is measured matters. There’s also a bit of a philosophical question here. If we want to measure how fast the ball was going at its fastest, we want to know the speed out of the pitcher’s hand. However, if we want to know how fast it seemed to the batter, we can find ourselves examining all sorts of other things. We could simply look at plate crossing speed, but that’s not all that matters to the batter. Probably more closely related to the batter’s perception of speed is the time from release to plate crossing. That time is affected by where the ball is released, where it crosses the plate, the speed of the baseball itself, and even the density of the air. I’m not sure there is one right answer for true pitch speed.

Once we agree on a standard for how to measure speed, if in fact we can agree, we come to our second important concept. Every measurement has an uncertainty attached to it. When we say that Aroldis Chapman’s pitch was moving 103.9 mph when it was 50 feet from home plate, what does that mean? How do we know it wasn’t really 102 mph or 105 mph, and how confident are we in that?

### Error in the PITCHf/x measurement

There are at least two sources of possible error to consider. The first is the measurement error inherent in a well-calibrated PITCHf/x camera system. Remember that PITCHf/x determines pitch speeds by measuring the position of the baseball 15 or 20 times during its flight and turning this information into a smooth, constant-acceleration trajectory. Based on the accuracy with which PITCHf/x is able to determine the location of the ball and based on the algorithm and assumptions they use to produce the trajectory, the measurement accuracy we can expect from PITCHf/x for the speed of a single pitch is 0.4 mph.

Since we expect that this type of measurement error is random, though we can only get within 0.4 mph on a single pitch, we can do much better for a large sample of pitches. For example, if we wanted to determine a pitcher’s average fastball velocity, we would only need 100 pitches to reduce the random measurement error to 0.04 mph. (Average error = sqrt (0.4^2 * 100) / 100 = 0.04.) After 1000 pitches, the random measurement error would be only 0.013 mph. So you see that we can basically reduce the effect of random measurement error to an arbitrarily small value by collecting additional data.

However, there is another source of error that is more difficult to deal with. If the PITCHf/x camera system is not well calibrated, systematic errors will be introduced. The effect of this systematic error is unlikely to decrease as the sample size increases.

What do I mean when I say that a PITCHf/x camera system is well calibrated? I mean that it does an accurate job of translating the pixel locations of the baseball in the images captured by the cameras into real-world, three-dimensional locations on the baseball diamond. Sportvision has calibration procedures o ensure that this translation is an accurate one.

Generally speaking, the calibration process seems to work pretty well, well enough that we have a large pitch data set that is trustworthy for analysis. We can identify pitch types, and what a pitcher throws on one day in one park matches up quite well with what he throws on another day in another park where his tosses are measured by a different PITCHf/x camera system.

But exactly how accurate is it? I have developed a process to measure the accuracy of the PITCHf/x system in each park over time. The process involves comparing a pitcher who pitches in one park with how he pitches in another park and assuming that certain things about his pitches should remain about the same from game to game. This kind of information is then collated for all the pitchers in the league. The result is a gauge of the accuracy of each PITCHf/x system. This gauge can also be applied to parameters other than speed, but here’s my assessment of the speed accuracy of the PITCHf/x systems in the 30 major league stadiums in 2010.

Park Code Error (mph) KCA 1.1 COL 1.1 CHA 0.7 CLE 0.6 BOS 0.5 SEA 0.2 ATL 0.2 SDN 0.2 MIN 0.2 TOR 0.2 WAS 0.1 ARI 0.1 DET 0.1 SLN 0.0 MIL 0.0 CHN 0.0 PHI 0.0 HOU 0.0 OAK -0.1 CIN -0.2 PIT -0.3 ANA -0.3 BAL -0.4 TBA -0.4 FLO -0.5 SFN -0.7 NYA -0.7 LAN -0.8 NYN -1.0 TEX -1.3

That is the systematic measurement error for pitch speed reported at 50 feet, averaged over the 2010 season. If you want to know the true speed of a pitch thrown in Kansas City this year, you should subtract about 1.1 mph. The calibration of a particular PITCHf/x system changes more often than once a season, so those numbers may not apply to a specific game, but they should give a general idea of the size of errors that may be observed.

(You may notice the numbers for all the parks don’t average to exactly zero. I believe that’s due to a seasonal effect that I am still trying to understand. The systems are baselined against the period 2008-2010. Pitchers seem to throw a little harder in the last half of the season than in the first half, even after accounting for temperature differences.)

### Fastest ever?

We see that Cincinnati’s PITCHf/x system has been pretty accurate for pitch speeds this year. This also seems to be true for the specific games in which Chapman pitched this week. So we can believe that he pitched a ball that was traveling 103.9 mph plus or minus 0.5 mph or so when it was 50 feet from home plate. It was probably going nearly 105 mph when it left his hand.

Is this the fastest ever? As far as I know, it’s the fastest pitch recorded by PITCHf/x. Some people have mentioned the 104.8 mph pitch that Joel Zumaya threw to Frank Thomas on Oct. 10, 2006. That was the pitch speed calculated by PITCHf/x at a distance of 55 feet from home plate. Translated to the 50-foot distance that is used by PITCHf/x today, that would be a 103.8-mph pitch, slower than the Chapman’s reported 103.9 mph, but the difference between the two is well within the margin of measurement error. I don’t have the data at hand to determine how well the Oakland PITCHf/x system was calibrated that day in 2006.

Chapman’s average fastball speed of 101.4 mph after two games is certainly higher than Zumaya’s average fastball speed of 98.8 mph from 2008 to 2010. We’ll see if he can keep up that velocity as the month of September unfolds, and hopefully, into next year as well.

Lucas Apostoleris said...

Very cool. Thanks very much for your park accuracy report. That seems to mesh pretty well with what I’ve eyeballed this year.

Jeff Z said...

Mike – Do you have any plans for releasing the correction values from previous seasons?

TCQ said...

Might the increase in last-half velocity be due to smoothed out mechanical flaws coming out of spring training? That’s the first thing that comes to my mind.

Mike Fast said...

Jeff, I don’t have any plans to release comprehensive correction values.

TCQ, it might be something like that. I’m not sure how to tell.

Tom M. Tango said...

Just asking if a pitcher throwing a ball at mile high as opposed to not will end up having a faster whatever (release and/or speed of ball).

Just looking for bias in the context, rather than bias in the machines. Seeing Colorado at #2 is always a sign for investigation. The two NY teams so close to each other as well. Florida / Tampa, the Cali teams, etc.

Tom M. Tango said...

Mike, it could be weather or park-related as well, no? (Think Usain with wind on his back, running at one mile above sea level.) When I see Colorado near the top, I’m always suspecting that.

Mike Fast said...

Tom, I’ve accounted for temperature and air density as best I can. I believe the wind has minimal effect on pitch speed.

Or are you suggesting that perhaps pitchers are able to throw harder at altitude?

Mike Fast said...

That’s a valid question, Tom. I don’t see any clues right now that point me in a direction of what might be going on, if it is something like that rather than random chance. I’ll think about it, though.