### Juiced baseballs, part II

Last post, I showed how MGL found the variation (SD) of MLB baseballs to be in the range of about 7 feet difference for a typical fly ball. I wondered if that were truly the case, or if some of it wasn't real, just imprecision due to measurement error.

After some Twitter conversations that led me to other sources, I'm leaning to the conclusion that the variance is real.

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Two of the three measurements in MGL's study (co-authored with Ben Lindbergh) were the circumference of the baseball and its average seam height. For both of those factors, the higher the measure, the more air resistance, and therefore shorter distance travelled.

It occurred to me -- why not measure distance directly, if that's what you're interested in? MGL told me, on Twitter, that that's been done. I found one study via a Google search (a study that Kevin later linked to in a comment).

That study took a box of one dozen MLB balls, fired them from a cannon one by one, and observed how far each travelled. Crucially, the authors adjusted that distance for the original speed and angle, because the cannon itself produces variations in intial conditions. So, what remains is mostly about the ball.

For one of the two boxes, the balls varied (SD) by 8 feet. For the second box, the SD was only 3 feet.

It's still possible that some of that variation is due to initial conditions that weren't controlled for, like small fluctuations in temperature, or air movement within the flight path, or whatever. Fortunately, the authors repeated the procedure, but for a single ball fired multiple times.

The SD for the single ball was 3 feet.

Using the usual method, we know

**SD for different balls ^ 2 = SD for a single ball ^ 2 + SD caused by ball differences ^ 2**

That means for the first box, we estimate that the balls vary by 7 feet. For the second box, it's 0 feet. That's a big difference. Fortunately again, the authors repeated the procedure for different types of balls.

NCAA balls have higher seams and therefore less carry. The study found an overall SD of 11 feet, and single ball variation of 2 feet. That means different balls vary by an expected 10.8 feet, which I'll round to 11.

For minor league balls, the study found an SD of 8 feet overall, but didn't test single balls. Taking 3 feet as a representative estimate for single-ball variation, we get that MiLB balls vary by 7 feet. (8 squared minus 3 squared equals 7 squared, roughly.)

So we have:

**-- MLB balls vary 0 feet in air resistance**

**-- MLB balls vary 7 feet in air resistance**

**-- MiLB balls vary 7 feet in air resistance**

**-- NCAA balls vary 11 feet in air resistance**

In that light, the 7 feet found in MGL's study doesn't seem out of line. Actually, that 7 feet is a bit of an overestimate. It includes variation in COR (bounciness), which doesn't factor into air resistance, as far as I can tell. Limiting only to air resistance, MGL's study found an SD of only 6 feet.

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One thing I noticed in the MGL data is that even for balls within the same era, the COR "bouciness" measure correlates highly to both circumference (-.46 overall) and seam height (-.35 overall). (For the 10 balls after the 2016 All-Star break, it's -.36 and -.56, respectively.)

I don't know if those measures are related on some kind of physics basis, or if it's just coincidence that they varied together that way.

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One thing I wonder: are balls within the same batch (whether the definition of "batch" is a box, a case, or a day's production) more uniform than balls from different batches? I haven't found a study that tells us that. From MGL's data, and treating day of use as a "batch," my eyeballs say batches are slightly more uniform than expected, but not much. My eyeballs could be wrong.

If batches *are* more uniform, teams could get valuable information by grabbing a few balls from today's batch, and getting them tested in advance. They'd be more likely to know, then, if they were dealing with livelier or deader balls that night.

Even if there's no difference within batches compared to between batches, it's still worth the testing. I don't know if any teams actually did this, but if any of them were testing balls in 2016, they'd have had advance knowledge that the balls were getting livelier.

I have no idea what a team would do with that information, that home runs were about to jump significantly over last year ... but you'd think it would be valuable some way.

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MGL tweeted, and I agreed, that it doesn't take much variation in a ball to make a huge difference to home run rates. He also thinks that any change in liveliness is likely to have been inadvertent on the part of the manufacturer, since it takes so little to make balls fly farther. I agree with that too.

But, why are MLB standards so lenient? As Lindbergh quotes from an earlier report,

" ... two baseballs could meet MLB specifications for construction but one ball could be theoretically hit 49.1 feet further."

Why doesn't MLB just put tighter control on the baseballs it uses? If the manufacturers can't make baseballs that precise, just put out a net at a standard distance, fire all the balls, and discard (or save for batting practice) all the balls that land outside the net. (That can't be so hard, can it? It can't be that the cannon would damage the balls too much, since MLB reuses balls that have been hit for line drives, which is a much more violent impact.)

You could even assign the balls to different liveliness groups, and require that different batches be stored at different humidor settings to equalize their bounciness.

Even if that's not practical, couldn't MLB, at least, test the balls regularly, so as to notice the variation before it shows up so obviously in the HR totals?

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Finally, one last thought I had. If a ball is hit for a deep fly ball, doesn't that suggest that, at least as a matter or probability, it's juicier than average? If I were the pitching team, I might not want to pitch that ball again. It might be an expected difference of only a foot or two, but every little bit helps.

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