THE BOOK--Playing The Percentages In Baseball

I had not previously read far enough down the thread about park sizes to see Tango’s calculation of average fence distance.  Now that I see it, I can make an additional commment.  If I “correct” Coors for the additional carry due to atmospheric effects, and if I cheat just a little, then the effective fence distance in Coors is about the same as in Fenway, as Tango has said.  Now if you look at the last figure in my article that Tango linked to, you will see a plot of average speed-off-bat for home runs in each park.  You will see Coors and Fenway at the bottom of the list.  That is more or less consistent with the effective fence distance being the same in both parks.

Of course, that consistency may all be an illusion, given the height of the Green Monster, but it is amusing to think about.  Might be worth some effort to figure out how to take that into account.

I think Greg already knows.  I think he even said that one foot of height is one foot of length.  So if the Green Monster is about 28 feet higher than your standard fence, that’s 28 feet of length to add in.  Just taking a wild guess that the Wall is about one-fourth of the fence in the OF, so that’s 7 extra feet of distance to add.

Then again, RF has a very short wall, so that removes a certain amount.

All in all, let’s say Boston’s fence height adds 3 or 4 feet in length.

I’d love for the data to be presented that way:
average distance of fence
+ adjustment for height of fence
+ adjustment for carry of park

I think the last time I checked, for a typical (median) home run trajectory, 1 foot of fence height corresponded to 0.84 feet of distance (i.e. the ball was coming down slightly steeper than 45 degrees.) This factor would probably be a bit different for a taller fence, though, as balls that clear the Green Monster are higher up and consequently descending at a shallower angle than they exhibit when they pass through the average fence height of about 11 feet.  So, 1 to 1 might be right for Fenway in left field.  I’ll see if I can figure that out (or email me if you want me to send you my fence distance and height table).

This appears to be 2009, so all that talk of the New Yankee Stadium being a wind tunnel is bunk.

Wait, what’s going on, I’m confused?

The balls carry less at NYS, but it’s easier to hit HR’s (less speed off bat). So it appears to be the shorter fences are making balls go over the fences, as Greg said earlier in the year.

Sometime shortly after Tango posted this, I made a quick addendeum to my article in which I said essentially what Dan (#5) just said.  Below average carry and well-below average SoB suggests that the large number of home runs might be the result of the park dimensions rather than atmospheric effects.  I won’t bet the farm on that conclusion (hence, the wishy-washy choice of the word “suggests"), but the data are leading me in that direction.  It would be nice to have more data to see if the early trends (based on home runs through about mid-May) hold up.

Meanwhile, Greg has an on-going project to do more direct studies of how wind might affect fly balls in YS.  Any progress on that?

Re Dan #4:  I would not exactly say the talk about wind is “bunk.” I would rather say that it is not confirmed by the data.  Subtle but important difference.  For example, given the short fence distances (esp, in RF), it might be true that very small wind currents (~few mph--too small to appreciably affect the “carry” in a statistically significant way) might have a large effect on balls that just clear the fence (and there have been a lot of those).  I would not rule out such an effect.  The data do seem to rule out a large effect due to wind. 

Anecdotally (and here I am just rambling):  Watching the NYY-Bos series last weekend, I saw a number of home runs to RF that were of a line-drive character and barely cleared the fence.  A line-drive (i.e., a ball hit with at a low trajectory) is typically not expected to be affected by the wind as much as a fly ball (or popup).

I was just looking at this again, and was struck suddenly by the preponderance of cool weather parks at the left side of the list, and warm weather parks at the right side.  So I decided to do something I hadn’t thought of before.  I pulled some historical data from 2002-06 on average game-time temperatures for each park in April, the altitude of each park, and the carry ratios from Alan’s plot, and I ran a linear regression model on them.  Here’s what I got:

Regression Analysis: Carry versus Temp, Altitude

The regression equation is
Carry = 0.901 + 0.00140 Temp + 0.000016 Altitude

Predictor Coef SE Coef T P
Constant 0.90076 0.02352 38.29 0.000
Temp 0.0014025 0.0003617 3.88 0.001
Altitude 0.00001600 0.00000266 6.01 0.000

S = 0.0135413 R-Sq = 64.2% R-Sq(adj) = 61.5%

Analysis of Variance

Source DF SS MS F P
Regression 2 0.0088732 0.0044366 24.20 0.000
Residual Error 27 0.0049509 0.0001834
Total 29 0.0138242

Source DF Seq SS
Temp 1 0.0022429
Altitude 1 0.0066303

Unusual Observations

Obs Temp Carry Fit SE Fit Residual St Resid
2 54.8 0.95600 0.98811 0.00422 -0.03211 -2.50R
11 60.4 1.07500 1.06843 0.01268 0.00657 1.38 X

R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large leverage.

The residuals all looked very good (i.e. normally distributed, no symmetry issues, etc.) - the outlier was Coors Field, of course…

From this look at it, temperature and altitude accounted for 61.5% of the variation in carry from one park to another, leaving at most 38.5% for wind.

I’m pointing this out not to suggest that what I’ve said here is conclusive of anything, but I would submit that the fact that this regresion equation is dominated by temp/altitude suggests that it will be more difficult to discern the impact of wind.

Alan, as you know I’ve got temperature (not just game time temperature but real-time temperature) data for each of these home runs, and of course altitude data for each.  It might be worthwhile to run a regression model on the indivisual data points to see how well temperature and altitude predict carry, and how much residual impact can be attributed to wind… perhaps then we can isolate those factors and look at just wind…

Greg (#8):  The idea of trying to remove temperature and altitude effects from the overall “carry” is a good one.  In fact, my initial approach tried to do just that (in a way that you and I have discussed), but that analysis is not as as “model-independent” as the carry analysis and is therefore less convincing.  So, I have decided to simply talk about “carry” and de-emphasize the specific issue of wind. 

I will give some thought to writing up the analysis that attempts to remove the temperature and altitude effects.  It will necessarily be more technical (and I am not sure that is a good thing).

An idea for Greg:

What matters as far as air drag is concerned is the air density.  Rather than do a regresssion analysis of carry vs. temperature and altitude, instead use the temperature and altitude to compute the air density, then do a regression analysis of carry vs. air density.

Yes, air density should do it, great idea.  I’ll let you know what I get (may have to use a middle of the road number for air pressure, as I didn’t capture that, and can’t spare the time to go get 800+ weather captures)

As for the wind study thing I had mentioned before, no action on that so far, I’ll check to see if any might be expected…

This is interesting stuff.  Does anyone know how much impact weather systems have (high pressure vs. low pressure) on air density?  Or how much impact humidity has?  Would they be significant or would they be drowned out by the impact of temperature and altitude?

I think Clay Davenport is a meteorologist.  I can ask him to stop by…

bsball - Air pressure and humidity have almost no measureable effect on drag.  Humidity may have an effect on the COR of the ball depending on how it is stored.

Thanks, Peter.

The reason I asked about air pressure is that the altitude effect is basically an air pressure effect.  The pressure is much less at high altitude, something like 85% in Denver vs sea level.  I don’t have a feel for how much the pressure changes in, say Houston, when there is a low pressure system over the area compared to when there is a high pressure system.

And I’m sure that I’ve read that humidity should have an effect, basically lowering the density of the air, and therefore creating less drag.  But again, I don’t have a sense for how Houston on an 85% humidity day compares to Phoenix on a 50% day.

Re Peter:  I don’t know how you can make the statement that air pressure has no measurable effect on drag.  The drag is directly proportional to the air density which is (nearly--exactly if air were a “perfect” gas) directly proportional to the pressure and inversely proportional to the temperature (in Kelvin).  Relative humidity certainly does affect the COR, as you have said.  However, it also affects the air density and therefore the drag.  A higher relative humidity means lower density (a water molecule weighs less than a nitrogen or oxygen molecule).  Some useful references:
http://wahiduddin.net/calc/calc_da_rh.htm
http://www.engineeringtoolbox.com/density-air-d_680.html

Alan - I should have known better than to have used the word measurable in the presence of a physicist.  Of course it is measurable.  But the effect is so small that it is well within the measurement errors of the other factors.

This is from Clay:

Greg’s equation for temperature and altitude is basically a shorthand for the effects of density. “Sea level pressure” has a standard value of 1013 millibars (mb); you lose 1 mb for every 10m or so of altitude you gain. Denver = 5280 ft = 1600 m has a typical surface pressure of around 850 mb. (Be careful with weather maps - they tend to show maps of pressure, where everything has been normalized to sea level, by introducing a fictitious column of air through the ground down to sea level; otherwise everything would turn into a terrain map).

Density of air is roughly equal to .35 * pressure/temperature, with pressure in mb, temperature in Kelvins, and density in kg per cubic meter, so it is directly proportional to changes in both temperature and pressure.

The temperature range over the course of a season between about 275 K (36F) and 315 K (108 F), which is roughly a 15% difference.

Pressure changes from local weather will vary from say 990 mb (you can go lower than that, but if you do you’re in a decently strong storm system and getting rained out) up to 1040 mb on the most cloudless, humidity-free day you can picture. That’s about a 5% variation, so 1/3 of the seasonal temperature range.

Note that the difference in altitude between parks, sea level to Coors, is about 15%. So the conditions of going from Baltimore to Denver are about the same - density-wise - as the diference between playing on the hottest day of the year and the coldest day of the year.

Lower density also has some interesting effects on ball spin. Curve balls are going to break a little less, hooks and slices are reduced so they’re a little more likely to stay fair, and the backspin on line drives won’t generate as much lift, so the ball won’t hang in the air as long. Reduced drag means the ball goes farther; reduced lift/hang time means it gets there faster.

Thanks to Clay for a precise and succinct explanation.  Regarding the effect of density on the Magnus force (which Clay was talking about in his last paragraph):  For balls hit with a small launch angle and lots of backspin, the “carry” as I have defined it can be greater than 1.  That is, the ball travels farther than it would in a vacuum, mainly because the enhancing effect of the Magnus force (which opposes gravity) is greater than the inhibiting effect of the drag (which slows the ball down).  Under such conditions, the ball will travel *less far* in Denver than at sea level.  With sufficient hitf/x and landing point data, it would be interesting to see if such an effect is realized in practice.  Or, said differently, if the batter can get enough backspin on the ball at a low trajectory to make this happen.

What is the minimum distance you have to hit the ball to get it over the green monster?

I just ran a trajectory that landed where the LF pole and the top of the wall meet (310 feet from home and +37 feet above field level), with a flight time of 6.5 seconds (i.e. a nice, high trajectory that would be descending steeply at the impact point.  I got 319 feet.

Greg (21).  Maybe OT, but how do you calculate what is “JE” for balls over for the green monster or CF wall.  For most parks I believe I read if it lands within 1 fence height from fence it is JE, but that seems a lot for Fenways monster or CF wall.

Also, wind speed is not constant as measured from the field surface to the apex of a balls flight. 
In some parks with the wind blowing out I could see where there is minimal wind velocity at home plate despite a brisk wind as seen from the flags at the top of the stadium, and presumably the reported wind speed for each game.  The ball might not be affected by the wind until it attains an undetermined height. How do you adjust for this?

JE’s are those that clear the fence by 10 feet or less, either by landing within 1 fence height of the fence, or by clearing the fence by 10 vertical feet or less.

As for the wind, for parks that are well-enclosed, I zero out the wind below a certain height - a good example of this would be a park like Rogers Centre, which is totally enclosed and which has a very high stadium wall.  However, aside from that, my trajectory modeling assumes a constant wind - I couldn’t do it any other way with the limited information we have for in situ weather.  Some day maybe we’ll have multiple weather stations inside the parks, and we can build a more complicated model for the wind.  Or we might get some serious CFD modeling of parks that will allow us to infer the wind currents inside the park from the outside wind.  Until then, the best we can do is pick a wind value that best represents the average wind in the park, and depend on the differences balancing out…

Didn’t know where this would best fit, but I got a kick out of it.

http://www.newyorker.com/humor/issuecartoons/2009/10/12/cartoons_20091005?slide=20#showHeader

Does the “carry” in Coors Field assume a normal ball or a “humidified” ball?

30 feet on a HR sounds like the “old” Coors Field.  I don’t think the current Coors Field with the humidified balls go nearly as far.  Greg?

Re mgl (#24):  Careful experiments have shown that the primary effect of a “humidified” ball is to lower the COR, which will affect the speed of the ball off the bat but not the “carry”, at least as I have defined carry, in which the batted ball speed is normalized out.  An increase in the mass or size of the ball due to the humidity would affect the carry.  However, these effects have been shown to be small.  Bottom line:  carry (again I emphasize, as I have defined it) is not affected by the humidified ball.  My guess is that the perception that balls don’t travel as far in Coors with the humidified ball is simply a statement that the batted ball speed is not as high as with the dry ball.

O.K.  Got it.  So the “carry” assumes a constant speed off bat.  So it is possible (likely in Colorado with the humidor) that environmental factors that affect the COR of the ball might alter the “effective” carry of the ball in each park?  For example, in a humid park, the COR of the ball might be lower, with a lower SOB, and in a dry park like Arizona, the COR could be higher with a corresponding higher SOB?

mgl (#26):  Yes, the “carry” (as I have defined it) is a measure of how far the ball goes for a given SOB and launch angle.  Changing the COR (and nothing else) changes the SOB but not the carry.  I agree with your last sentence:  “For example, in a humid park, the COR of the ball might be lower, with a lower SOB, and in a dry park like Arizona, the COR could be higher with a corresponding higher SOB?”