"Oh, great!" you're thinking, "here goes Tommy on another one of his ill-fated mathematical boondoggles."

Well, yeah, but calm down. This one should be relatively painless. There is only a very little bit of math. And a tiny bit of calculus. But PLENTY of errors, I promise. Still with me? Okay, then, let's go.

Table of Contents

A Curious Mind
Groundballs
Discussion Question of the Day

A Curious Mind

I was (am) the kind of student in school who was always thinking about math during economics class, philosophy during history class, physics during biology class, and (dear lord!) anything else during 11th grade English.

Anyway, in Econ 1, they teach you about all of the basic tools of economics (usually both micro and macro). One of these tools is called elasticity. Put simply, elasticity is the sensitivity of one variable to changes in another variable. The most common type of elasticity that economists refer to is the price elasticity of demand. Price elasticity of demand is expressed as:

E = (ΔQ/Q) / (ΔP/P)

Where Q is quantity demanded and P is price. This is a cool little formula, and it can tell you a lot about the market for a good. Demand for items that are price inelastic (where elasticity is less than one) are not very sensitive to changes in price. A good example here would be gasoline. Demand for items that are price elastic (where elasticity is greater than one) are particularly sensitive to changes in price. A good example would be overseas vacations.

So, as I said, my mind wanders. And this was definitely true in Econ 1. I began to think about the relationship between the slope of the demand curve (which just plots price vs. quantity demanded of a good) and price elasticity of demand. It should be clear that they are not the same, but it should also be clear that they are related. When you do economics with calculus, it turns out the right answer for price elasticity of demand as a derivative is:

via upload.wikimedia.org

This formula just tells us that elasticity is the slope of the demand curve times the ratio of price to quantity (the negative sign is just a convention, you can more or less ignore it). What do we learn from this exercise? We learn that, although elasticity often looks like a straight line (because you can read it from the often-linear demand curve), it is in fact not a linear formula at all. In fact it's sort of messy. So you can have different elasticities for different starting points on the demand curve, even though the demand curve itself is perfectly linear.

Groundballs

Now that I've thoroughly confused you, let's turn to groundball ratios.

Groundball rates are typically reported in two seemingly similar ways. The first is groundball rate (or GB%), which is just the percentage of all balls in play that were on the ground. You can think of this as equivalent to K% or BB%. 

The second way is as a ratio with fly balls. This, more common, formulation is expressed simply as GB/FB. 

Dave Cameron, over at Fangraphs, takes issue with the latter metric:

Take Lenny DiNardo, for instance. He returned to the majors last week, being called up by Kansas City to fill a hole in their rotation down the stretch. DiNardo’s primary skill has always been an ability to rack up groundballs, and his first start of 2009 was no exception. In fact, if we looked at his 8.0 G/F rate, we would think that hitters were pounding the ball into the ground all day.

They weren’t, however. There’s an additional batted ball type – line drives, and DiNardo gave up six of them. When you include those in the balls in play denominator, his GB% is 53.3%, which highlights the fact that he got a majority of groundballs but doesn’t suggest the same crazy performance than an 8.0 G/F rate implies.

This is a good point. Line drives matter. So do infield flies, for that matter.

Building off of Cameron's point, Tom Tango jumped in to dump on GB/FB for a different reason, though:

My bigger problem is the non-symmetry of a ratio. For example, if the "average" GB/FB ratio is 1.1, then if you have a linear equation that uses GB/FB ratio, it will give equal impact to a ratio that is 2.0 or 0.1 (both are 0.9 "ratios" from the average). As you can see, this is mathematical nonsense. If the average GB/FB ratio is 1.1, then the GB/(GB+FB) percentage is 52.4%. And a 2.0 GB/FB ratio is a 66.7% percentage. Symetrically, that’s a GB/(GB+FB) percentage of 38%, or a GB/FB ratio is 0.6. That is, a 2.0 GB/FB ratio is as far away from 1.1, as 0.6 is from 1.1.

Using GB/FB in a linear formula would definitely be a big mistake. I'd like to illustrate Tango's point graphically. (You'll forgive me for not having the wizardry of Justin and Walter.) Here is GB/FB versus GB/(GB+FB):

Now, it shouldn't be surprising that there is a limit on GB/(GB+FB). In fact this function has an asymptote at 1. But that is what we want--groundballs aren't the kind of result you can keep racking up. The best a pitcher can do is have every ball in play be a groundball, which would be 100% GB%. 

All we're saying here is that GB/FB isn't linear. A 0.5 mark isn't half as good as a 1.0 mark. And a 2.0 mark isn't twice as good as a 1.0 mark. If you're curious, the equation for the simplified formula graphed above is:

GB/(GB+FB) = (GB/FB) / (1 + GB/FB)

Anyway, all this reminds me of the lesson I learned from elasticity. It may look as though GB/FB is a reliable statistic that can be easily pigeonholed and compared, but it isn't. That doesn't mean it's not useful, it just means the math is a little more complicated. 

Discussion Question of the Day

What explanations can we come up with for why GB/FB has persisted so much?