Walk Like a Sabermetrician: Calculus

Pickover Calculus and Pizza? I can’t decide.

What prompted this post was not a calculus barb directed at sabermetrics, but one of the reactions to such a barb in a Baseball Think Factory thread: a flat out statement that “calculus has no place in baseball statistics”. On one hand, I really should just ignore this. The statement itself is so outlandish as to be difficult to respond to. It’s akin to saying that “cymbals have no place in music” or that “rice has no place in one’s diet”. Calculus is obviously not used directly by most sabermetricians, and one can certainly be a practice high-level sabermetrics without using any calculus. But to simply write off the possibility of using an entire branch of mathematics in the discipline is absurd.

...I do not wish to give the impression that I think the application of calculus is central to the current practice of sabermetrics. Clearly it is not, given the paucity of work applying it to sabermetric questions. But it is another tool at our disposal, and one that is perfectly suited to assist in the types of sabermetric questions that have always interested me. Calculus certainly has vast applications in understanding the mathematical relationships between sabermetric formulas. Why can you predict team runs scored fairly accurately (at least in a normal team context) using a dynamic equation like Base Runs or a linear weights equation? Why does any variant of the Pythagorean family of win estimators match up so well in practice with linear equations that follow the rule that ten runs = one win? Calculus is also inherent in any sort of exercise involving hypothesis testing, even if it is only implicit. After all, the normal distribution is defined as an integral of a particular function.

I will close with a list of links to articles on this blog that have used calculus in some manner. As you will see, the scope of topics that I have applied calculus to are fairly limited—mostly to understand how events are valued in various offensive measures and to estimate runs per win from non-linear win estimators. Hopefully those of you with more imagination and a broader range of research interests can come up with other applications. Even if what I’ve written about did represent the full extent of possible applications of calculus in sabermetrics, it should be clear that there is a place for it. And if there wasn’t a place for a branch of mathematics which has countless applications in the sciences, statistics, and probability in sabermetrics, I’d suggest it would be time to re-evaluate how we practice sabermetrics.