Durbin’s infinite ERA

A pitcher’s Earned Run Average for a given game is defined as runs earned against him divided by the fraction of a game that he pitched.

To calculate a season ERA, sabrmatricians take the mean of this stat over all games played.

Phillies pitcher J.D. Durbin has a high ERA.

Over the course of the 2007 season, Phillies pitcher J.D. Durbin has accumluated an ERA of 6.27. But yesterday, Durbin accomplished something truly extraordinary. He gave up seven runs in the first inning without recording an out. In other words, he gave up seven runs without pitching one inning — or even a fraction of an inning.

Take another look at the formula above. Plug in seven runs, and zero innings. But wait: remember anything spooky from high school about dividing by zero?

You’ll never hear a mathematician say that Durbin’s ERA equals infinity; infinity is not even a real number. Instead, they’d say that for any integer N you care to choose, Durbin’s ERA for that game is larger than N.

So Durbin’s ERA is larger than N = 100,000,000.

It is larger than N = 999,999,999,999,999,999,999,999,999.

It is even larger than O’s revliever Paul Shuey’s ERA from last week’s record setting 30-3 loss to the Rangers (40.5).

If we plotted Durbin’s ERA as a function of time, there would be a point way up at infinity at t = yesterday’s game. Mathematicians would say that at t = yesterday’s game, his ERA “blows up”.

Luckily for Durbin, the Lebesgue theory of integration says that an integral can still have finite value even if it blows up at certain times, as long as those times occur on “sets of measure zero.” Since it appears that we count innings as discrete units, Durbin’s recent pitching debacle indeed occurred on such a set. Thus he owes a few beers to turn-of-the-century french mathematicians for his bloated yet finite 6.27.

Suz Tolwinski, who is getting her PhD in applied mathematics at the University of Arizona and is also my girlfriend, did most of the heavy lifting for this post.