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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.

ERA formula

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.

J.D. DurbinSo 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.

13 Responses to “Durbin’s infinite ERA”

  1. Paul Moro says:

    I never quite understood the practice of retiring numbers. Maybe this is just a personal taste, but I think it’s far more meaningful when players choose numbers that their idols wore. Even when Jackie Robinson’s # got retired, I didn’t understand why you would deprive players who wanted to pay homage to him by wearing 42. It’s just odd to me. It was basically just MLB saying “Hey, look! We’re not racist anymore! Now let’s just let bygones be bygones, shall we?”

  2. Outstanding work, Suz-n-Ward. Outstanding.

    Very intriguing stuff.

  3. I don’t see what the big deal is…There has definately been an occurence of pitchers throughout history to end a season with an era of infinity; mostly late-season callups who only faced a few batters, but still. And does it matter that Durbin allowed seven runs? (No batters retired) + (any amount of runs) equals an era of infinity, so this happens almost every day to a relief pitcher or two, and to starters once in a blue moon (which in itself is not that uncommon – the occurence of two full moons in the same month that is).

  4. Interesting read but seriously .. football season starts in 2 days so let’s put baseball to bed.

  5. Wilson Alvarez had a career ERA of “infinity” after his first career start in 1989 for the Rangers.

    Two years later, in his second career start (for the White Sox), he pitched a no-hitter.

  6. Any Twins fan reading this post will not be surprised.

  7. I think that technically Durbin has no ERA for the game, because calculating it involves dividing by zero, which is an invalid function. But I’ve never really understood math.

  8. I like the use of Lebesgue measure on a sports blog, but I would have to agree with kurtsy that this must happen all the time with relief pitchers. Also, plotting his instanteneous ERA versus time doesn’t really make sense, because it would always be either 0 or infinite. If you plot his cumulative ERA versus time then there would be jumps when runs were scored or outs were made, but points at infinity would pretty much only happen before his first out of the season.

  9. His ERA in yesterdays game was infinity. There, I said it. And I’m a mathematician.

  10. The point is not that Durbin is some sort of crazy anomaly, but that it’s really fun to make fun of bad pitching with math.

    Does anyone know to what degree statistical measures in baseball are standardized, by the way? ie. do SABR and BP (and any other relvant groups) define all their stats uniformly?

  11. Sarah Green says:

    Suz, yes, all the stats are standardized—that is to say, no matter which source you consult, any player N will have the same average A. (Or to put it in less math-y terms, all baseball statisticians will have someone, let’s call him Jerk McDouchebag hitting at, say, .287.)

    I don’t quite know how it works, I just rely on the brainiacs to come up with the numbers, on which I then slavishly rely. :)

    Thus I accept on blind faith that somewhere out there, some number-cruncher is coming up with a finite number for a pitcher’s ERA, even when that pitcher has recorded zero outs. How this strange alchemy occurs is as mysterious to me as the turning of water into wine. And I’m comfortable with that.

    I predict that some humorless nerd will now attempt to kill our buzz by actually answering this question (or suggesting that our baseball blog now talk about football, or unintentionally recapping what it says in the post, but in a dumber way). Humorless nerd, over to you!

  12. Well, actually, not all stats are standardized. Sarah is right that for most of the standard statistics, the formulas are the same, but not always. A good example is the “hold.” Two of the major stat companies (which supply media outlets with their stats) – SportsTicker and STATS Inc. – have competing definitions of what a hold is, so different sites and newspapers can award players different amounts of holds depending on whom they get their stats from (and annoyingly, generally don’t tell you which). With some of the newer Sabermetric stats, different web sites all have slightly different formulas. For example “runs created” totals will all be roughly the same, but slightly different depending on the exact formula used. But generally, for classic stats like .avg, .obp, .slg, etc, the formulas are set in stone, and a run or an RBI is always going to be 1=1.

  13. Sarah Green says:

    Holds and saves are the most craptastic stats ever. “If the tying run is on the top step of the dugout and the moon is full, then the pitcher shall be credited with a save.” Posh!

    Humorless nerd, took you long enough. :)

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