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Dismal Rocket ScienceCrunch the numbers: Roger Clemens might be worth every penny.

By Jordan Ellenberg
Posted Tuesday, May 15, 2007, at 12:37 PM ET

(Continued from page 1)

But what we really need to know is how many runs Clemens won't give up—in other words, how many runs would Yankee opponents score in those 130 innings if the Bombers keep sending out the pitchers they're currently using? And now we come to the crux of the matter. The Yankees pitching, already thin, has been brutalized by injury, and the resulting parade of last-minute replacements to last-minute replacements was certainly a factor in Steinbrenner's willingness to open his wallet for a season-saving ace.

Then again, Chien-Ming Wang and Mike Mussina are already off the disabled list, and by the time Clemens is ready to pitch, only one, not all, of the replacements—Kei Igawa, Matt DeSalvo, Philip Hughes, Chase Wright, and Jeff Karstens—will be starting regularly for the Yankees. Some of these pitchers have shown great promise and one (Igawa) has a long record of success, at least in Japan. It's hard to imagine there won't be one of them who's maintaining, at worst, an ERA of 6.00.

If that's the case, that pitcher would give up 87 runs in the same 130 innings we're expecting from Clemens. Assume Clemens saves the Yankees 40 runs over the course of the season. How much does this boost the Yankees in the standings? For this, we need to use a remarkable tool developed by Bill James in the early days of sabermetrics, the Pythagorean formula. Though it has the same name as one, it isn't a mathematical theorem, but it's about as close as you get in baseball statistics—and it allows you to predict the effect a change in runs scored or runs allowed will have on winning percentage. James' formula reads

PCT = RS2 / (RS2 + RA2)

where PCT is the team's winning percentage, RS is runs scored, and RA is runs against.

This season, the Yankees are on pace to lead the league in runs scored with 900. If we take this 900-runs-scored figure as fixed, then for the Yankees to end up winning 86.6 games, they'd be expected to allow about 840 runs. Take away the 40 runs saved by Clemens and you would get a record of 90-72. In other words, Clemens would get the Yankees those four wins. Thus, in our admittedly rough approximation, Clemens would help the Yankees to the tune of just about what they're paying him.

Of course, whenever you carry out probabilistic computations like this, you're dealing with what Don Rumsfeld would call "known unknowns." If any one of the Yankee spare parts (or a new one picked up cheaply along the way) steps forward and pitches adequately, the benefit of having Clemens could be cut in half. (Though any serious injury to oldsters like Mike Mussina or Andy Pettitte could make adding Clemens seem genius.) If you think Davenport's simulation places too much weight on the Yankees' bad start and expect New York would have won 90 games without Clemens, that brings the break-even point down closer to three extra wins. And if you're certain, as some are, that Clemens would be pitching for division-rival Boston if he weren't in New York, you need to take that into account, too. All of these factors are hard to get a quantitative grasp on.

And then there's the fact that the dollar value of Clemens might not be what's ultimately in question. If Steinbrenner is willing to spend however much it takes to increase the Yankees' chances of winning the AL East, even by a fraction, he may simply not care all that much about the expected cash benefit of signing Clemens. Indeed, the Yankees, for all their on-field success and huge fan base, lost money last year. Adding one of the best pitchers in baseball certainly makes the Yankees better; whether it will make them richer remains to be seen.

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Jordan Ellenberg is an associate professor of mathematics at the University of Wisconsin. His first novel is The Grasshopper King.
Photograph of Roger Clemens by Jim McIsaac/Getty Images; photograph of Clemens on Slate's home page by Steve Nesius/AP.
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