"Scorecasting" on players gunning for .300

A few months ago, I wrote about a study by two psychology researchers, Devin Pope and Uri Simonsohn. The study found that, for players hitting .299 in their last at-bat of the season, they wound up hitting well over .400 in that last at-bat. The authors concluded that it's because .299 hitters really want to get to .300, and, therefore, they try extra hard (and succeed).

But, really, that isn't the case. It's really just an illusion caused by selective sampling. When a player hitting .299 gets a hit to push him over .300, he is much more likely to be taken out (or held out) of the lineup, to preserve the .300. Therefore, it's not that they're more likely to get a hit in their last at-bat -- it's that their last at-bat is more likely to be one that results in a hit.

(For an analogy: when a game ends with less than 3 outs, the last batter probably hits well over .500 (since the winning run must have scored on the play). But that's not because the player rises to the situation; it's because, as it were, the situation rises to the player. When he gets a hit, he's the last batter because the game ends. When he doesn't, he's not the last batter.)

Since the original study and article, the authors have modified their paper a bit, saying that the batting average effect is "likely to be at least partially explained" by selective sampling. However, the data given in the previous posts does suggest that almost the *entire* effect is explained by selective sampling. (PDFs: Old paper; new paper.)

There is one part of the study's findings that's probably partially real, and that's the issue of walks. None of the .299 hitters walked in their last at-bat. That's partially selective sampling -- if they walked, they're still at .299, and stayed in the game, so it's not their last at-bat -- but probably partially real, in that .299 hitters were more likely to swing away.

(My results are in previous posts here and here.)

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The study is given featured status in "Scorecasting," in the chapter on round numbers. However, while the authors of the original paper mention the selective sampling issue, the authors of "Scorecasting" do not:

"What's more surprising is that when these .299 hitters swing away, they are remarkably successful. According to Pope and Simonsohn, in that final at-bat of the season, .299 hitters have hit almost .430. ... (Why, you might ask, don't *all* batters employ the same strategy of swinging wildly? ... if every batter swung away liberally throughout the season, pitchers would probably adjust accordingly and change their strategy to throw nothing but unhittable junk.) ...

"Another way to achieve a season-ending average of .300 is to hit the goal and then preserve it. Sure enough, players hitting .300 on the season's last day are much more likely to take the day off than are players hitting .299."

"Scorecasting" treats these two paragraphs as two separate effects. In reality, the second causes the first.

You can read an excerpt -- almost the entire thing, actually -- at Deadspin, here.

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One thing that interested me in the chapter was this:

"But no benchmark is more sacred than hitting .300 in a season. It's the line of demarcation between All-Stars and also-rans. It's often the first statistic cited when making a case for or against a position player in arbitration. Not surprisingly, it carries huge financial value. By our calculations, the difference between two otherwise comparable players, one hitting .299 and the other .300, can be as high as two percent of salary, or, given the average major league salary, $130,000."

The authors don't say how they calculated that, but it seems reasonable. A free-agent win is worth $4.5 million, according to Tom Tango and others. That means a run is worth $450,000. One point of batting average, in 500 AB, is turning half an out into half a hit. Assuming the average hit is worth about 0.6 runs and an out is worth negative 0.25 runs, that means the single point of batting average is worth a bit over 0.4 runs. That's close to $200,000.

That figure is higher than the authors' figure of $130,000. The difference is probably just that the authors used the average MLB salary, which includes players not yet free agents (arbs and slaves). However, they imply that the difference between .299 and .300 is worth more than other one-point differences. That might be true, but it would be nice to know how they figured it out and what they found.

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Finally, two bloggers weigh in. Tom Scocca, at Slate, criticizes the original study. Then, Christopher Shea, at the Wall Street Journal, criticizes Scocca.