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Sunday, March 7, 2010


This blog post has absolutely nothing to do with politics, philosophy, economics, life, the meaning of the universe, or even Paris. It is about Stephen Jay Gould, Joe Dimaggio, statistics, and streaks. It was inspired by the fact that I am currently on a twenty-one game winning streak of Spider Solitaire on my computer, which reminded me of something that has been eating away at me for years. Do you ever shout at the tv screen when someone says something with which you disagree? Do you ever wish that you could reach through the pages of a book and grab Plato or Macchiavelli or Jane Austen by the collar and say, "Now, listen!"? Well, then you may understand why I am writing this post. Even when he was alive, there was no chance that I would ever have a one-on-one with Gould, although I did hear him speak once. Now that he is long gone, I am left with my daydreams -- and with my blog. So here goes.

Stephan Jay Gould, now tragically and much too soon deceased, was a marvelous biologist and science writer, tenured at Harvard along with Richard Lewontin and E. O. Wilson. He was, among many other things, a fanatic baseball lover. In 1988, he wrote a review for the NY Review of Books of a book about Joe Dimaggio's famous 1941 streak -- hitting safely in fifty-six straight games. You can find the review here:

In the course of the review, Gould, drawing on detailed analyses by a number of other famous scientists and mathematicians who were also sports fans, undertook to debunk the popular view -- a myth, Gould claims -- that hitters, basketball players, and other professional sports greats "get hot," "get in the zone," and then are able to sink strings of baskets or get strings of hits or whatever because they are momentarily elevated to a higher place of sports excellence. None of this is true, Gould insists. The occurrence of runs or streaks is -- with the singular exception of Dimaggio's streak -- nothing more nor less than what statistical probability predicts will happen.

Now, I learned long ago that it is almost always a mistake for an amateur to try to tell a professional his or her business. If I shop for antiques occasionally on a Sunday, it is not likely that I am going to spot a valuable piece of furniture in a shop whose owner -- who buys and sells furniture all day long for a living -- has failed to appraise correctly. Inasmuch I could not sink a set shot even if Shaq picked me up and held me over the basket, I would think twice about telling Michael Jordan that he is wrong when he says that he was "in the zone" and couldn't miss. When professional athletes claim that they are sometimes hot or are in the zone, we fans ought at the very least to accord their report some evidentiary value.

The core of Gould's argument is a fact well known to mathematicians but usually misunderstood by the innumerate, namely that in random distributions of some property in a sequence, runs or streaks are much more common than one might intuitively anticipate. So, if you are a .300 hitter, Gould says, sheer chance dictates that every so often you will have a hitting streak that strikes everyone in the stands but the statisticians as a sign that you are "hot." Not a bit of it, Gould replies. That is just what you would expect of a .300 hitter. Obviously, he observes, a .250 hitter will have such a streak less often. The player's claim that he got the hits because he was hot or in the zone is just false, for all his conviction.

I think that this argument [you have to read Gould's lovely review to get all the details] is based on a rather simple but fundamental logical error. [Now you see why I wanted to reach through the page and grab him by the lapel.] Here is the point: the statistical argument is based on the assumption that there is such a thing as being a .300 hitter, which is logically different from being a batter who hits .300. Being a .300 hitter, for the purposes of the statistical analysis, is like being a bag of marbles 30% of which are white and 70% of which are black. If you repeatedly choose marbles from the bag at random, you will indeed get some long runs of white marbles more often than intuition tells you that you ought to. That is the sum and substance of Gould's point.

But being a .300 hitter is not like being six feet tall or having naturally curly hair. It is not a property you have prior to, and independently of, playing the game of baseball. It is at least possible -- and the pros repeatedly claim that it is true -- that the great players have the ability to concentrate, shut out every irrelevant stimulus, sharpen their reflexes, and raise their level of performance. That ability is a part of what makes them great players.

There is a great story about Ty Cobb that illustrates this point. At the age of sixty, Cobb participated in an Oldtimers' Game, organized as a feel good exercise in fan adoration. When Cobb stepped to the plate to hit, he turned to the oldtimer who was catching and politely asked him to step back a bit, because he could not trust his hands to keep hold of the bat and he didn't want to hurt him. The catcher stepped back a few paces, whereupon Cobb laid down a perfect bunt and beat it out to first. Even in an exhibition, he cared more about winning than about anything else. [It is that feature of Cobb's character that leads the ghostly players in Field of Dreams to vote not to ask him to join them in their pick-up games on Kevin Costner's old-time field.]

The players themselves claim that there are times when they can call on that extra level of performance, and times when they cannot. Furthermore, it is at least plausible that an initial statistically random series of successes -- hits, or baskets -- can trigger that extra effort in them, so that they feel themselves to be in the zone. That is what can enable to hitter whose natural talents might lead a talent scout to expect a .280 batting average to pull himself up to .300. If we could measure a player's "natural" batting average, just as we can count the proportion of white balls in a sack, and then compare it with the average he actually achieves, we could judge whether he is "falling short," "living up to his potential," or "developing a hot bat." But in fact neither we nor Gould can carry out that measurement ex ante. So we are left to judge a player's
ability" by his batting average ex post, and that simply does not permit us to test the claim that he sometimes "gets hot." We are left to rely on the reports of the people who really know best, namely the players themselves.

So, Stephen Jay, if there is a heaven for wonderful writers, I hope you are reading this. Dimaggio's streak is still a marvel, but I insist that sometimes Michael had hot hands.


Unknown said...

On the contrary, this issue is of great importance, related to the ability to know in advance the probability of a particular event.

As I understand it, you make a such a connection in your recent blog, "HCR, The Underlying Issue" of 2/25/10. It is also relevant to the "precautionary principle" and identification of a prudent course of action related to global warming. That is, how are we to know a priori whether global warming is induced by human activity, and to decide how much to invest in reducing CO2 accumulation in the atmosphere.

So instructions for your readers in probability are quite relevant, you see! :-)

As are any corrections to the above comments.

Arthur Mervyn said...

I recently read The Drunkard's Walk: How Randomness Rules Our Lives, by Leonard Mlodinow, which discusses this topic (with due credit to Gould's original discussion) and other misconceptions about randomness, chance, and probability. A fascinating, quick read for the layman.

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