Statistical Wisdom and Weldon’s Dice

I went to a library talk, by my friend Jonathan Skinner, reviewing a book, The Seven Pillars of Statistical Wisdom, by Stephen Stigler. Jonathan was a professional statistician. One thing I enjoyed was his quoting Christopher Hitchens: “What can be asserted without evidence can also be dismissed without evidence.”

I also learned how the Arctic and Antarctic got their names. Skinner said Aristotle named them, based on the Greek word for “bear.” That surprised me; could Aristotle have been aware of the poles’ existence? And how could he have known about polar bears? But when I mentioned this to my (smarter) wife, she suggested the “bear” reference was to a constellation. I checked Wikipedia and while the Greek origin is correct, there was no mention of Aristotle. And of course my wife was right.
Skinner discussed a basic statistical concept: probability. He talked about it in connection with dice as an example. This reminded me of the Tom Stoppard play, Rosencrantz and Guildenstern Are Dead, and Guildenstern’s repeated coin flips. They come up heads every time. Not statistically impossible, but increasingly unlikely as the number of flips mounts. Stoppard is jesting with the laws of probability.

Of course they tell us heads and tails should be 50-50. But I also remembered a guy who wrote in to Numismatic News, doubting that theory, and reporting his own test. He flipped a coin 600 times and got 496 heads! Of course, the probability of that result is not zero. But I actually calculated it, and the answer is one divided by 6.672 times 10 to the 61st power. For readers not mathematically inclined, that’s an exceedingly tiny probability. Ten to the 61st power means 1 followed by 61 zeroes.

However, that guy, as if to flaunt his scientific rigor, explained his procedure: on each of his 600 tosses, he methodically started with the coin in the heads-up position, and then . . . well, enough said.

But Skinner related a similar tale, of Frank Weldon who (in 1894) really did try to put the theory to a rigorous test. He rolled a dozen dice 26,306 times, and recorded the results. That huge effort would make him either a martyr to science, or a fool (like the Numismatic News guy) because, after all, what is there to test? Is there any sane reason to doubt what such a simple probability calculation dictates?

However, Skinner quoted Yogi Berra: “In theory, theory and practice are the same. In practice they are not.”

Well, guess what. Weldon found the numbers five and six over-represented. With six faces to each die, you should expect any two numbers to come up one-third of the time, or 33.33%. But Weldon got 33.77%. You might think that’s a minor deviation, down to random chance. But statisticians have mathematical tools to test for that, i.e., whether a result is “statistically significant.” And the odds against Weldon’s result were calculated to be 64,499 to one.

So another fool (er, researcher), Zacariah Labby, decided to repeat Weldon’s experiment, but this time using machinery to roll the dice, and a computer to tabulate the results. He got 33.43%, a smaller deviation, but still significantly significant.

How can this be explained? It had been suggested that the small concave “pips” on the die faces denoting the numbers might affect the results. And then Labby measured his die faces with highly accurate equipment and found the dice were not absolutely perfect cubes.

But don’t rush out to a casino to try to capitalize on the Weldon/Labby deviation. Labby concluded his paper by noting that casinos use dice without concave pips and more precisely engineered, to scotch any such bias.
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