Tuesday, August 14, 2007

I do not think that word means what you think it means

As reported in the Financial Times:
"We were seeing things that were 25-standard deviation moves, several days in a row,” said David Viniar, Goldman’s chief financial officer.

By Chebyshev's inequality, no matter what the probability distribution, the fraction of points which lie within k standard deviations of the mean must be at least (1 - 1/k2). For a normal (Gaussian) distribution, the probability of a 25 standard deviation outlier is much less, so much so as to be negligible. But even for the less stringent bound, the probability of being 25 or more standard deviations from the mean is at most 1/625, or 0.0016. To do that for three days in a row is at most (1/625)3 = 1/244140625 = 4.096 * 10-9.

Put another way: There are about 250 trading days in a year (weekdays, excluding holidays). Even at the Chebyshev limit, it would take an expected million years or so to get three straight days of 25 standard deviation outliers. And Viniar's comment does not rule out a longer run of outliers.

Either these guys have botched the calculation, or they do not understand what standard deviation actually means.

(h/t Brad DeLong)

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