> though. So outside about 14 sigmas you should be able to say the
> probability is below 10e-40. The problem is that if there are small
> deviations from "Gaussian-ness" way out on the wings of your distribution,
> the REAL probability of a certain result is not well approximated by the
> Error Function result.
This is a real good point - if we are assuming a Gaussian distribution, then
we are assuming the best case. The worst case is given by Tchebycheff's
theorem, which states that, given a probability distribution where only the
mean and standard deviation is known, then the probability that an
observation will fall more than x standard deviations from the mean is
bounded above *only* by 1/x^2. (It's a tight bound for one value of x, but
with a very unlikely distribution). In other words, if you have no
guarantees about the distribution, "counting sigmas" is going to give you a
false sense of security, and if the distribution is even slightly deviant
from Gaussian, then the result can be very wrong indeed.
However we do have a little comfort. George's function - maximum convolution
error from a single iteration - does have some distribution. It "appears"
bell-shaped, and looks like a normal distribution, the samples George has
made are also reasonably-sized. Not only that, each observation is actually
the maximum of the deviations from integer over many channels in the FFT -
in other words, the observations are already coming from a "smoothed"
distribution. I'm sure the distribution of a maximum of several observations
is something which there is a statistical test for. (The resulting
distribution may not be normal, but may have an analogous test).
Thanks too Todd for backing up the heuristic calculation of "14 sigmas" that
I got the hard way :)
Chris
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