Rich Ulrich wrote:
>
> >
> But what if I have a population of numbers that is made up of
> 1 billion draws from a Cauchy distribution? No one has ever
> defined this for me, and I have never tested it, but if you *have*
> a sample in hand, then you *can* compute a standard deviation
> even though in theory the variance is "undefined" or infinite.
>
> I does seem to me that you could take your billion numbers, and
> standardize them to the mean of 78 and SD of <whatever> and
> it won't change the nature, that the numbers belong to a Cauchy -
> and repeated sampling will *not* give a standard deviation-of-
> the-mean that is smaller than the standard deviation of the draw.
>
> I'm afraid that I am not giving an answer here - I am raising a
> question that I don't know the answer to.
>
> But -- without being Cauchy -- it must be true that you can always
> imagine an outlier extreme enough that your means will not
> be normal, or be very useful compared to some other description.
*Numbers* do not belong to any distribution whatsoever. Numbers in the
context of a method of generation do.
If you standardize your billion numbers to a specified mean and
standard deviation, and then apply the same scaling to further samples
the standard deviations of your other samples will not resemble that of
your first sample. Resampling from the original set *will* give SD's
that converge to that of the original if the derived samples are large
enough.
-Robert Dawson
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