MathGroup Archive 1997

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Re: statistics

<much interesting part snip>
>experiment seems to show that:
>Expected Mean                       -> m  +/- 1.00 s/Sqrt[n]
>Expected Sqrt(Variance)         -> s  +/- 0.38 s/Sqrt[n]
>Expected Skewness                  -> 0  +/- 2.32 1/Sqrt[n]
>Expected Kurtosis                   -> 0  +/- 4.31 1/Sqrt[n]
>If this assumption is correct (in form), then what are the 0.38, 2.32 and
>4.31  analytically?

Comment on the first two.  The first one is well known in sampling stat. 
The coefficient is 1 since the population is the whole real line which
makes population size infinite.  The second one is a bit tricky. 
Essentially this is chi-square dist, but you demand the stat for the Sqrt
of chi-square.  If you can live with the stat for Variance alone, then

Expected Variance -> s^2 +/- Sqrt(2/(n-1)) s^2

Now, I don't know if I can do this.  But the above can be transformed as
below, using the well known formula Sqrt(1+x)=1+(1/2)x+... assuming small

Expected Sqrt(Variance) -> s +/- 1/Sqrt(2(n-1)) s

In this case, I get the numerical coefficient 1/Sqrt(2)=0.7.  The apparent
discrepancy comes from my expanding the Sqrt(1+x) above.

To be a real masochist, however, and to carry out the computation of the
third and fourth, you need to determine the Jacobian and integrate it to
find out the stat's dist function.  Then further integrate it with the stat
as the integrand (you get the mean) and then with (x-u)^2 (you get the
variance for the stat x).

I don't know how efficiently you can do this with Mathematica.  My feeling
is that you will be stomped already at the second case.  Try the expected
Variance (not its Sqrt) to see if it agrees with the analytic sol'n above. 
And make sure you use unbiased def for s.

Reference: Statistics for Physicists, BR Martin, Acedemic Press, 1971.

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