How can I calculate cut-off points from a normal distribution?

Question

I'm trying to calculate the upper percentage points for the 0.99 percentile for samples drawn from a normal distribution, with a sample size of 500. How can I calculate the expected values for skewness, kurtosis, studentized range, which would be expected for a drawing of 500 values?

Answer 1

You can always simulate this kind of thing first, e.g. in R

set.seed(1)
samat <- matrix(rnorm(1000000), ncol=500)
means <- rowMeans(samat)
varis <- rowMeans((samat-means)^2)
skews <- rowMeans((samat-means)^3) / varis^(3/2)
kurts <- rowMeans((samat-means)^4) / varis^2  -  3

and find the quantiles

> quantile(means, probs=c(0.01, 0.5, 0.99) )
          1%          50%          99% 
-0.096977914 -0.001302285  0.109797211 
>
> quantile(varis, probs=c(0.01, 0.5, 0.99) )
       1%       50%       99% 
0.8494985 0.9970259 1.1495887 
>
> quantile(skews, probs=c(0.01, 0.5, 0.99) )
          1%          50%          99% 
-0.257665499 -0.002052328  0.260948808 
>
> quantile(kurts, probs=c(0.01, 0.5, 0.99) )
        1%        50%        99% 
-0.4286435 -0.0320487  0.5399296 

and while it is clear that you can probably only treat the first couple of decimal places as useful, it also shows that skewness of 0.60 would be extremely high. It also suggests that the "skewness 0.129" you quote is rather low for the 99th percentile; I make it nearer 0.35 or 0.36 for a sample of 250 and about 0.26 for a sample of 500.