One can also look more closely at the error distribution for the standard deviation by generating m datasets of n random numbers for a known probability distribution and compute a standard deviation for each dataset.
The computed distribution assuming the standard deviation of the sum is reduced by the square root of n gives a calculated distribution that is slightly offset from the observed distribution.
Using Bessel's correction and assuming two additional degrees of freedom gives a better fit to the observed distribution. The observed distribution appears to have a slightly different shape with a small reduction in the peak value and slightly larger values at the sides sides indicating a broadening of the peak. The sum of the probabilities for histogram intervals in both cases is 1 as expected. One has to use a large number of datasets to notice the differences.
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