Standard Deviation Distribution

The standard deviation distribution is the statistical distribution of the sample standard deviation

(1)

for independent observations taken from a population with a normal distribution, where is an integer and is the sample mean. The probability density function of is zero for and, for , is given by

(2)

where is a gamma function and is the fixed population variance,

(3)

(Kenney and Keeping 1951, pp. 161 and 171). The function is plotted above for (red), 4 (orange), ..., 10 (blue), and 12 (violet).

The mean is given by

			(4)
			(5)

where

(6)

(Kenney and Keeping 1951, p. 171). In the usual statistical process control convention, , where is the bias factor for the sample standard deviation with denominator (Duncan 1986, pp. 62 and 134; National Institute of Standards and Technology). Romanovsky (1925) gave the asymptotic expansion, as ,

(7)

(OEIS A088801 and A088802, Romanovsky 1925, Pearson 1935; Kenney and Keeping 1951, p. 171).

For real , the raw moments are given by

(8)

and the variance of is

			(9)
			(10)

is an unbiased estimator of (Kenney and Keeping 1951, p. 171).