Standard Deviation Distribution

Consider the sample standard deviation

(1)

for  samples taken from a population with a normal distribution. The distribution of  is then given by

(2)

where  is a gamma function and

(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). The function  is known as  in statistical process control (Duncan 1986, pp. 62 and 134). Romanovsky showed that

(7)

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

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).

REFERENCES:

Duncan, A. J. Quality Control and Industrial Statistics, 5th ed. New York: McGraw-Hill, 1986.

Kenney, J. F. and Keeping, E. S. "The Distribution of the Standard Deviation." §7.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 170-173, 1951.

Pearson, E. The Application of Statistical Methods to Industrial Standardization and Quality Control. British Standards House, 1935.

Romanovsky, V. "On the Moments of the Standard Deviation and of the Correlation Coefficient in Samples from Normal." Metron 5, 3-46, 1925.

Sloane, N. J. A. Sequences A088801 and A088802 in "The On-Line Encyclopedia of Integer Sequences."