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
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
is an unbiased estimator of (Kenney and Keeping 1951, p. 171).
See also
Sample Variance,
Sample
Variance Distribution,
Standard Deviation
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References
Duncan, A. J. Quality Control and Industrial Statistics, 5th ed. New York: McGrawHill, 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. 170173,
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, 346, 1925.Sloane, N. J. A. Sequences
A088801 and A088802
in "The OnLine Encyclopedia of Integer Sequences."Referenced
on WolframAlpha
Standard Deviation Distribution
Cite this as:
Weisstein, Eric W. "Standard Deviation Distribution."
From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/StandardDeviationDistribution.html
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