StandardDeviationDistribution

Consider the sample standard deviation

 s=sqrt(1/Nsum_(i=1)^N(x_i-x^_)^2)

(1)

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

 f_N(s)=2((N/(2sigma^2))^((N-1)/2))/(Gamma(1/2(N-1)))e^(-Ns^2/(2sigma^2))s^(N-2),

(2)

where Gamma(z)
is a gamma function and

 sigma^2=(Ns^2)/(N-1)

(3)

(Kenney and Keeping 1951, pp. 161 and 171). The function f_N(s) is plotted above for N=2 (red), 4 (orange), …, 10 (blue), and 12 (violet).

The mean is given by

where

 b(N)=sqrt(2/N)(Gamma(N/2))/(Gamma((N-1)/2))

(6)

(Kenney and Keeping 1951, p. 171). The function b(N) is known as c_4 in statistical process control (Duncan 1986, pp. 62
and 134). Romanovsky showed that

 b(N)=1-3/(4N)-7/(32N^2)-9/(128N^3)+...

(7)

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

The raw moments are given by

 mu_r^'=(2/N)^(r/2)(Gamma((N-1+r)/2))/(Gamma((N-1)/2))sigma^r,

(8)

and the variance of s is

s/b(N)
is an unbiased estimator of sigma (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: 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.”

Referenced
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Standard Deviation Distribution

Cite this as:

Weisstein, Eric W. “Standard Deviation Distribution.”
From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/StandardDeviationDistribution.html

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