Standard Deviation Distribution
المؤلف:
Duncan, A. J
المصدر:
Quality Control and Industrial Statistics, 5th ed. New York: McGraw-Hill, 1986.
الجزء والصفحة:
...
26-2-2021
3468
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
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).
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."
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