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Population Standard Deviation

The population standard deviation of a real random variable X with finite second moment is

mu=E[X]
(1)
sigma=sqrt(E[(X-mu)^2]),
(2)

where E denotes the expectation value with respect to the statistical distribution of X. For a finite population x_1,...,x_N, this becomes

mu=1/Nsum_(i=1)^(N)x_i
(3)
sigma=sqrt(1/Nsum_(i=1)^(N)(x_i-mu)^2).
(4)

The divisor is N, in contrast to the divisor n-1 in the usual sample standard deviation.

See also

Expectation Value, Sample Standard Deviation, Standard Deviation, Variance

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References

Casella, G. and Berger, R. L. Statistical Inference, 2nd ed. Pacific Grove, CA: Duxbury, 2002.

Cite this as:

Weisstein, Eric W. "Population Standard Deviation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PopulationStandardDeviation.html

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