Sample Variance

The sample variance m_2 (commonly written s^2 or sometimes s_N^2) is the second sample central moment and is defined by

where m=x^_ the sample mean and N is the sample size.

To estimate the population variance mu_2=sigma^2 from a sample of N elements with a priori unknown mean (i.e., the mean is estimated from the sample itself), we need an unbiased estimator mu^^_2 for mu_2. This estimator is given by k-statistic k_2, which is defined by

(Kenney and Keeping 1951, p. 189). Similarly, if N samples are taken from a distribution with underlying central moments mu_n, then the expected value of the observed sample variance m_2 is

Note that some authors (e.g., Zwillinger 1995, p. 603) prefer the definition

since this makes the sample variance an unbiased estimator for the population variance. The distinction between s_N^2 and s_(N-1)^2 is a common source of confusion, and extreme care should be exercised when consulting the literature to determine which convention is in use, especially since the uninformative notation s is commonly used for both. The unbiased sample variance s_(N-1)^2 is implemented as Variance [list].

Also note that, in general, sqrt(sigma^^^2) is not an unbiased estimator of the standard deviation sigma even if sigma^^^2 is an unbiased estimator for sigma^2.

See also

k-Statistic, Sample, Sample Central Moment, Sample Mean, Sample Size, Sample Variance Computation, Sample Variance Distribution, Standard Deviation, Unbiased Estimator, Variance

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References

Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, p. 16, 2000.Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.

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Sample Variance

Cite this as:

Weisstein, Eric W. "Sample Variance." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SampleVariance.html

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