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Adjusted sample variance

by , PhD

The adjusted sample variance is a measure of the dispersion of a sample around its mean. It is obtained by summing the squared deviations from the mean and dividing the result thus obtained by the number of observations minus one.

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It is also often called unbiased sample variance, because, under appropriate conditions, it is an unbiased estimator of the population variance.


It is defined as follows.

Definition Given a sample [eq1] of n observations, their adjusted sample variance is[eq2]where $\overline{x}_{n}$ is their sample mean:[eq3]

The adjective "adjusted" refers to the fact that the sum of squared deviations is divided by $n-1$ rather than by n.


Suppose you have observed the following sample of six observations:[eq4]Their sample mean is[eq5]Their unadjusted sample variance is[eq6]

More details

The lecture entitled Variance estimation presents more details about the adjusted sample variance, including its properties as an estimator of the population variance (unbiasedness, consistency, etc.). You can also find a brief introduction to another commonly used estimator of variance in the glossary entry entitled Unadjusted sample variance.

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