Unbiased estimation of standard deviation

In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the use of significance tests and confidence intervals, or by using Bayesian analysis.

However, for statistical theory, it provides an exemplar problem in the context of estimation theory which is both simple to state and for which results cannot be obtained in closed form. It also provides an example where imposing the requirement for unbiased estimation might be seen as just adding inconvenience, with no real benefit.

In statistics, the standard deviation of a population of numbers is often estimated from a random sample drawn from the population. The most common measure used is the sample standard deviation, which is defined by

where ${\displaystyle \{x_{1},x_{2},\ldots ,x_{n}\}}$ is the sample (formally, realizations from a random variable X) and ${\displaystyle {\overline {x}}}$ is the sample mean.

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