Unbiased Estimation of Standard Deviation - Background

Background

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

s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}\,,

where is the sample (formally, realizations from a random variable X) and is the sample mean.

One way of seeing that this is a biased estimator of the standard deviation of the population is to start from the result that s2 is an unbiased estimator for the variance σ2 of the underlying population if that variance exists and the sample values are drawn independently with replacement. The square root is a nonlinear function, and only linear functions commute with taking the expectation. Since the square root is a concave function, it follows from Jensen's inequality that the square root of the sample variance is an underestimate.

The use of n − 1 instead of n in the formula for the sample variance is known as Bessel's correction, which corrects the bias in the estimation of the population variance, and some, but not all of the bias in the estimation of the sample standard deviation.

It is not possible to find an estimate of the standard deviation which is unbiased for all population distributions, as the bias depends on the particular distribution. Much of the following relates to estimation assuming a normal distribution.

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