The standard deviation 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 In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...

where is the sample (formally, realizations from the distribution of a random variable X) and is the sample mean. A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

The reason for this definition is 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. However, s estimates the population standard deviation σ with negative bias; that is, s tends to underestimate σ. In statistics, a biased estimator is one that for some reason on average over- or underestimates what is being estimated. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... In statistics, the difference between an estimators expected value and the true value of the parameter being estimated is called the bias. ...

A simple explanation why the square root of the sample variance is a biased estimator of the standard deviation is that the standard deviation is a nonlinear function of the variance, so the property of being unbiased does not carry over. This is because the operations of taking the mean and applying a function in general do not commute unless that function is linear, and, by definition, an estimate is unbiased when its mean equals to the quantity being estimated.

When the random variable is normally distributed, a minor correction exists to eliminate the bias. The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ...

To derive the correction, note that for normally distributed X, Cochran's theorem implies that has a chi distribution with n − 1 degrees of freedom. Consequently, In statistics, Cochrans theorem is used in the analysis of variance. ... In probability theory and statistics, the chi distribution is a continuous probability distribution. ...

where c₄ is a constant that depends on the sample size n as follows:

where is the gamma function. The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

Thus an unbiased estimator of σ is had by dividing s by c₄. Tables giving the value of c₄ for selected values of n may be found in most text books on statistical quality control. As n grows large it approaches 1, and even for smaller values the correction is minor. For example, for n = 10 the value of c₄ is about 0.9727. It is important to keep in mind this correction only produces an unbiased estimator for normally distributed X. When this condition is satisfied, another result about s involving c₄ is that the standard deviation of s is . The application of statistical techniques (SPC and Statistical Sampling) to control a product characteristic to ensure it meets product specifications or the engineering design specifications. ...

See also

• Estimation of covariance matrices
• Sample mean and covariance

In multivariate statistics, the importance of the Wishart distribution stems in part from the fact that it is the probability distribution of the maximum likelihood estimator of the covariance matrix of a multivariate normal distribution. ...

References

• What are Variables Control Charts?

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