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Unbiased estimation of standard deviation
From Wikipedia, the free encyclopedia
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
where 
is the sample (formally, realizations from a random variable X) and 
is the sample mean.
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.
An intuitive 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. 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,
![\operatorname{E}[s] = c_4\sigma](../../../math/5/0/4/50488b095edad9805e29de45dce969be.png)
where c4 is a constant that depends on the sample size n as follows:
and 
is the gamma function.
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 textbooks 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 
.
This article incorporates text from the National Institute of Standards and Technology, which is in the public domain.