Unbiased estimation of standard deviation
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In statistics, 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 explanation why the square root of the sample variance is a biased estimator of the standard deviation is that the square root is a nonlinear function, and only linear functions commute with taking the mean. 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.
Bias correction
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](/images/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 
.
See also
References
- What are Variables Control Charts?
- Douglas C. Montgomery and George C. Runger, Applied Statistics and Probability for Engineers, 3rd edition, Wiley and sons, 2003. (see Sections 7-2.2 and 16-5)
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