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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.
An estimate of the standard deviation for N > 100 data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values R represents four standard deviations so that s ≈ R/4.
Firstly, while the sample variance (using Bessel's correction) is an unbiased estimator of the population variance, its square root, the sample standard deviation, is a biased estimate of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality.
The theory of median-unbiased estimators was revived by George W. Brown in 1947: [8]. An estimate of a one-dimensional parameter θ will be said to be median-unbiased, if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates.
A conventional choice is to add noise with a standard deviation of / for a sample size n; this noise is often drawn from a Student-t distribution with n-1 degrees of freedom. [47] This results in an approximately-unbiased estimator for the variance of the sample mean. [48]
Gurland and Tripathi (1971) provide a correction and equation for this effect. [4] ... See also unbiased estimation of standard deviation for more discussion. See also
To estimate μ based on the first n observations, one can use the sample mean: T n = (X 1 + ... + X n)/n. This defines a sequence of estimators, indexed by the sample size n . From the properties of the normal distribution, we know the sampling distribution of this statistic: T n is itself normally distributed, with mean μ and variance σ 2 / n .
The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n − 1.5 yields an almost unbiased estimator. The unbiased sample variance is a U-statistic for the function f ( y 1 , y 2 ) = ( y 1 − y 2 ) 2 /2 , meaning that it is obtained by averaging a 2-sample statistic ...