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For correlated random variables the sample variance needs to be computed according to the Markov chain central limit theorem.. Independent and identically distributed random variables with random sample size
The standard deviation σ of X is defined as which can be shown to equal. Using words, the standard deviation is the square root of the variance of X . The standard deviation of a probability distribution is the same as that of a random variable having that distribution. Not all random variables have a standard deviation.
Unbiased estimation of standard deviation. 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 ...
The use of the term "error" as discussed in the sections above is in the sense of a deviation of a value from a hypothetical unobserved value. At least two other uses also occur in statistics, both referring to observable prediction errors :
where represents the standard deviation of the function , represents the standard deviation of , represents the standard deviation of , and so forth.. It is important to note that this formula is based on the linear characteristics of the gradient of and therefore it is a good estimation for the standard deviation of as long as ,,, … are small enough.
Huber-White standard errors assume is diagonal but that the diagonal value varies, while other types of standard errors (e.g. Newey–West, Moulton SEs, Conley spatial SEs) make other restrictions on the form of this matrix to reduce the number of parameters that the practitioner needs to estimate.
Therefore, a naïve algorithm to calculate the estimated variance is given by the following: Let n ← 0, Sum ← 0, SumSq ← 0. For each datum x : n ← n + 1. Sum ← Sum + x. SumSq ← SumSq + x × x. Var = (SumSq − (Sum × Sum) / n) / (n − 1) This algorithm can easily be adapted to compute the variance of a finite population: simply ...
Measurement uncertainty. In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a measured quantity. All measurements are subject to uncertainty and a measurement result is complete only when it is accompanied by a statement of the associated uncertainty, such as the standard deviation.