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In the empirical sciences, the so-called three-sigma rule of thumb (or 3 σ rule) expresses a conventional heuristic that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% probability as near certainty.
Most analytical instruments produce a signal even when a blank (matrix without analyte) is analyzed.This signal is referred to as the noise level. The instrument detection limit (IDL) is the analyte concentration that is required to produce a signal greater than three times the standard deviation of the noise level.
The theorem refines Chebyshev's inequality by including the factor of 4/9, made possible by the condition that the distribution be unimodal. It is common, in the construction of control charts and other statistical heuristics, to set λ = 3 , corresponding to an upper probability bound of 4/81= 0.04938..., and to construct 3-sigma limits to ...
A little algebra shows that the distance between P and M (which is the same as the orthogonal distance between P and the line L) (¯) is equal to the standard deviation of the vector (x 1, x 2, x 3), multiplied by the square root of the number of dimensions of the vector (3 in this case).
A vector X ∈ R k is multivariate-normally distributed if any linear combination of its components Σ k j=1 a j X j has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions.
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In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori Milstein who first published it in 1974. [1] [2]