Abstract
Abstract. For a normal distribution, the probability density p(x) is everywhere positive, so in principle, all real numbers are possible. In reality, the probability that a random variable is far away from the mean is so small that this possibility can be often safely ignored. Usually, a small real number k is picked (e.g., 2 or 3); then, with a probability Po(k)≈ 1 (depending on k), the normally distributed random variable with mean a and standard deviation σ belongs to the interval \(a = [a - k\cdot\sigma ,a + k\cdot\sigma \)
The actual error distribution may be non-Gaussian; hence, the probability P(k) that a random variable belongs to a differs from Po(k). It is desirable to select k for which the dependence of P o (k) on the distribution is the smallest possible. Empirically, this dependence is the smallest for k є [1.5, 2.5]. In this paper, we give a theoretical explanation for this empirical result.
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Nguyen, H.T., Kreinovich, V., Solopchenko, G.N., Tao, CW. (2003). Why Two Sigma? A Theoretical Justification. In: Reznik, L., Kreinovich, V. (eds) Soft Computing in Measurement and Information Acquisition. Studies in Fuzziness and Soft Computing, vol 127. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36216-6_2
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DOI: https://doi.org/10.1007/978-3-540-36216-6_2
Publisher Name: Springer, Berlin, Heidelberg
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