Why Two Sigma ? A

For a normal distribution, the probability density (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 P0(k) 1 (depending on k), the normally distributed random variable with mean a and standard deviation belongs to the interval a = a ? k ; a + k ]. The actual error distribution may be non-Gaussian; hence, the probability P (k) that a random variable belongs to a diiers from P0(k). It is desirable to select k for which the dependence of P0(k) on the distribution is the smallest possible. Empirically, this dependence is the smallest for k 2 1:5; 2:5]. In this paper, we give a theoretical explanation for this empirical result.