Why Two Sigma? A Theoretical Justification

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.