Why 95 % and Two Sigma ? A Theoretical Justi cation for anEmpirical Measurement

| The probability p(k) that the value of a random variable is far away from the mean (e.g. further than k standard deviations away) is so small that this possibility can be often safely ignored. It is desirable to select k for which the dependence of the probability p(k) on the distribution is the smallest possible. Empirically, this dependence is the smallest for k between 1.5 and 2.5. In this paper, we give a theoretical explanation for this empirical result. For many measuring instruments, the measurement error is normally distributed; see, e.g., 7, 8]. For a normal distribution, the probability density (x) is positive for all x, so in principle, all real numbers are possible. In reality, however, the probability of a random variable to be far away from the mean is so small that in many practical applications, this possibility can be safely ignored. So, the values of a normally distributed random variable are located, with a reasonably high probability, within a nite interval. To implement this idea, in practice, usually , a small real number k is picked (typically, k = 2 or k = 3). Then, with a probability P 0 (k) 1 (depending on k), the values of a normally distributed random variable with mean a and standard deviation belong to the interval a ? k ; a + k ]. For a normal distribution, this probability does not depend on a and , only on k. For k = 2, we have P 0 (k) 0:95; for k = 3, we have P 0 (k) 0:999; for k = 6, we have P 0 (k) 1 ? 10 ?6 , etc. In many real-life situations, the actual error distribution is close to Gaussian but not exactly normal 7, 8]. The deviation from a Gaussian distribution can be characterized by one or several parameters " (so that Gaussian distribution corresponds to " = 0). For a non-Gaussian distribution characterized by a parameter ", the probability P (k; ") that a random variable belongs to the interval a = a ? k ; a + k ] is, in general, diierent from P 0 (k). It is therefore desirable to select k for which the dependence of P (k; ") on " is the smallest possible, i.e., for which we can guarantee that P (2 a) P 0 (k) irrespective of whether is normally distributed or not. …