A generalization of the normal error law

Abstract

The author replaces the classical distribution function of the Laplace-Gauss law by the function

$$\eta (x) = \frac{a}{{e^{\alpha ^2 x^2 } - b}},$$

which corresponds to Brillouin's generalized quantum statistics and gives a detailed method for determining the constants a, α and b. Analyzing 14 sets of different measurements he shows that the discrepancy existing between these measurements and the normal Gauss distribution can be brought into agreement with the above function by giving b the values between −5 and +0.71. This is witnessed by the calculation of the probable error for which the quantum distribution function in agreement with experiment gives values which differ by up to 16% from those of the classical error law.