Fitting a Normal Distribution to Interval and Fuzzy Data

Abstract

In traditional statistical analysis, if we know that the distribution is normal, then the most popular way to estimate its mean a and standard deviation sigma from the data sample x1,..., xn is to equate a and sigma to the arithmetic mean and sample standard deviation of this sample. After this equation, we get the cumulative distribution function F(x) = phi (x-a/sigma) of the desired distribution. In many practical situations, we only know intervals [xi, xi] that contain the actual (unknown) values of xi or, more generally, a fuzzy number that describes xt. Different values of xt lead, in general, to different values of F(x). In this paper, we show how to compute, for every x, the resulting interval [F_(x),F(x)] of possible values of F(x) -or the corresponding fuzzy numbers.