ON THE DISTRIBUTION OF Z-SCORES

Abstract

Let X ,..., Xn 1 be a random sample from a distribution with sample mean X and samplevariance S 2. In this paper we consider certain very general properties of the so-called “Z-scores"X X S i n i ( − )/ : = 1,...., . A representation theorem is then given for Z-scores obtained from an underlyingnormal population, together with a theorem for their limiting distribution as the sample size tends to infinity.Finally, two applications involving grading and testing for an outlier are presented.