ON THE DISTRIBUTION OF Z-SCORES

Document Type: Research Note

Authors

1 Department of Mathematics, Islamic Azad University, Shiraz, I. R. of Iran

2 Department of Statistics, Faculty of Basic Sciences, Mazandaran University, Babolsar, I. R. of Iran

Abstract

Let X ,..., Xn 1 be a random sample from a distribution with sample mean X and sample
variance 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 underlying
normal 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.

Keywords