Better Articulating Normal Curve Theory for Introductory Mathematical Statistics Students: Power Transformations and Their Back-Transformations

This article addresses a gap in many, if not all, introductory mathematical statistics textbooks, namely, transforming a random variable so that it better mimics a normal distribution. Virtually all such textbooks treat the subject of variable transformations, which furnishes a nice opportunity to introduce and study this transformation-to-normality topic, a topic students frequently encounter in subsequent applied statistics courses. Accordingly, this article reviews variable power transformations of the Box--Cox type within the context of normal curve theory, as well as addresses their corresponding back-transformations. It presents four theorems and a conjecture that furnish the basics needed to derive equivalent results for all nonnegative values of the Box--Cox power transformation exponent. Results are illustrated with the exponential random variable. This article also includes selected pedagogic tools created with R code.