A Rich Learning Statistical Lesson in the Cauchy

Abstract

Cauchy distribution is a good example of a distribution that has many strange properties and is usually used as a counter example to show that some statistical properties are not always true.  In this paper, some hidden properties of the Cauchy distribution are highlighted and studied in details. The focus is to closely look at some very interesting properties of the Cauchy distribution that are rarely mentioned in classrooms. Mentioning these properties in classrooms will have a positive effect on students’ attitude toward learning. One of the main properties of the Cauchy distribution is that it has undefined or infinite moments; the odd moments are undefined while the even moments are infinite. For this distribution, the only way to summarize the data is by ordering them (order Statistics); it is the minimal sufficient statistic. The relation of Cauchy to other distributions such as the Normal, and Uniform are highlighted. The above properties several other properties are discussed in details in this paper. We believe that the content of this manuscript will be a real contribution to educational statistics.