Modes, Medians and Means: A Unifying Perspective

CRANK

Introduction / WarningAny traditional introductory statistics course will teach students the definitions of modes, medians and means. But, because introductory courses can’t assume that students have much mathematical maturity, the close relationship between these three summary statistics can’t be made clear. This post tries to remedy that situation by making it clear that all three concepts arise as specific parameterizations of a more general problem.To do so, I’ll need to introduce one non-standard definition that may trouble some readers. In order to simplify my exposition, let’s all agree to assume that \(0^0 = 0\). In particular, we’ll want to assume that \( 0 ^0 = 0\), even though \( \epsilon ^0 = 1\) for all \(\epsilon > 0\). This definition is non-standard, but it greatly simplifies what follows and emphasizes the conceptual unity of modes, medians and means.Constructing a Summary StatisticTo see how modes, medians and means arise, let’s assume that we have a list of nu…