Modes, Medians and Means: A Unifying Perspective

Any 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 00=0

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${0}^{0}=0$

. In particular, we’ll want to assume that |0|0=0

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$|0{|}^{0}=0$

, even though |ϵ|0=1

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$|ϵ{|}^{0}=1$

for all ϵ>0

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$ϵ>0$

. This definition is non-standard, but it greatly simplifies what follows and emphasizes the conceptual unity of modes, medians and means.

Constructing a Summary Statistic

To see how modes, medians and means arise, let’s assume that we have a list of numbers, (x1,x2,,xn)

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$\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$

, that we want to summarize. We want our summary to be a single number, which we’ll call s

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$s$

. How should we select s

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$s$

so that it summarizes the numbers,  (x1,x2,,xn)

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$\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$

, effectively?

To answer that, we’ll assume that s

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$s$

is an effective summary of the entire list if the typical discrepancy between s

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$s$

and each of the xi

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${x}_{i}$

is small. With that assumption in place, we only need to do two things: (1) define the notion of discrepancy between two numbers, xi

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${x}_{i}$

and s

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$s$

; and (2) define the notion of a typical discrepancy. Because each number xi

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${x}_{i}$

produces its own discrepancy, we’ll need to introduce a method for aggregating the individual discrepancies to order to say something about the typical discrepancy.