*On the possible inequalities involving the mean, the median* *and the*

*mode in a divisible ordered abelian group*

**David E. Dobbs **

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA.

Pages 1-13 | Received 15 September 2022, Accepted 02 November 2022, Published 16 June 2023

**Abstract**

Let (G,\leq) be a nonzero additive divisible ordered abelian group. By a list L (of length n) in G , we mean a finite multiset x_1,\ldots ,x_n with each x_i \in G (and where possibly x_i = x_j for some i \neq j) for some n \geq 2 such that L has a unique mode. There are exactly 13 (pairwise distinct) possible continued inequalities involving the mean, the median and the mode, along with the relations < and/or = (such as â€śmean < median = mode”) that a given list in G may satisfy. For each of these 13 situations, it is proved that there exists a list in G that satisfies that inequality, the minimal length of such a list is determined and is at most 6, infinitely many lists in G are constructed that each satisfy that minimal length (for the given inequality), at least one of those lists also has the property that its minimal entry is any preassigned element of G, and the minimal length of a satisfying list (for the given inequality) is independent of G. Because of the first, second and fifth of these facts, the search for a suitable list (satisfying a given inequality) can be restricted to the case G = \mathbb{Q} (under addition). For that setting, a proof, programming assignment, or discovery activity could be carried out in courses at various levels.

**Keywords**: Mean, median, mode, ordered abelian group, torsionfree, divisible abelian group.

**MSC numbers**: Primary 06F20, 62A01; Secondary 13C11, 20K99.

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