{"id":1685,"date":"2023-06-14T11:33:21","date_gmt":"2023-06-14T11:33:21","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=1685"},"modified":"2023-10-12T21:59:53","modified_gmt":"2023-10-12T21:59:53","slug":"on-the-possible-inequalities-involving-the-mean-the-median-and-themode-in-a-divisible-ordered-abelian-group","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/on-the-possible-inequalities-involving-the-mean-the-median-and-themode-in-a-divisible-ordered-abelian-group\/","title":{"rendered":"On the possible inequalities involving the mean, the median and the mode in a divisible ordered abelian group"},"content":{"rendered":"\n
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Moroccan Journal of Algebra and Geometry with Applications<\/strong><\/p>\n\n\n\n

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On the possible inequalities involving the mean, the median<\/em> and the<\/em>
mode in a divisible ordered abelian group<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

David E. Dobbs <\/strong><\/i>
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA.<\/p>\n\n\n\n

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Pages 1-13 | Received 15 September 2022, Accepted 02 November 2022, Published 16 June 2023 <\/span><\/p>\n\n\n\n

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Abstract<\/span><\/strong><\/p>\n\n\n\n

Let (G,\\leq) <\/span> be a nonzero additive divisible ordered abelian group. By a list L<\/span> (of length n<\/span>) in G <\/span>, we mean a finite multiset x_1,\\ldots ,x_n<\/span> with each x_i \\in G<\/span> (and where possibly x_i = x_j<\/span> for some i \\neq j<\/span>) for some n \\geq 2<\/span> such that L<\/span> has a unique mode. There are exactly 13<\/span> (pairwise distinct) possible continued inequalities involving the mean, the median and the mode, along with the relations <<\/span> and\/or =<\/span> (such as \u201cmean < median = mode”) that a given list in G<\/span> may satisfy. For each of these 13 situations, it is proved that there exists a list in G<\/span> 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 (<\/span>for the given inequality)<\/span>, at least one of those lists also has the property that its minimal entry is any preassigned element of G<\/span>, and the minimal length of a satisfying list (for the given inequality) is independent of G<\/span>. Because of the first, second and fifth of these facts, the search for a suitable list (<\/span>satisfying a given inequality)<\/span> can be restricted to the case G = \\mathbb{Q}<\/span> (<\/span>under addition)<\/span>. For that setting, a proof, programming assignment, or discovery activity could be carried out in courses at various levels. <\/p>\n\n\n\n

Keywords<\/strong>:<\/span>  Mean, median, mode, ordered abelian group, torsionfree, divisible abelian group. <\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 06F20, 62A01; Secondary 13C11, 20K99.<\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles 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 Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-page-builder.php","meta":{"footnotes":""},"class_list":["post-1685","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1685","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/comments?post=1685"}],"version-history":[{"count":18,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1685\/revisions"}],"predecessor-version":[{"id":1977,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1685\/revisions\/1977"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=1685"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}