{"id":15712,"date":"2022-04-22T13:14:25","date_gmt":"2022-04-22T12:14:25","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=15712"},"modified":"2022-04-22T14:27:11","modified_gmt":"2022-04-22T13:27:11","slug":"tutoriel-sur-la-moyenne-geometrique-et-harmonique","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/descriptive-analysis\/tutorial-on-geometric-and-harmonic-mean\/","title":{"rendered":"Geometric Mean and Harmonic Tutorial"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"15712\" class=\"elementor elementor-15712\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d011e01 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d011e01\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5755304\" data-id=\"5755304\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ea2fd87 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"ea2fd87\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/descriptive-analysis\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Descriptive analysis<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-804b560\" data-id=\"804b560\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-aa75127 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"aa75127\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Home page<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-21a319e\" data-id=\"21a319e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8e5291b elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"8e5291b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/fr.wikipedia.org\/wiki\/Statistique_descriptive\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">WIki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-25751ac elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"25751ac\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2008db9\" data-id=\"2008db9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c5483d8 elementor-widget elementor-widget-text-editor\" data-id=\"c5483d8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>This page shows several examples on the use cases of the different means, in particular the geometric mean and the harmonic mean.<\/p><p><img decoding=\"async\" class=\"aligncenter wp-image-11096 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/09\/cropped-Capture.png\" alt=\"geometric mean and harmonic mean\" width=\"97\" height=\"97\" title=\"\"><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b6767c2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b6767c2\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-43a78af\" data-id=\"43a78af\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-aa3c20b elementor-widget elementor-widget-heading\" data-id=\"aa3c20b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/descriptive-analysis\/tutorial-on-geometric-and-harmonic-mean\/#Moyenne-arithmetique\" >Arithmetic average<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/descriptive-analysis\/tutorial-on-geometric-and-harmonic-mean\/#La-moyenne-geometrique\" >The geometric mean<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/descriptive-analysis\/tutorial-on-geometric-and-harmonic-mean\/#Exemple-de-moyenne-geometrique\" >Example of geometric mean<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/en\/descriptive-analysis\/tutorial-on-geometric-and-harmonic-mean\/#Moyenne-geometrique-et-changement-dechelle\" >Geometric mean and scale change<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/complex-systems-ai.com\/en\/descriptive-analysis\/tutorial-on-geometric-and-harmonic-mean\/#Et-la-moyenne-harmonique\" >And the harmonic mean?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/complex-systems-ai.com\/en\/descriptive-analysis\/tutorial-on-geometric-and-harmonic-mean\/#Exemple-de-moyenne-harmonique\" >Example of harmonic mean<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Moyenne-arithmetique\"><\/span>Arithmetic average<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a85d084 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a85d084\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e3f56cd\" data-id=\"e3f56cd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6d9dfe4 elementor-widget elementor-widget-text-editor\" data-id=\"6d9dfe4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p id=\"af2e\" class=\"pw-post-body-paragraph lk ll jd lm b ln lo ke lp lq lr kh ls lt lu lv lw lx ly lz ma mb mc md me mf iw gc\" data-selectable-paragraph=\"\">The arithmetic mean is named appropriately: we find it by adding all the numbers in the dataset, then dividing by the number of numbers in the dataset (to bring the sum down to the scale of the numbers original).<\/p><blockquote class=\"nd ne nf\"><p id=\"e7a7\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">3 + 8 + 10 = 21<br \/>21 \u00f7 3 = 7<br \/><strong class=\"lm je\">Arithmetic mean<\/strong>\u00a0=\u00a0<strong class=\"lm je\">7<\/strong><\/code><\/p><\/blockquote><p id=\"68f0\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Notice, what we&#039;re basically saying here is: if every number in our dataset was the same number, what number would it have to be to have the same sum as our actual dataset?<\/p><p id=\"c0bb\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">But there&#039;s nothing particularly special about the addition. It&#039;s just a fairly simple mathematical operation. The arithmetic mean works well to produce an &quot;average&quot; number of a data set when there is an additive relationship between the numbers.<\/p><p class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Such a relationship is often called &quot;linear&quot;, because when graphed in ascending or descending order, numbers tend to fall on or around a straight line. A simple idealized example would be a dataset where each number is produced by adding 3 to the previous number:<\/p><blockquote class=\"nd ne nf\"><p id=\"e236\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">1, 4, 7, 10, 13, 16, 19...<\/code><\/p><\/blockquote><figure class=\"mi mj mk ml gz mm gn go paragraph-image\"><div class=\"gn go xh\"><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter size-medium wp-image-15715\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_nr24D4m2hTL7cGjg1uThyw-300x259.png\" alt=\"\" width=\"300\" height=\"259\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_nr24D4m2hTL7cGjg1uThyw-300x259.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_nr24D4m2hTL7cGjg1uThyw-14x12.png 14w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_nr24D4m2hTL7cGjg1uThyw.png 576w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div><\/figure><p id=\"ea4a\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">The arithmetic mean therefore gives us a perfectly reasonable median value:<\/p><blockquote class=\"nd ne nf\"><p id=\"ffe7\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">(1 + 4 + 7 + 10 + 13 + 16 + 19) \u00f7 7 =\u00a0<strong class=\"lm je\">10<\/strong><\/code><\/p><\/blockquote><figure class=\"mi mj mk ml gz mm gn go paragraph-image\"><div class=\"gn go xh\"><img decoding=\"async\" class=\"aligncenter size-medium wp-image-15716\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_iqmOO-Ar-2A9PnkXTQ-wgA-300x259.png\" alt=\"\" width=\"300\" height=\"259\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_iqmOO-Ar-2A9PnkXTQ-wgA-300x259.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_iqmOO-Ar-2A9PnkXTQ-wgA-14x12.png 14w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_iqmOO-Ar-2A9PnkXTQ-wgA.png 576w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div><\/figure><p id=\"3595\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">But not all data sets are better described by this linear regression. Some have a multiplicative or exponential relationship, for example if we multiply each consecutive number by 3 rather than adding by 3 as we did above:<\/p><blockquote class=\"nd ne nf\"><p id=\"16a7\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">1, 3, 9, 27, 81, 243, 729\u2026<\/code><\/p><\/blockquote><p id=\"0590\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">This produces what is called a geometric series. When drawn in order, these numbers look more like a curve than a straight line.<\/p><p id=\"bc28\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">In this situation, the arithmetic mean is ill-suited to produce an &quot;average&quot; number to summarize this data.<\/p><blockquote class=\"nd ne nf\"><p id=\"f953\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">(1 + 3 + 9 + 27 + 81 + 243 + 729) \u00f7 7 =\u00a0<strong class=\"lm je\">156.1<\/strong><\/code><\/p><\/blockquote><figure class=\"mi mj mk ml gz mm gn go paragraph-image\"><div class=\"gn go xh\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-15717\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_3FQYgskPEfwodxSVgrGMlg-300x254.png\" alt=\"\" width=\"300\" height=\"254\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_3FQYgskPEfwodxSVgrGMlg-300x254.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_3FQYgskPEfwodxSVgrGMlg-14x12.png 14w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_3FQYgskPEfwodxSVgrGMlg.png 576w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div><\/figure><p id=\"4b97\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">156 is not particularly close to most numbers in our data set. In fact, it&#039;s more than 5x the median (middle number), which is 27.<\/p><p class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">What to do in this case ?<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bc5dade elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bc5dade\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ad742a5\" data-id=\"ad742a5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ad02b57 elementor-widget elementor-widget-heading\" data-id=\"ad02b57\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"La-moyenne-geometrique\"><\/span>The geometric mean<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7eba54b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7eba54b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-825016e\" data-id=\"825016e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c94a366 elementor-widget elementor-widget-text-editor\" data-id=\"c94a366\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p id=\"1b7b\" class=\"pw-post-body-paragraph lk ll jd lm b ln lo ke lp lq lr kh ls lt lu lv lw lx ly lz ma mb mc md me mf iw gc\" data-selectable-paragraph=\"\">Since the relationship is multiplicative, to find the geometric mean we multiply rather than add all the numbers. Then, to scale the product to the range of the dataset, we need to take the root of the item count, rather than just dividing.<\/p><p id=\"f10b\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Thus, the geometric mean of our dataset is:<\/p><blockquote class=\"nd ne nf\"><p id=\"de57\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">1 * 3 * 9 * 27 * 81 * 243 * 729 = 10,460,353,203<br \/>7th root of 10,460,353,203 = 27<br \/><strong class=\"lm je\">geometric mean<\/strong>\u00a0=\u00a0<strong class=\"lm je\">27<\/strong><\/code><\/p><\/blockquote><figure class=\"mi mj mk ml gz mm gn go paragraph-image\"><div class=\"gn go xh\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-15718\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_7sQflp6iyrOK2IX3Lhg68A-300x254.png\" alt=\"\" width=\"300\" height=\"254\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_7sQflp6iyrOK2IX3Lhg68A-300x254.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_7sQflp6iyrOK2IX3Lhg68A-14x12.png 14w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_7sQflp6iyrOK2IX3Lhg68A.png 576w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div><\/figure><p id=\"a970\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">In this case, our geometric mean closely resembles the median value of our data set. In fact, it equals the median.<\/p><p id=\"9aa0\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">The geometric mean will not always equal the median, only in cases where there is an exact and consistent multiplicative relationship between all the numbers (for example, multiplying each previous number by 3, as we have done).<\/p><p class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Real-world datasets rarely contain such exact relationships, but for those that approximate this type of multiplicative relationship, the geometric mean will yield a closer &quot;median number&quot; than the arithmetic mean.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a981dab elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a981dab\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4d6b871\" data-id=\"4d6b871\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-54b5336 elementor-widget elementor-widget-heading\" data-id=\"54b5336\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exemple-de-moyenne-geometrique\"><\/span>Example of geometric mean<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-371315d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"371315d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1800c94\" data-id=\"1800c94\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-080f1c4 elementor-widget elementor-widget-text-editor\" data-id=\"080f1c4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p id=\"2bf6\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Suppose we have 100,000 $ which generates a variable interest rate every year for 5 years:<\/p><blockquote class=\"nd ne nf\"><p id=\"bcb4\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">annual interest rates: 1%, 9%, 6%, 2%, 15%<\/code><\/p><\/blockquote><p id=\"7d7a\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">We&#039;d like to take a shortcut to find our average annual interest rate, and therefore our total amount of money after 5 years, so we try to &quot;average&quot; these rates:<\/p><blockquote class=\"nd ne nf\"><p id=\"75a6\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">(.01 + .09 + .06 + .02 + .15) \u00f7 5 = .066 =\u00a0<strong class=\"lm je\">6.6%<\/strong><\/code><\/p><\/blockquote><p id=\"21af\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Next, we insert this average percentage into a compound interest formula:<\/p><blockquote class=\"nd ne nf\"><p id=\"5801\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">Total interest earned = $100,000 * (1.066\u2075 - 1) = $37,653.11<br \/>Interest + principal = $37,653.11 + 100,000 = $137,653.11<br \/><strong class=\"lm je\">final total<\/strong>\u00a0=\u00a0<strong class=\"lm je\">$137,653.11<\/strong><\/code><\/p><\/blockquote><p id=\"1c43\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Let&#039;s compare the results:<\/p><blockquote class=\"nd ne nf\"><p id=\"900c\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Year 1<\/strong>: 100,000 + (100,000 * .01) = 100,000 * 1.01 = $101,000<\/code><br \/><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Year 2<\/strong>: 101,000 * 1.09 = $110,090<\/code><br \/><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Year 3<\/strong>: 110,090 * 1.06 = $116,695.40<\/code><br \/><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Year 4<\/strong>: 116,695.40 * 1.02 = $119,029.31<\/code><br \/><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Year 5<\/strong>: 119,029.31 * 1.15 = $136,883.70<\/code><br \/><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Actual final total<\/strong>\u00a0=\u00a0<strong class=\"lm je\">$136,883.70<\/strong><\/code><\/p><\/blockquote><p id=\"ef3e\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Our shortcut overestimated our actual earnings by nearly 1,000 $.<\/p><p>We made a common mistake: we applied an additive operation to a multiplicative process and got an inaccurate result.<\/p><p>Let&#039;s try again with the geometric mean:<\/p><blockquote class=\"nd ne nf\"><p id=\"9685\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">1.01 * 1.09 * 1.06 * 1.02 * 1.15 = 1.368837042<br \/>5th root of 1.368837042 = 1.064805657<br \/><strong class=\"lm je\">geometric mean<\/strong>\u00a0=\u00a0<strong class=\"lm je\">1.064805657<\/strong><\/code><\/p><\/blockquote><p id=\"e8d8\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Plug the geometric mean of interest rates into our compound interest formula:<\/p><blockquote class=\"nd ne nf\"><p id=\"9eed\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\">Total interest earned = $100,000 * (1.0648\u2075 - 1) = $36,883.70<br \/>Interest + principal = $36,883.70 + 100,000 = $136,883.70<br \/><strong class=\"lm je\">final total<\/strong>\u00a0=\u00a0<strong class=\"lm je\">$136,883.70\u00a0<\/strong>exactly the same as the long method above<\/code><\/p><\/blockquote><p id=\"8ba9\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Which corresponds to reality!<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f1546d4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f1546d4\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c9e620c\" data-id=\"c9e620c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d7b52fd elementor-widget elementor-widget-heading\" data-id=\"d7b52fd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Moyenne-geometrique-et-changement-dechelle\"><\/span>Geometric mean and scale change<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ef865cc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ef865cc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0caa537\" data-id=\"0caa537\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-edf0dc1 elementor-widget elementor-widget-text-editor\" data-id=\"edf0dc1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p id=\"bfc0\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">A cool feature of geometric mean is that you can actually average numbers on completely different scales.<\/p><p id=\"b7c7\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">For example, we want to compare the online ratings of two coffeeshops using two different sources. The problem is that source 1 uses a 5 star scale and source 2 uses a 100 point scale:<\/p><blockquote class=\"nd ne nf\"><p id=\"3d3b\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><strong class=\"lm je\"><em class=\"jd\">Coffeeshop A<\/em><\/strong><em class=\"jd\"><br \/><\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">source 1<\/em><\/code><em class=\"jd\">\u00a0rating:\u00a0<\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">4.5<\/em><\/code><em class=\"jd\"><br \/><\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">spring 2<\/em><\/code><em class=\"jd\">\u00a0rating:\u00a0<\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">68<\/em><\/code><\/p><p id=\"3306\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><strong class=\"lm je\"><em class=\"jd\">Coffeeshop B<\/em><\/strong><em class=\"jd\"><br \/><\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">source 1<\/em><\/code><strong class=\"lm je\"><em class=\"jd\">\u00a0<\/em><\/strong><em class=\"jd\">rating:\u00a0<\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">3<\/em><\/code><em class=\"jd\"><br \/><\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">spring 2<\/em><\/code><strong class=\"lm je\"><em class=\"jd\">\u00a0<\/em><\/strong><em class=\"jd\">rating:\u00a0<\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">75<\/em><\/code><\/p><\/blockquote><p id=\"7eac\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">If we naively take the arithmetic mean of the raw scores of each coffeeshop:<\/p><blockquote class=\"nd ne nf\"><p id=\"a745\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><strong class=\"lm je\"><em class=\"jd\">Coffee shop A =<\/em>\u00a0<\/strong><code class=\"fr xd xe xf xg b\">(4.5 + 68)\u00a0<em class=\"jd\">\u00f7<\/em>\u00a02<strong class=\"lm je\">\u00a0= 36.25<br \/><\/strong><\/code><strong class=\"lm je\"><em class=\"jd\">Coffeeshop B =<\/em><\/strong>\u00a0<code class=\"fr xd xe xf xg b\">(3 + 75)\u00a0<em class=\"jd\">\u00f7<\/em>\u00a02 =<strong class=\"lm je\">\u00a039<\/strong><\/code><\/p><\/blockquote><p id=\"cbbd\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">We would conclude that Coffeeshop B was the winner.<\/p><p id=\"dc0f\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">If we knew the numbers a little better, we would know that we need to normalize our values on the same scale before averaging them with the arithmetic mean, to get an accurate result. So we multiply the ratings from Source 1 by 20 to scale them from a 5-star scale to the 100-star scale from Source 2:<\/p><blockquote class=\"nd ne nf\"><p id=\"c1cb\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><strong class=\"lm je\"><em class=\"jd\">Coffeeshop A<\/em><br \/><\/strong><code class=\"fr xd xe xf xg b\">4.5 * 20 = 90<br \/>(90 + 68) \u00f7 2 =\u00a0<strong class=\"lm je\">79<\/strong><\/code><\/p><p id=\"3ef7\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><strong class=\"lm je\"><em class=\"jd\">Coffeeshop B<br \/><\/em><\/strong><code class=\"fr xd xe xf xg b\">3 * 20 = 60<br \/>(60 + 75) \u00f7 2 =\u00a0<strong class=\"lm je\">67.5<\/strong><\/code><\/p><\/blockquote><p id=\"7108\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">So we find that Coffeeshop A is the true winner, unlike the naive application of the arithmetic mean above.<\/p><p id=\"d995\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">The geometric mean, however, allows us to reach the same conclusion without having to worry about scale or units of measurement:<\/p><blockquote class=\"nd ne nf\"><p id=\"37be\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><strong class=\"lm je\"><em class=\"jd\">Coffeeshop A<\/em><\/strong><em class=\"jd\">\u00a0=\u00a0<\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">square root of (4.5 * 68) =\u00a0<\/em><strong class=\"lm je\"><em class=\"jd\">17.5<br \/><\/em><\/strong><\/code><strong class=\"lm je\"><em class=\"jd\">Coffeeshop B<\/em><\/strong><em class=\"jd\">\u00a0=\u00a0<\/em><code class=\"fr xd xe xf xg b\"><em class=\"jd\">square root of (3 * 75) =\u00a0<\/em><strong class=\"lm je\"><em class=\"jd\">15<\/em><\/strong><\/code><\/p><\/blockquote><p id=\"f12a\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">And There you go !<\/p><p id=\"84ed\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">The arithmetic mean is dominated by larger scale numbers, which makes us think that Coffeeshop B is the top rated store. This is because the arithmetic mean expects an additive relationship between numbers and ignores scales and proportions. Hence the need to put the numbers on the same scale before applying the arithmetic mean.<\/p><p id=\"a004\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">The geometric mean, on the other hand, can easily handle varying proportions, due to its multiplicative nature. This is an extremely useful property, but notice what we&#039;re losing: we have no interpretable scale at all. The geometric mean is effectively unitless in such situations.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-54d1d04 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"54d1d04\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6065fc3\" data-id=\"6065fc3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3bbc901 elementor-widget elementor-widget-heading\" data-id=\"3bbc901\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Et-la-moyenne-harmonique\"><\/span>And the harmonic mean?<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ceac652 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ceac652\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9d22746\" data-id=\"9d22746\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-81ab43a elementor-widget elementor-widget-text-editor\" data-id=\"81ab43a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p id=\"68a1\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">While the arithmetic mean requires addition and the geometric mean uses multiplication, the harmonic mean uses inverses.<\/p><p id=\"dc37\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">As you may recall, the inverse of a number n is simply 1\/n. (for example, the reciprocal of 5 is 1\/5). For numbers that are already fractions, this means you can simply &quot;flip&quot; the numerator and denominator: inverse of 4\/5 = 5\/4. This is true because 1 divided by a fraction gives the inverse of that fraction, e.g. 1 \u00f7 (4\/5) = 5\/4.<\/p><p id=\"5516\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Thus, the harmonic mean can be described in words: the inverse of the arithmetic mean of the inverses of the data set.<\/p><p id=\"91ab\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">It&#039;s really just a few simple steps:<\/p><blockquote class=\"nd ne nf\"><p id=\"77a8\" class=\"lk ll mg lm b ln my ke lp lq mz kh ls ng na lv lw nh nb lz ma ni nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\"><em class=\"jd\">1. Take the reciprocal of all numbers in the dataset<br \/>2. Find the arithmetic mean of those reciprocals<br \/>3. Take the reciprocal of that number<\/em><\/code><\/p><\/blockquote><p id=\"a4a3\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">In mathematical notation, this looks like:<\/p><figure class=\"mi mj mk ml gz mm gn go paragraph-image\"><div class=\"gn go yc\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-15719 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_Q4U3tTSsNKPNp2weVxa93Q.png\" alt=\"\" width=\"700\" height=\"207\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_Q4U3tTSsNKPNp2weVxa93Q.png 700w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_Q4U3tTSsNKPNp2weVxa93Q-300x89.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_Q4U3tTSsNKPNp2weVxa93Q-18x5.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/04\/1_Q4U3tTSsNKPNp2weVxa93Q-600x177.png 600w\" sizes=\"(max-width: 700px) 100vw, 700px\" \/><\/div><\/figure><p id=\"5c33\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">The harmonic mean of 1, 4 and 4 is 2.<\/p><p id=\"b0ac\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Note: Because 0 has no inverse (nothing can be multiplied by 0 for = 1), the harmonic mean also cannot handle data sets that contain 0s, like the geometric mean.<\/p><p id=\"ecf9\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">But what is that for ?<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7807de7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7807de7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-99bcc10\" data-id=\"99bcc10\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cc8a925 elementor-widget elementor-widget-heading\" data-id=\"cc8a925\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exemple-de-moyenne-harmonique\"><\/span>Example of harmonic mean<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b943552 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b943552\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3a7340c\" data-id=\"3a7340c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-edab0ad elementor-widget elementor-widget-text-editor\" data-id=\"edab0ad\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p id=\"acfa\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Again, similar to using the geometric mean as the counterpart of the arithmetic mean for multiplicative or nonlinear relationships, the harmonic mean helps us find multiplicative\/dividing relationships between fractions without worrying about common denominators.<\/p><p id=\"c50e\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">As such, the harmonic mean naturally accepts another multiplication\/division layer on top of the geometric mean. It is therefore useful when dealing with data sets of rates or ratios (i.e. fractions) over different lengths or time periods.<\/p><div class=\"o dz\"><div class=\"eo cf fc fd fe ff fg fh fi fj fk\"><article class=\"meteredContent\"><div class=\"l\"><div class=\"l\"><section><div class=\"iw ix iy iz ja\"><p id=\"070a\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">The canonical example of using harmonic means in the real world involves traveling through physical space at different speeds, i.e. speeds.<\/p><p>Consider a trip to the grocery store and back:<\/p><ul><li>On the way, you drove 30 mph the whole way<\/li><li>On the way back, traffic was slow and you drove 16 km\/h the whole way<\/li><li>You took the same route and covered the same amount of ground (5 miles) each way.<\/li><\/ul><p>What was your average speed for the entire duration of this trip?<\/p><p>Again, we could naively apply the arithmetic mean to 30 mph &amp; 10 mph, and proudly declare &quot;20 mph!&quot; \u00bb<\/p><p id=\"f81e\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Since you traveled faster in one direction, you traveled that 8 km faster and spent less time traveling at that speed, so your average travel speed over the entire duration of your trip n It&#039;s not the midpoint between 30 mph and 10 mph, it should be closer to 10 mph because you&#039;ve spent more time riding at that speed.<\/p><p id=\"c765\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">In order to properly apply the arithmetic mean here, we need to determine the time spent traveling at each fare and then weight our arithmetic mean calculation appropriately:<\/p><p id=\"ca80\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\"><strong class=\"lm je\">Trip There: (at 30 mph)<\/strong><code class=\"fr xd xe xf xg b\"><br \/>30 miles per 60 mins = 1 mile every 2 minutes = 1\/2 mile every minute<br \/>5 miles at 1\/2 mile per minute = 5 \u00f7 1\/2 = 10 minutes<br \/><strong class=\"lm je\">&quot;Trip There&quot; time<\/strong>\u00a0=\u00a0<strong class=\"lm je\">10 minutes<\/strong><\/code><\/p><p id=\"8f97\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\"><strong class=\"lm je\">Trip Back: (at 10 mph)<\/strong><code class=\"fr xd xe xf xg b\"><br \/>10 miles per 60 mins = 1 mile every 6 minutes = 1\/6 mile every minute<br \/>5 miles at 1\/6 mile per minute = 5 \u00f7 1\/6 = 30 minutes<br \/><strong class=\"lm je\">&quot;Trip Back&quot; time<\/strong>\u00a0=\u00a0<strong class=\"lm je\">30 minutes<\/strong><\/code><\/p><p id=\"4a69\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Total trip time<\/strong>\u00a0= 10 + 30 =\u00a0<strong class=\"lm je\">40 minutes<\/strong><\/code><\/p><p id=\"e9f5\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">\u201cTrip There\u201d % of total trip<\/strong>\u00a0= 10 \/ 40 minutes = .25 = 25%<br \/><strong class=\"lm je\">\u201cTrip Back\u201d % of total trip<\/strong>\u00a0= 30 \/ 40 minutes = .75 = 75%<\/code><\/p><p id=\"589c\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Weighted Arithmetic Mean<\/strong>\u00a0= (30mph * .25)+(10mph * .75) = 7.5 + 7.5 = 15<br \/><strong class=\"lm je\">Average rate of travel = 15 mph<\/strong><\/code><\/p><p id=\"43c2\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">So we see that our true average rate of travel was 15 mph, which is 5 mph (or 25 %) less than our na\u00efve claim of 20 mph using an unweighted arithmetic mean.<\/p><p id=\"2116\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">Let&#039;s try again using the harmonic mean.<\/p><p id=\"5169\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\"><code class=\"fr xd xe xf xg b\"><strong class=\"lm je\">Harmonic mean<\/strong>\u00a0of 30 and 10 = ...<br \/><strong class=\"lm je\">Arithmetic mean<\/strong>\u00a0of\u00a0<strong class=\"lm je\">reciprocals<\/strong>\u00a0= 1\/30 + 1\/10 = 4\/30 \u00f7 2 = 4\/60 = 1\/15<br \/><strong class=\"lm je\">Reciprocal<\/strong>\u00a0of\u00a0<strong class=\"lm je\">arithmetic mean<\/strong>\u00a0= 1 \u00f7 1\/15 = 15\/1 =\u00a0<strong class=\"lm je\">15<\/strong><\/code><\/p><p id=\"561f\" class=\"pw-post-body-paragraph lk ll jd lm b ln my ke lp lq mz kh ls lt na lv lw lx nb lz ma mb nc md me mf iw gc\" data-selectable-paragraph=\"\">And There you go !<\/p><\/div><\/section><\/div><\/div><\/article><\/div><\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Descriptive analysis WIki home page This page shows several examples on the use cases of the different means, in particular the geometric mean and the harmonic mean. \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":15506,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-15712","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/15712","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=15712"}],"version-history":[{"count":3,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/15712\/revisions"}],"predecessor-version":[{"id":15722,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/15712\/revisions\/15722"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/15506"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=15712"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}