Mean | Definition, Formula, & Facts

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<div class="drawer d-flex flex-column open"> <div class="left-rail-section-content"> <div class="topic-left-rail-header text-truncate bg-gray-50 position-relative text-right d-flex align-items-center"> <div class="tlr-title px-20 py-15 text-left"> <em class="material-icons text-gray-400 d-lg-none" data-icon="toc"></em> <a class="font-serif font-weight-bold text-black link-blue" href="https://www.britannica.com/science/mean">mean</a> </div> <button aria-label="Close" class="js-sections-close-button btn-link btn-sm btn d-lg-none position-absolute top-0 p-10 right-0" > <em class="material-icons font-26" data-icon="close"></em> </button> </div> <div class="section-content pl-10 pr-20 pl-sm-50 pr-sm-60 pl-lg-5 pr-lg-10 pt-10 pt-lg-0 bg-gray-50 clear-catfish-ad"> <div class="toc mb-20"> <div class="font-serif font-14 font-weight-bold mx-15 mb-15 mt-20"> Table of Contents </div> <span class="toc-extra-link selected link-gray-900 mt-15"> Introduction </span> <a class="toc-extra-link link-gray-900" href="https://www.britannica.com/science/mean/additional-info">References & Edit History</a> <a class="toc-extra-link link-gray-900" href="/facts/mean">Related Topics</a> </div> <div class="mb-30 tlr-discover"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Discover </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/inventors-and-inventions-of-the-industrial-revolution"> <img loading="lazy" src="https://cdn.britannica.com/34/193634-131-F5FF783D/factories-Industrial-Revolution-workers-house-machines.jpg?w=200&h=200&c=crop" alt="A Factory Interior, watercolor, pen and gray ink, graphite, and white goache on wove paper by unknown artist, c. 1871-91; in the Yale Center for British Art. Industrial Revolution England" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/inventors-and-inventions-of-the-industrial-revolution" >Inventors and Inventions of the Industrial Revolution</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/the-assassination-of-leon-trotsky"> <img loading="lazy" src="https://cdn.britannica.com/35/28235-131-19392D4F/Leon-Trotsky.jpg?w=200&h=200&c=crop" alt="Leon Trotsky" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/the-assassination-of-leon-trotsky" >The Assassination of Leon Trotsky</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/10-best-hockey-players-of-all-time"> <img loading="lazy" src="https://cdn.britannica.com/13/134913-131-D9520745/Ice-Sledge-Hockey-Canada-Cup-USA-ice-2010.jpg?w=200&h=200&c=crop" alt="Ice Sledge Hockey, Hockey Canada Cup, USA (left) vs Canada, 2009. UBC Thunderbird Arena, Vancouver, BC, competition site for Olympic ice hockey and Paralympic ice sledge hockey. Vancouver 2010 Olympic and Paralympic Winter Games, Vancouver Olympics" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/10-best-hockey-players-of-all-time" >10 Best Hockey Players of All Time</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/americas-5-most-notorious-cold-cases-including-one-you-may-have-thought-was-already-solved"> <img loading="lazy" src="https://cdn.britannica.com/71/196471-131-8FEA8DDD/Daily-Police-Bulletin-Elizabeth-Short-Black-Dahlia-January-1947.jpg?w=200&h=200&c=crop" alt="Los Angeles Police Department wanted flyer on Elizabeth Short, aka the "Black Dahlia," who was brutally murdered in January 1947. The FBI supported the Los Angeles Police Department in the case, including by identifying Short through her fingerprints that" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/americas-5-most-notorious-cold-cases-including-one-you-may-have-thought-was-already-solved" >America’s 5 Most Notorious Cold Cases (Including One You May Have Thought Was Already Solved)</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/6-of-the-worlds-most-dangerous-birds"> <img loading="lazy" src="https://cdn.britannica.com/58/152058-131-0EB52C8F/Ostriches.jpg?w=200&h=200&c=crop" alt="Close-up of ostriches (Struthio camelus) necks and heads; location unknown." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/6-of-the-worlds-most-dangerous-birds" >6 of the World's Most Dangerous Birds</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/10-famous-artworks-by-leonardo-da-vinci"> <img loading="lazy" src="https://cdn.britannica.com/06/200006-131-ABB681CF/Leonardo-da-Vinci-Italian-Renaissance-Florence-Engraving-1500.jpg?w=200&h=200&c=crop" alt="Leonardo da Vinci (1452-1519), Italian Renaissance painter from Florence. Engraving by Cosomo Colombini (d. 1812) after a Leonardo self portrait. Ca. 1500." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/10-famous-artworks-by-leonardo-da-vinci" >10 Famous Artworks by Leonardo da Vinci</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/the-rise-of-the-machines-pros-and-cons-of-the-industrial-revolution"> <img loading="lazy" src="https://cdn.britannica.com/17/167717-131-94C99B30/boys-mill-Macon-Georgia-1909.jpg?w=200&h=200&c=crop" alt="Young boys working in a thread spinning mill in Macon, Georgia, 1909. Boys are so small they have to climb onto the spinning frame to reach and fix broken threads and put back empty bobbins. Child labor. Industrial revolution" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/the-rise-of-the-machines-pros-and-cons-of-the-industrial-revolution" >The Rise of the Machines: Pros and Cons of the Industrial Revolution</a> </div> </div> </div> </div> </div> </div> <button class="drawerToggle btn position-sticky border btn-xs btn-white btn-circle rounded-sm d-none d-lg-flex " type="button" aria-label="Toggle Drawer"> <em class="material-icons font-18 text-blue" data-icon="keyboard_arrow_left"></em> </button> </div> </div> <div class="col"> <div class="h-100 ml-0 pr-lg-0 "> <div class="h-100 grid gx-0 gx-lg-20"> <div class="h-100 col-sm"> <div class="h-100 infinite-pagination-container d-flex flex-column position-relative"> <div class="position-absolute top-0 h-100 w-100"> <div class="toc-sticky-header d-none d-lg-none bg-gray-50 px-10 px-sm-30 position-sticky w-100 "> <div class="toc-sticky-header-inner-container align-items-center d-flex mx-auto h-100 w-100"> <button class="d-flex d-lg-none btn btn-sm btn-white text-blue border-2 border-gray-100 gtm-mobile-toc-header-button js-sections-button d-lg-none p-10"> <em class="material-icons my-n5 md-icon" data-icon="toc"></em> Contents </button> <div class="header-ai-simplify-button-placeholder"></div> <div class="header-ai-ask-button-placeholder"></div> </div> </div> </div> <div class="grey-box w-100 grey-box-top grey-box-bottom"> <div class="grey-box-content mx-auto w-100"> <script type="application/ld+json"> { "@context" : "https://schema.org", "@type" : "BreadcrumbList", "itemListElement" : [ { "@type" : "ListItem", "position" : 1, "item" : { "@id" : "https://www.britannica.com/browse/Science", "name": "Science" } } , { "@type" : "ListItem", "position" : 2, "item" : { "@id" : "https://www.britannica.com/browse/Mathematics", "name": "Mathematics" } } ] } </script> <nav class="breadcrumb mt-20"> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Science">Science</a> </span> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Mathematics">Mathematics</a> </span> </nav> <div class="page2ref-true topic-content topic-type-REGULAR" data-student-article="false"> <script class="page-description-json" type="application/json"> { "url": "/science/mean", "shareUrl": "https://www.britannica.com/science/mean", "browserTitle": "Mean | Definition, Formula, & Facts", "firstTopicPage": true, "topicId":371524 } </script> <div class="reading-channel"> <div class="topic-header"> <div class="d-flex align-items-top justify-content-between"> <div class="d-flex flex-column"> <div> <div> <h1>mean</h1></div> </div> <div class="topic-identifier font-16 font-md-20">mathematics</div> </div> </div> <div class="d-none d-sm-flex flex-row"> <div class="mr-10 mb-15"> <button class="ai-ask-button btn btn-sm border-2 btn-outline-red-400 border-red-400 btn-sm btn-outline-red-400 border-red-400 px-sm-5 px-md-10 js-inline-ai-ask-button"> Ask the Chatbot a Question </button> </div> <div class="d-block md-topic-tools qa-action-buttons mb-15" data-topic-id="371524"> <button class="js-tooltip btn btn-sm btn-outline-blue border pr-10 border-2 text-nowrap" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="more_vert"></em> More Actions </button> <div class="md-more-popover popover popover-sm p-0 font-14 z-1"> <div> <a class="js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" href="/print/article/371524" target="_blank" > <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="print"></em> Print </a> </div> <div> <button class="js-modal qa-cite-modal-button btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=371524] .md-cite-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="verified"></em> Cite </button> <div class="md-cite-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom mb-15"> <em class="material-icons text-blue mr-10">verified</em>Cite </div> <div class="font-serif"> While every effort has been made to follow citation style rules, there may be some discrepancies. 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They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <a class="byline-edit-history" href="https://www.britannica.com/science/mean/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn btn-sm border-2 btn-outline-red-400 border-red-400 module-spacing js-inline-ai-ask-button p-10 ml-5"> Ask the Chatbot </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/science/geometric-mean" topicid="229814">geometric mean</a></dd> <dd><a href="/science/quadratic-mean" topicid="485886">quadratic mean</a></dd> <dd><a href="/science/harmonic-mean" topicid="255514">harmonic mean</a></dd> <dd><a href="/science/arithmetic-mean" topicid="34759">arithmetic mean</a></dd> <dd><a href="/science/weighted-arithmetic-mean" topicid="638974">weighted arithmetic mean</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/mean"> See all related content </a> </div> </div> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref165437"></span>mean</strong>, in <a href="https://www.britannica.com/science/mathematics" class="md-crosslink " data-show-preview="true">mathematics</a>, a quantity that has a <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="value" href="https://www.britannica.com/dictionary/value" data-type="EB">value</a> intermediate between those of the extreme members of some set. Several kinds of means exist, and the method of calculating a mean depends upon the relationship known or assumed to govern the other members. The <span id="ref165438"></span><a href="https://www.britannica.com/science/arithmetic-mean" class="md-crosslink ">arithmetic mean</a>, denoted <span class="md-overbar"><em>x</em></span>, of a set of <em>n</em> numbers <em>x</em><sub>1</sub>, <em>x</em><sub>2</sub>, …, <em>x</em><sub><em>n</em></sub> is defined as the sum of the numbers divided by <em>n</em>:<span class="md-formula"><span id="ref-15242"></span><img src="https://cdn.britannica.com/42/15242-004-6D9A7A12/Equation.jpg" alt="Equation." class="inline-image-baseline"></span></p><span class="marker MOD1 mod-inline"></span><span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">The arithmetic mean (usually synonymous with average) represents a point about which the numbers balance. For example, if unit masses are placed on a line at points with coordinates <em>x</em><sub>1</sub>, <em>x</em><sub>2</sub>, …, <em>x</em><sub><em>n</em></sub>, then the arithmetic mean is the coordinate of the <a href="https://www.britannica.com/science/centre-of-gravity" class="md-crosslink autoxref " data-show-preview="true">centre of gravity</a> of the system. In <a href="https://www.britannica.com/science/statistics" class="md-crosslink " data-show-preview="true">statistics</a>, the arithmetic mean is commonly used as the single value typical of a set of <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="data" href="https://www.britannica.com/dictionary/data" data-type="EB">data</a>. For a system of particles having unequal masses, the centre of gravity is determined by a more general average, the <span id="ref165439"></span>weighted arithmetic mean. If each number (<em>x</em>) is assigned a corresponding positive weight (<em>w</em>), the weighted arithmetic mean is defined as the sum of their products (<em>w</em><em>x</em>) divided by the sum of their weights. In this case, <span class="md-formula"><span id="ref-15241"></span><img src="https://cdn.britannica.com/41/15241-004-DA8AF7A8/Equation.jpg" alt="Equation." class="inline-image-baseline"></span></p><span class="marker MOD2 mod-inline"></span><span class="marker PREMOD3 mod-inline"></span><p class="topic-paragraph">The weighted arithmetic mean also is used in statistical analysis of grouped data: each number <em>x</em><sub><em>i</em></sub> is the midpoint of an interval, and each corresponding value of <em>w</em><sub><em>i</em></sub> is the number of data points within that interval.</p><div class="module-spacing"> </div><span class="marker MOD3 mod-inline"></span><span class="marker PREMOD4 mod-inline"></span><p class="topic-paragraph">For a given set of data, many possible means can be defined, depending on which features of the data are of interest. For example, suppose five squares are given, with sides 1, 1, 2, 5, and 7 cm. Their average area is (1<sup>2</sup> + 1<sup>2</sup> + 2<sup>2</sup> + 5<sup>2</sup> + 7<sup>2</sup>)/5, or 16 square cm, the area of a square of side 4 cm. The number 4 is the <span id="ref165440"></span>quadratic mean (or root mean square) of the numbers 1, 1, 2, 5, and 7 and differs from their arithmetic mean, which is 3 <span class="md-fraction md-fraction-oblique"><span><span class="md-frac-num">1</span><span class="md-fraction-symbol">/</span><span class="md-frac-den">5</span></span></span>. In general, the quadratic mean of <em>n</em> numbers <em>x</em><sub>1</sub>, <em>x</em><sub>2</sub>, …, <em>x</em><sub><em>n</em></sub> is the <a href="https://www.britannica.com/science/square-root" class="md-crosslink autoxref " data-show-preview="true">square root</a> of the arithmetic mean of their squares,<span class="md-formula"><span id="ref-15240"></span><img src="https://cdn.britannica.com/40/15240-004-C96EA533/Depiction-root-arithmetic-mean-numbers-xn-squares.jpg" alt="Depiction of the square root of the arithmetic mean of n numbers x1, x2, ?, xn squares." class="inline-image-baseline"></span> The arithmetic mean gives no indication of how widely the data are spread or <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="dispersed" href="https://www.britannica.com/dictionary/dispersed" data-type="EB">dispersed</a> about the mean. Measures of the dispersion are provided by the arithmetic and quadratic means of the <em>n</em> differences <em>x</em><sub>1</sub> − <span class="md-overbar"><em>x</em></span>, <em>x</em><sub>2</sub> − <span class="md-overbar"><em>x</em></span>, …, <em>x</em><sub><em>n</em></sub> − <span class="md-overbar"><em>x</em></span>. The quadratic mean gives the <a href="https://www.britannica.com/topic/standard-deviation-statistics" class="md-crosslink autoxref " data-show-preview="true">standard deviation</a> of <em>x</em><sub>1</sub>, <em>x</em><sub>2</sub>, …, <em>x</em><sub><em>n</em></sub>.</p><span class="marker MOD4 mod-inline"></span><span class="marker PREMOD5 mod-inline"></span><p class="topic-paragraph">The arithmetic and quadratic means are the special cases <em>p</em> = 1 and <em>p</em> = 2 of the <em>p</em>th-power mean, <em>M</em><sub><em>p</em></sub>, defined by the formula<span class="md-formula"><span id="ref-15239"></span><img src="https://cdn.britannica.com/39/15239-004-63F93435/Equation.jpg" alt="Equation." class="inline-image-baseline"></span>where <em>p</em> may be any <a href="https://www.britannica.com/science/real-number" class="md-crosslink autoxref " data-show-preview="true">real number</a> except zero. The case <em>p</em> = −1 is also called the <span id="ref165441"></span>harmonic mean. Weighted <em>p</em>th-power means are defined by<span class="md-formula"><span id="ref-15238"></span><img src="https://cdn.britannica.com/38/15238-004-5614E58E/Equation.jpg" alt="Equation." class="inline-image-baseline"></span></p><span class="marker MOD5 mod-inline"></span><span class="marker PREMOD6 mod-inline"></span><p class="topic-paragraph">If <em>x</em> is the arithmetic mean of <em>x</em><sub>1</sub> and <em>x</em><sub>2</sub>, the three numbers <em>x</em><sub>1</sub>, <em>x</em>, <em>x</em><sub>2</sub> are in arithmetic progression. If <em>h</em> is the harmonic mean of <em>x</em><sub>1</sub> and <em>x</em><sub>2</sub>, the numbers <em>x</em><sub>1</sub>, <em>h</em>, <em>x</em><sub>2</sub> are in harmonic progression. A number <em>g</em> such that <em>x</em><sub>1</sub>, <em>g</em>, <em>x</em><sub>2</sub> are in geometric progression is defined by the condition that <em>x</em><sub>1</sub>/<em>g</em> = <em>g</em>/<em>x</em><sub>2</sub>, or <em>g</em><sup>2</sup> = <em>x</em><sub>1</sub><em>x</em><sub>2</sub>; hence <span class="md-formula"><span id="ref-74780"></span><img src="https://cdn.britannica.com/80/74780-004-C64CD468/Geometric.jpg" alt="Geometric mean 1" class="inline-image-baseline"></span>This <em>g</em> is called the <span id="ref165442"></span>geometric mean of <em>x</em><sub>1</sub> and <em>x</em><sub>2</sub>. The geometric mean of <em>n</em> numbers <em>x</em><sub>1</sub>, <em>x</em><sub>2</sub>, …, <em>x</em><sub><em>n</em></sub> is defined to be the <em>n</em>th root of their product: <span class="md-formula"><span id="ref-74781"></span><img src="https://cdn.britannica.com/81/74781-004-083350EE/Geometric.jpg" alt="Geometric mean 2" class="inline-image-baseline"></span></p><span class="marker MOD6 mod-inline"></span><span class="marker PREMOD7 mod-inline"></span><p class="topic-paragraph">All the means discussed are special cases of a more general mean. If <em>f</em> is a <a href="https://www.britannica.com/science/function-mathematics" class="md-crosslink " data-show-preview="true">function</a> having an <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="inverse" href="https://www.britannica.com/dictionary/inverse" data-type="EB">inverse</a> <em>f</em> <sup>−1</sup> (a function that “undoes” the original function), the number <span class="md-formula"><span id="ref-15237"></span><img src="https://cdn.britannica.com/37/15237-004-7F3C9872/Inverse-function.jpg" alt="Inverse function." class="inline-image-baseline"></span>is called the mean value of <em>x</em><sub>1</sub>, <em>x</em><sub>2</sub>, …, <em>x</em><sub><em>n</em></sub> associated with <em>f</em>. When <em>f</em>(<em>x</em>) = <em>x</em><sup><em>p</em></sup>, the inverse is <em>f</em> <sup>−1</sup>(<em>x</em>) = <em>x</em><sup>1/<em>p</em></sup>, and the mean value is the <em>p</em>th-power mean, <em>M</em><sub><em>p</em></sub>. When <em>f</em>(<em>x</em>) = ln <em>x</em> (the <a href="https://www.britannica.com/science/natural-logarithm" class="md-crosslink " data-show-preview="true">natural logarithm</a>), the inverse is <em>f</em> <sup>−1</sup>(<em>x</em>) = <em>e</em><sup><em>x</em></sup> (the <a href="https://www.britannica.com/science/exponential-function" class="md-crosslink " data-show-preview="true">exponential function</a>), and the mean value is the geometric mean.</p><div class="module-spacing"> <DIV class="marketing-INLINE_SUBSCRIPTION marketing-content" data-marketing-id="INLINE_SUBSCRIPTION"><style> .student-promo-banner-wrapper { container-type: inline-size; margin-bottom: 15px; } @container (min-width: 475px) { .student-promo-banner { flex-direction: row; } .student-promo-banner-img-wrapper { margin-bottom: 0; margin-right: 10px; justify-content: flex-start; } .student-promo-banner-text-wrapper { text-align: left; margin-bottom: 0px; margin-left: 10px; } .student-promo-banner-button-wrapper { margin-right: 0; } }</style> <div class="student-promo-banner-wrapper"> <div class="student-promo-banner d-flex flex-column align-items-center bg-blue rounded p-20"> <div class="student-promo-banner-img-wrapper mb-20 mr-0 d-flex justify-content-center"> <img class="rounded" style="max-width: 100px; min-width: 80px" src="https://cdn.britannica.com/marketing/BlueThistle.webp" /> </div> <div class="student-promo-banner-text-wrapper ml-0 mb-10 text-center text-white"> <div class="h2 mb-10">Get Unlimited Access</div> <div class="h4 font-weight-semi-bold">Try Britannica Premium for free and discover more.</div> </div> <div class="student-promo-banner-button-wrapper d-flex justify-content-center align-items-center ml-auto mr-auto"> <a class="btn btn-m btn-orange" href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=inline-cta&utm_campaign=august-2024">Subscribe</a> </div> </div> </div> </DIV></div><div class="one-good-fact-module"> </div><span class="marker MOD7 mod-inline"></span><span class="marker PREMOD8 mod-inline"></span><p class="topic-paragraph">For information on the development of various definitions of the mean, <em>see</em> <a href="https://www.britannica.com/science/probability" class="md-crosslink " data-show-preview="true">probability and statistics</a>. For further technical information, <em>see</em> <a href="https://www.britannica.com/science/statistics" class="md-crosslink " data-show-preview="true">statistics</a> and <a href="https://www.britannica.com/science/probability-theory" class="md-crosslink " data-show-preview="true">probability theory</a>.</p><span class="marker MOD8 mod-inline"></span></section> <span class="md-signature font-12"><a href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419">The Editors of Encyclopaedia Britannica</a></span><span class="md-signature font-12">This article was most recently revised and updated by <a href="/editor/Erik-Gregersen/6723">Erik Gregersen</a>.</span><span class="marker end-of-content"></span><span class="marker after-article"></span></div> <div id="chatbot-simplify-root"></div> <div id="chatbot-root"></div> </div> </div> </div> <div class="ai-dialog-placeholder"></div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article> </div> </div> </div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>  <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"DESKTOP","pageNumber":1,"pagesTotal":1,"pageId":371524,"pageLength":664,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-133/js/libs/jquery-3.5.0.min.js?v=3.133.48"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-133/dist/topic-page.js?v=3.133.48' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"C","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"MEDIUM","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":true,"hasAiTopQuestions":false,"hasSimplifyButton":false} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>

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Mean | Definition, Formula, & Facts | Britannica