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class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Witryna"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Spis treści" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Spis treści</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ukryj</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Początek</div> </a> </li> <li id="toc-Dla_rozkładu_dyskretnego" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dla_rozkładu_dyskretnego"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Dla rozkładu dyskretnego</span> </div> </a> <ul id="toc-Dla_rozkładu_dyskretnego-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dla_rozkładu_ciągłego" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dla_rozkładu_ciągłego"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Dla rozkładu ciągłego</span> </div> </a> <ul id="toc-Dla_rozkładu_ciągłego-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Przykłady_zastosowań" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Przykłady_zastosowań"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Przykłady zastosowań</span> </div> </a> <button aria-controls="toc-Przykłady_zastosowań-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Przełącz podsekcję Przykłady zastosowań</span> </button> <ul id="toc-Przykłady_zastosowań-sublist" class="vector-toc-list"> <li id="toc-W_matematyce" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#W_matematyce"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>W matematyce</span> </div> </a> <ul id="toc-W_matematyce-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-W_rachunku_błędów" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#W_rachunku_błędów"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>W rachunku błędów</span> </div> </a> <ul id="toc-W_rachunku_błędów-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-W_teorii_kinetycznej_gazów" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#W_teorii_kinetycznej_gazów"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>W teorii kinetycznej gazów</span> </div> </a> <ul id="toc-W_teorii_kinetycznej_gazów-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-W_teorii_sygnałów" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#W_teorii_sygnałów"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>W teorii sygnałów</span> </div> </a> <ul id="toc-W_teorii_sygnałów-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Zobacz_też" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zobacz_też"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Zobacz też</span> </div> </a> <ul id="toc-Zobacz_też-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uwagi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uwagi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Uwagi</span> </div> </a> <ul id="toc-Uwagi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Przypisy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Przypisy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Przypisy</span> </div> </a> <ul id="toc-Przypisy-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Spis treści" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Przełącz stan spisu treści" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Przełącz stan spisu treści</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Średnia kwadratowa</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Przejdź do artykułu w innym języku. Treść dostępna w 39 językach" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-39" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">39 języków</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%B0%D8%B1_%D9%85%D8%AA%D9%88%D8%B3%D8%B7_%D9%85%D8%B1%D8%A8%D8%B9" title="جذر متوسط مربع – arabski" lang="ar" hreflang="ar" data-title="جذر متوسط مربع" data-language-autonym="العربية" data-language-local-name="arabski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%97%E0%A6%AE%E0%A7%82%E0%A6%B2_%E0%A6%97%E0%A6%A1%E0%A6%BC_%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%97" title="বর্গমূল গড় বর্গ – bengalski" lang="bn" hreflang="bn" data-title="বর্গমূল গড় বর্গ" data-language-autonym="বাংলা" data-language-local-name="bengalski" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A1%D1%8F%D1%80%D1%8D%D0%B4%D0%BD%D1%8F%D0%B5_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D0%B5" title="Сярэдняе квадратычнае – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Сярэдняе квадратычнае" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%80%D0%B5%D0%B4%D0%BD%D0%BE_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%BE" title="Средно квадратично – bułgarski" lang="bg" hreflang="bg" data-title="Средно квадратично" data-language-autonym="Български" data-language-local-name="bułgarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mitjana_quadr%C3%A0tica" title="Mitjana quadràtica – kataloński" lang="ca" hreflang="ca" data-title="Mitjana quadràtica" data-language-autonym="Català" data-language-local-name="kataloński" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvadratick%C3%BD_pr%C5%AFm%C4%9Br" title="Kvadratický průměr – czeski" lang="cs" hreflang="cs" data-title="Kvadratický průměr" data-language-autonym="Čeština" data-language-local-name="czeski" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Effektiv_v%C3%A6rdi" title="Effektiv værdi – duński" lang="da" hreflang="da" data-title="Effektiv værdi" data-language-autonym="Dansk" data-language-local-name="duński" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quadratisches_Mittel" title="Quadratisches Mittel – niemiecki" lang="de" hreflang="de" data-title="Quadratisches Mittel" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ruutkeskmine" title="Ruutkeskmine – estoński" lang="et" hreflang="et" data-title="Ruutkeskmine" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Root_mean_square" title="Root mean square – angielski" lang="en" hreflang="en" data-title="Root mean square" data-language-autonym="English" data-language-local-name="angielski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Media_cuadr%C3%A1tica" title="Media cuadrática – hiszpański" lang="es" hreflang="es" data-title="Media cuadrática" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvadrata_avera%C4%9Do" title="Kvadrata averaĝo – esperanto" lang="eo" hreflang="eo" data-title="Kvadrata averaĝo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Batezbesteko_koadratiko" title="Batezbesteko koadratiko – baskijski" lang="eu" hreflang="eu" data-title="Batezbesteko koadratiko" data-language-autonym="Euskara" data-language-local-name="baskijski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%B0%D8%B1_%D9%85%DB%8C%D8%A7%D9%86%DA%AF%DB%8C%D9%86_%D9%85%D8%B1%D8%A8%D8%B9%D8%A7%D8%AA" title="جذر میانگین مربعات – perski" lang="fa" hreflang="fa" data-title="جذر میانگین مربعات" data-language-autonym="فارسی" data-language-local-name="perski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Moyenne_quadratique" title="Moyenne quadratique – francuski" lang="fr" hreflang="fr" data-title="Moyenne quadratique" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%9C%EA%B3%B1%ED%8F%89%EA%B7%A0%EC%A0%9C%EA%B3%B1%EA%B7%BC" title="제곱평균제곱근 – koreański" lang="ko" hreflang="ko" data-title="제곱평균제곱근" data-language-autonym="한국어" data-language-local-name="koreański" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97_%E0%A4%AE%E0%A4%BE%E0%A4%A7%E0%A5%8D%E0%A4%AF_%E0%A4%AE%E0%A5%82%E0%A4%B2" title="वर्ग माध्य मूल – hindi" lang="hi" hreflang="hi" data-title="वर्ग माध्य मूल" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rata-rata_kuadrat" title="Rata-rata kuadrat – indonezyjski" lang="id" hreflang="id" data-title="Rata-rata kuadrat" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezyjski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%95%D7%A8%D7%A9_%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%94%D7%A8%D7%99%D7%91%D7%95%D7%A2%D7%99%D7%9D" title="שורש ממוצע הריבועים – hebrajski" lang="he" hreflang="he" data-title="שורש ממוצע הריבועים" data-language-autonym="עברית" data-language-local-name="hebrajski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D1%82%D1%8B%D2%9B_%D0%BE%D1%80%D1%82%D0%B0%D1%88%D0%B0%D0%BB%D0%B0%D1%80" title="Квадраттық орташалар – kazachski" lang="kk" hreflang="kk" data-title="Квадраттық орташалар" data-language-autonym="Қазақша" data-language-local-name="kazachski" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/N%C3%A9gyzetes_k%C3%B6z%C3%A9p" title="Négyzetes közép – węgierski" lang="hu" hreflang="hu" data-title="Négyzetes közép" data-language-autonym="Magyar" data-language-local-name="węgierski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwadratisch_gemiddelde" title="Kwadratisch gemiddelde – niderlandzki" lang="nl" hreflang="nl" data-title="Kwadratisch gemiddelde" data-language-autonym="Nederlands" data-language-local-name="niderlandzki" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E4%B9%97%E5%B9%B3%E5%9D%87%E5%B9%B3%E6%96%B9%E6%A0%B9" title="二乗平均平方根 – japoński" lang="ja" hreflang="ja" data-title="二乗平均平方根" data-language-autonym="日本語" data-language-local-name="japoński" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvadratisk_gjennomsnitt" title="Kvadratisk gjennomsnitt – norweski (bokmål)" lang="nb" hreflang="nb" data-title="Kvadratisk gjennomsnitt" data-language-autonym="Norsk bokmål" data-language-local-name="norweski (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Media_quadr%C3%A0tica" title="Media quadràtica – piemoncki" lang="pms" hreflang="pms" data-title="Media quadràtica" data-language-autonym="Piemontèis" data-language-local-name="piemoncki" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Valor_eficaz" title="Valor eficaz – portugalski" lang="pt" hreflang="pt" data-title="Valor eficaz" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Medie_p%C4%83tratic%C4%83" title="Medie pătratică – rumuński" lang="ro" hreflang="ro" data-title="Medie pătratică" data-language-autonym="Română" data-language-local-name="rumuński" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%80%D0%B5%D0%B4%D0%BD%D0%B5%D0%B5_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5" title="Среднее квадратическое – rosyjski" lang="ru" hreflang="ru" data-title="Среднее квадратическое" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Root_mean_square" title="Root mean square – Simple English" lang="en-simple" hreflang="en-simple" data-title="Root mean square" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kvadratick%C3%BD_priemer" title="Kvadratický priemer – słowacki" lang="sk" hreflang="sk" data-title="Kvadratický priemer" data-language-autonym="Slovenčina" data-language-local-name="słowacki" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B0_%D1%81%D1%80%D0%B5%D0%B4%D0%B8%D0%BD%D0%B0" title="Квадратна средина – serbski" lang="sr" hreflang="sr" data-title="Квадратна средина" data-language-autonym="Српски / srpski" data-language-local-name="serbski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Neli%C3%B6llinen_keskiarvo" title="Neliöllinen keskiarvo – fiński" lang="fi" hreflang="fi" data-title="Neliöllinen keskiarvo" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a 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class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Narzędzia dla stron"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Wygląd"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Wygląd</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ukryj</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Z Wikipedii, wolnej encyklopedii</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="pl" dir="ltr"><p><b>Średnia kwadratowa</b> (niekiedy <b>RMS</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[a]</a></sup>, od <a href="/wiki/J%C4%99zyk_angielski" title="Język angielski">ang.</a> <em lang="en">root mean square</em>) – przykład <a href="/wiki/Statystyka_(funkcja)" title="Statystyka (funkcja)">miary statystycznej</a> pozwalającej oszacować <a href="/wiki/Rz%C4%85d_wielko%C5%9Bci" title="Rząd wielkości">rząd wielkości</a> serii danych liczbowych lub funkcji ciągłej, użyteczny zwłaszcza w przypadku, gdy wielkości różnią się znakiem. </p><p>Średnia kwadratowa jest szczególnym przypadkiem innej miary, jest to mianowicie <a href="/wiki/%C5%9Arednia_pot%C4%99gowa" title="Średnia potęgowa">średnia potęgowa</a> rzędu 2, jednak ze względu na jej znaczenie praktyczne ma odrębną nazwę. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Dla_rozkładu_dyskretnego"><span id="Dla_rozk.C5.82adu_dyskretnego"></span>Dla rozkładu dyskretnego</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=1" title="Edytuj sekcję: Dla rozkładu dyskretnego" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Dla rozkładu dyskretnego"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Średnia kwadratowa</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> liczb <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\dots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\dots ,a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc2b55ae65455992fe3ab08c90c55a0cc2e7709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.229ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\dots ,a_{n}}"></span> jest to pierwiastek ze <a href="/wiki/%C5%9Arednia_arytmetyczna" title="Średnia arytmetyczna">średniej arytmetycznej</a> kwadratów tych liczb<sup id="cite_ref-epwn_2-0" class="reference"><a href="#cite_note-epwn-2">[1]</a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}:={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\ldots +a_{n}^{2}}{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mi>n</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}:={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\ldots +a_{n}^{2}}{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ea904a64705e2ef01421721a93853d4373dc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.121ex; height:7.676ex;" alt="{\displaystyle M_{2}:={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\ldots +a_{n}^{2}}{n}}}.}"></span></dd></dl> <p>Na przykład średnia kwadratowa liczb 2, 2, 5 i 7 jest równa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}={\sqrt {\frac {2^{2}+2^{2}+5^{2}+7^{2}}{4}}}\approx 4{,}53.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </msqrt> </mrow> <mo>≈</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>53.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}={\sqrt {\frac {2^{2}+2^{2}+5^{2}+7^{2}}{4}}}\approx 4{,}53.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a37552c58b13541dd97a76fc6fe1cc631a90e67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.834ex; height:7.676ex;" alt="{\displaystyle M_{2}={\sqrt {\frac {2^{2}+2^{2}+5^{2}+7^{2}}{4}}}\approx 4{,}53.}"></span></dd></dl> <p><b>Ważona średnia kwadratowa</b> jest to średnia kwadratowa z uwzględnieniem wag poszczególnych składników: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2W}={\sqrt {\frac {w_{1}\cdot a_{1}^{2}+w_{2}\cdot a_{2}^{2}+\ldots +w_{n}\cdot a_{n}^{2}}{w_{1}+w_{2}+\ldots +w_{n}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>W</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2W}={\sqrt {\frac {w_{1}\cdot a_{1}^{2}+w_{2}\cdot a_{2}^{2}+\ldots +w_{n}\cdot a_{n}^{2}}{w_{1}+w_{2}+\ldots +w_{n}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e27e1eb8fe4207d517d5fa324550b19e891e34d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:43.553ex; height:7.509ex;" alt="{\displaystyle M_{2W}={\sqrt {\frac {w_{1}\cdot a_{1}^{2}+w_{2}\cdot a_{2}^{2}+\ldots +w_{n}\cdot a_{n}^{2}}{w_{1}+w_{2}+\ldots +w_{n}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Dla_rozkładu_ciągłego"><span id="Dla_rozk.C5.82adu_ci.C4.85g.C5.82ego"></span>Dla rozkładu ciągłego</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=2" title="Edytuj sekcję: Dla rozkładu ciągłego" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Dla rozkładu ciągłego"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Średnią kwadratową</b> <a href="/wiki/Funkcja_ci%C4%85g%C5%82a" title="Funkcja ciągła">funkcji ciągłej</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span> określonej w przedziale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [T_{1},T_{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [T_{1},T_{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43fb98e73a3a95714c697a4b2e0a097ffcfe5f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.151ex; height:2.843ex;" alt="{\displaystyle [T_{1},T_{2}]}"></span> określamy według wzoru: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{SK}={\sqrt {{\frac {1}{T_{2}-T_{1}}}\int \limits _{T_{1}}^{T_{2}}[x(t)]^{2}\,dt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>K</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </munderover> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{SK}={\sqrt {{\frac {1}{T_{2}-T_{1}}}\int \limits _{T_{1}}^{T_{2}}[x(t)]^{2}\,dt}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2cc1ab084d880295f1faef6660350341272ba2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:30.517ex; height:10.343ex;" alt="{\displaystyle x_{SK}={\sqrt {{\frac {1}{T_{2}-T_{1}}}\int \limits _{T_{1}}^{T_{2}}[x(t)]^{2}\,dt}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Przykłady_zastosowań"><span id="Przyk.C5.82ady_zastosowa.C5.84"></span>Przykłady zastosowań</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=3" title="Edytuj sekcję: Przykłady zastosowań" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=3" title="Edytuj kod źródłowy sekcji: Przykłady zastosowań"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="W_matematyce">W matematyce</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=4" title="Edytuj sekcję: W matematyce" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=4" title="Edytuj kod źródłowy sekcji: W matematyce"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Średnia kwadratowa różnic wartości zmiennej i <a href="/wiki/Warto%C5%9B%C4%87_oczekiwana" title="Wartość oczekiwana">wartości oczekiwanej</a> jest <a href="/wiki/Odchylenie_standardowe" title="Odchylenie standardowe">odchyleniem standardowym</a> tej zmiennej (dla populacji skończonej): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {\frac {\sum \limits _{i=1}^{n}\left(x_{i}-x_{0}\right)^{2}}{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\sqrt {\frac {\sum \limits _{i=1}^{n}\left(x_{i}-x_{0}\right)^{2}}{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a695d97fd8bdf675adc8b399768c731b7299728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.99ex; height:9.509ex;" alt="{\displaystyle \sigma ={\sqrt {\frac {\sum \limits _{i=1}^{n}\left(x_{i}-x_{0}\right)^{2}}{n}}}}"></span></dd></dl> <p>gdzie: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> – liczebność populacji</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> – wartość oczekiwana zmiennej.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="W_rachunku_błędów"><span id="W_rachunku_b.C5.82.C4.99d.C3.B3w"></span>W rachunku błędów</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=5" title="Edytuj sekcję: W rachunku błędów" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=5" title="Edytuj kod źródłowy sekcji: W rachunku błędów"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Odchylenie_standardowe" title="Odchylenie standardowe">Odchylenie standardowe</a> jest miarą <a href="/wiki/Niepewno%C5%9B%C4%87_pomiaru" title="Niepewność pomiaru">niepewności pomiarowej</a> przy wielokrotnym pomiarze tej samej wielkości. <a href="/wiki/Estymator" title="Estymator">Estymatorem</a> wartości oczekiwanej jest wówczas wartość średnia. Aby średnia kwadratowa odchyleń od średniej mogła być średnim błędem kwadratowym, liczba pomiarów powinna być większa od 30. </p> <div class="mw-heading mw-heading3"><h3 id="W_teorii_kinetycznej_gazów"><span id="W_teorii_kinetycznej_gaz.C3.B3w"></span>W teorii kinetycznej gazów</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=6" title="Edytuj sekcję: W teorii kinetycznej gazów" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=6" title="Edytuj kod źródłowy sekcji: W teorii kinetycznej gazów"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Średnia <a href="/wiki/Pr%C4%99dko%C5%9B%C4%87" title="Prędkość">prędkość</a> kwadratowa cząsteczek <a href="/wiki/Gaz_doskona%C5%82y" title="Gaz doskonały">gazu doskonałego</a> w <a href="/wiki/Kinetyczno-molekularna_teoria_gaz%C3%B3w" title="Kinetyczno-molekularna teoria gazów">teorii kinetycznej gazów</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\overline {V^{2}}}}={\sqrt {\frac {\sum \limits _{i}^{n}{V_{i}^{2}}}{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mover> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mrow> <mi>n</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\overline {V^{2}}}}={\sqrt {\frac {\sum \limits _{i}^{n}{V_{i}^{2}}}{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6dfff4c207a94fb4025e0ed0e01eae43c1c96f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.61ex; height:9.509ex;" alt="{\displaystyle {\sqrt {\overline {V^{2}}}}={\sqrt {\frac {\sum \limits _{i}^{n}{V_{i}^{2}}}{n}}}}"></span></dd></dl> <p>jest ściśle związana z temperaturą tego gazu. Dla gazu jednoatomowego zależność ta wyraża się wzorem: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\overline {V^{2}}}}={\sqrt {\frac {3kT}{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mover> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>3</mn> <mi>k</mi> <mi>T</mi> </mrow> <mi>m</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\overline {V^{2}}}}={\sqrt {\frac {3kT}{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd651704b08ddf1138543abc63410cb34ebe34b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.678ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\overline {V^{2}}}}={\sqrt {\frac {3kT}{m}}}}"></span></dd></dl> <p>gdzie: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> – <a href="/wiki/Sta%C5%82a_Boltzmanna" title="Stała Boltzmanna">stała Boltzmanna</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> – temperatura w <a href="/wiki/Kelwin" title="Kelwin">kelwinach</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> – <a href="/wiki/Masa_(fizyka)" title="Masa (fizyka)">masa</a> pojedynczej cząsteczki.</dd></dl> <p>Wzór ten można wyprowadzić z rozkładu prędkości cząsteczek (<a href="/wiki/Rozk%C5%82ad_Maxwella" title="Rozkład Maxwella">rozkład Maxwella</a>). </p> <div class="mw-heading mw-heading3"><h3 id="W_teorii_sygnałów"><span id="W_teorii_sygna.C5.82.C3.B3w"></span>W teorii sygnałów</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=7" title="Edytuj sekcję: W teorii sygnałów" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=7" title="Edytuj kod źródłowy sekcji: W teorii sygnałów"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Warto%C5%9B%C4%87_skuteczna" title="Wartość skuteczna">Wartość skuteczna</a> <a href="/wiki/Nat%C4%99%C5%BCenie_pr%C4%85du_elektrycznego" title="Natężenie prądu elektrycznego">natężenia</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{SK}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{SK}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b025709ab54f746f2ddf6e50fd99e03235f0962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.776ex; height:2.509ex;" alt="{\displaystyle I_{SK}}"></span> <a href="/wiki/Pr%C4%85d_przemienny" title="Prąd przemienny">prądu przemiennego</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06931e7bcffd32010f83c3d5dbef2d9bcbdcb670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.451ex; height:2.843ex;" alt="{\displaystyle i(t)}"></span> jest taką wartością natężenia <a href="/wiki/Pr%C4%85d_sta%C5%82y" title="Prąd stały">prądu stałego</a>, która w ciągu czasu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ee86b741f542feb8f95f3c81fd53b043a25e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.283ex; height:2.509ex;" alt="{\displaystyle T,}"></span> równego <a href="/wiki/Funkcja_okresowa" title="Funkcja okresowa">okresowi</a> prądu przemiennego, spowoduje ten sam efekt cieplny, co dany sygnał prądu przemiennego. Wartość ta jest średnią kwadratową tego sygnału: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{SK}={\sqrt {{\frac {1}{T}}\int \limits _{t_{0}}^{t_{0}+T}{i^{2}(t)}dt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>K</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>T</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{SK}={\sqrt {{\frac {1}{T}}\int \limits _{t_{0}}^{t_{0}+T}{i^{2}(t)}dt}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0ce9988cc947f3782e89c267dc8be721949b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:24.23ex; height:10.343ex;" alt="{\displaystyle I_{SK}={\sqrt {{\frac {1}{T}}\int \limits _{t_{0}}^{t_{0}+T}{i^{2}(t)}dt}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Zobacz_też"><span id="Zobacz_te.C5.BC"></span>Zobacz też</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=8" title="Edytuj sekcję: Zobacz też" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=8" title="Edytuj kod źródłowy sekcji: Zobacz też"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Nier%C3%B3wno%C5%9Bci_mi%C4%99dzy_%C5%9Brednimi" title="Nierówności między średnimi">nierówności między średnimi</a></li> <li><a href="/wiki/Nier%C3%B3wno%C5%9B%C4%87_mi%C4%99dzy_%C5%9Brednimi_pot%C4%99gowymi" title="Nierówność między średnimi potęgowymi">nierówność między średnimi potęgowymi</a></li> <li><a href="/wiki/Wariancja" title="Wariancja">wariancja</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Uwagi">Uwagi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=9" title="Edytuj sekcję: Uwagi" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=9" title="Edytuj kod źródłowy sekcji: Uwagi"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection refsection-uwagi ll-script ll-script-uwagi"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Nazwa stosowana głównie w żargonie elektronicznym w znaczeniu: wartość skuteczna.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Przypisy">Przypisy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&veaction=edit&section=10" title="Edytuj sekcję: Przypisy" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_kwadratowa&action=edit&section=10" title="Edytuj kod źródłowy sekcji: Przypisy"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-epwn-2"><span class="mw-cite-backlink"><a href="#cite_ref-epwn_2-0">↑</a></span> <span class="reference-text"><cite class="citation web open-access"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3984213"><i>średnia kwadratowa</i></a>, [w:] <i><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">Encyklopedia PWN</a></i> [online], <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Wydawnictwo Naukowe PWN</a><span class="accessdate"> [dostęp 2021-10-01]</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=unknown&rft.atitle=%C5%9Brednia+kwadratowa&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.jtitle=%5B%5BWydawnictwo+Naukowe+PWN%5D%5D&rft_id=https%3A%2F%2Fencyklopedia.pwn.pl%2Fhaslo%2F%3B3984213" style="display:none"> </span>.</cite></span> </li> </ol></div></div> <div class="navbox do-not-make-smaller mw-collapsible mw-collapsed" data-expandtext="pokaż" data-collapsetext="ukryj"><style data-mw-deduplicate="TemplateStyles:r74983602">.mw-parser-output .navbox{border:1px solid var(--border-color-base,#a2a9b1);margin:auto;text-align:center;padding:3px;margin-top:1em;clear:both}.mw-parser-output table.navbox:not(.pionowy){width:100%}.mw-parser-output .navbox+.navbox{border-top:0;margin-top:0}.mw-parser-output .navbox.pionowy{width:250px;float:right;clear:right;margin:0 0 0.4em 1.4em}.mw-parser-output .navbox.pionowy .before,.mw-parser-output .navbox.pionowy .after{padding:0.5em 0;text-align:center}.mw-parser-output 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