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<div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>In modeling and statistics</span> </div> </a> <ul id="toc-In_modeling_and_statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Synonyms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Synonyms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Synonyms</span> </div> </a> <ul id="toc-Synonyms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_variables" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other variables</span> </div> </a> <ul id="toc-Other_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span 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Available in 26 languages" > <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-26" 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">26 languages</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/%D9%85%D8%AA%D8%BA%D9%8A%D8%B1_%D9%85%D8%B3%D8%AA%D9%82%D9%84_%D9%88%D9%85%D8%AA%D8%BA%D9%8A%D8%B1_%D8%AA%D8%A7%D8%A8%D8%B9" title="متغير مستقل ومتغير تابع – Arabic" lang="ar" hreflang="ar" data-title="متغير مستقل ومتغير تابع" data-language-autonym="العربية" data-language-local-name="Arabic" 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%B8%E0%A7%8D%E0%A6%AC%E0%A6%BE%E0%A6%A7%E0%A7%80%E0%A6%A8_%E0%A6%9A%E0%A6%B2%E0%A6%95_%E0%A6%93_%E0%A6%85%E0%A6%A7%E0%A7%80%E0%A6%A8_%E0%A6%9A%E0%A6%B2%E0%A6%95" title="স্বাধীন চলক ও অধীন চলক – Bangla" lang="bn" hreflang="bn" data-title="স্বাধীন চলক ও অধীন চলক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BB%D0%B5%D0%B6%D0%BD%D1%8B%D1%8F_%D1%96_%D0%BD%D0%B5%D0%B7%D0%B0%D0%BB%D0%B5%D0%B6%D0%BD%D1%8B%D1%8F_%D0%B7%D0%BC%D0%B5%D0%BD%D0%BD%D1%8B%D1%8F" title="Залежныя і незалежныя зменныя – Belarusian" lang="be" hreflang="be" data-title="Залежныя і незалежныя зменныя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Variables_dependents_i_independents" title="Variables dependents i independents – Catalan" lang="ca" hreflang="ca" data-title="Variables dependents i independents" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Abh%C3%A4ngige_und_unabh%C3%A4ngige_Variable" title="Abhängige und unabhängige Variable – German" lang="de" hreflang="de" data-title="Abhängige und unabhängige Variable" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Variables_dependientes_e_independientes" title="Variables dependientes e independientes – Spanish" lang="es" hreflang="es" data-title="Variables dependientes e independientes" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aldagai_independente_eta_dependenteak" title="Aldagai independente eta dependenteak – Basque" lang="eu" hreflang="eu" data-title="Aldagai independente eta dependenteak" data-language-autonym="Euskara" data-language-local-name="Basque" 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/%D9%85%D8%AA%D8%BA%DB%8C%D8%B1%D9%87%D8%A7%DB%8C_%D9%88%D8%A7%D8%A8%D8%B3%D8%AA%D9%87_%D9%88_%D9%85%D8%B3%D8%AA%D9%82%D9%84" title="متغیرهای وابسته و مستقل – Persian" lang="fa" hreflang="fa" data-title="متغیرهای وابسته و مستقل" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Athr%C3%B3ga_neamhsple%C3%A1cha_agus_athr%C3%B3ga_sple%C3%A1cha" title="Athróga neamhspleácha agus athróga spleácha – Irish" lang="ga" hreflang="ga" data-title="Athróga neamhspleácha agus athróga spleácha" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8F%85%EB%A6%BD%EB%B3%80%EC%88%98%EC%99%80_%EC%A2%85%EC%86%8D%EB%B3%80%EC%88%98" title="독립변수와 종속변수 – Korean" lang="ko" hreflang="ko" data-title="독립변수와 종속변수" data-language-autonym="한국어" data-language-local-name="Korean" 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%AA%E0%A4%B0%E0%A4%A4%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0_%E0%A4%94%E0%A4%B0_%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%A4%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0_%E0%A4%9A%E0%A4%B0" title="परतंत्र और स्वतंत्र चर – Hindi" lang="hi" hreflang="hi" data-title="परतंत्र और स्वतंत्र चर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Variabili_dipendenti_e_indipendenti" title="Variabili dipendenti e indipendenti – Italian" lang="it" hreflang="it" data-title="Variabili dipendenti e indipendenti" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Varyab_endepandan" title="Varyab endepandan – Haitian Creole" lang="ht" hreflang="ht" data-title="Varyab endepandan" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Variabele_(onderzoek)" title="Variabele (onderzoek) – Dutch" lang="nl" hreflang="nl" data-title="Variabele (onderzoek)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zmienne_zale%C5%BCna_i_niezale%C5%BCna" title="Zmienne zależna i niezależna – Polish" lang="pl" hreflang="pl" data-title="Zmienne zależna i niezależna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Vari%C3%A1veis_dependentes_e_independentes" title="Variáveis dependentes e independentes – Portuguese" lang="pt" hreflang="pt" data-title="Variáveis dependentes e independentes" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B7%D0%B0%D0%B2%D0%B8%D1%81%D0%B8%D0%BC%D0%B0%D1%8F_%D0%B8_%D0%B7%D0%B0%D0%B2%D0%B8%D1%81%D0%B8%D0%BC%D0%B0%D1%8F_%D0%BF%D0%B5%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%BD%D1%8B%D0%B5" title="Независимая и зависимая переменные – Russian" lang="ru" hreflang="ru" data-title="Независимая и зависимая переменные" data-language-autonym="Русский" data-language-local-name="Russian" 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/Dependent_and_independent_variables" title="Dependent and independent variables – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dependent and independent variables" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Riippuvat_ja_riippumattomat_muuttujat" title="Riippuvat ja riippumattomat muuttujat – Finnish" lang="fi" hreflang="fi" data-title="Riippuvat ja riippumattomat muuttujat" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Beroende_variabel" title="Beroende variabel – Swedish" lang="sv" hreflang="sv" data-title="Beroende variabel" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AE%BF%E0%AE%AF%E0%AF%81%E0%AE%AE%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AE%BE_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AE%BF%E0%AE%AF%E0%AF%81%E0%AE%AE%E0%AF%8D" title="சார் மாறியும் சாரா மாறியும் – Tamil" lang="ta" hreflang="ta" data-title="சார் மாறியும் சாரா மாறியும்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Ba%C4%9F%C4%B1ml%C4%B1_ve_ba%C4%9F%C4%B1ms%C4%B1z_de%C4%9Fi%C5%9Fkenler" title="Bağımlı ve bağımsız değişkenler – Turkish" lang="tr" hreflang="tr" data-title="Bağımlı ve bağımsız değişkenler" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%BB%D0%B5%D0%B6%D0%BD%D0%B0_%D1%96_%D0%BD%D0%B5%D0%B7%D0%B0%D0%BB%D0%B5%D0%B6%D0%BD%D0%B0_%D0%B7%D0%BC%D1%96%D0%BD%D0%BD%D1%96" title="Залежна і незалежна змінні – Ukrainian" lang="uk" hreflang="uk" data-title="Залежна і незалежна змінні" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Bi%E1%BA%BFn_ph%E1%BB%A5_thu%E1%BB%99c_v%C3%A0_bi%E1%BA%BFn_%C4%91%E1%BB%99c_l%E1%BA%ADp" title="Biến phụ thuộc và biến độc lập – Vietnamese" lang="vi" hreflang="vi" data-title="Biến phụ thuộc và biến độc lập" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%87%AA%E8%AE%8A%E6%95%B8%E5%90%8C%E6%87%89%E8%AE%8A%E6%95%B8" title="自變數同應變數 – Cantonese" lang="yue" hreflang="yue" data-title="自變數同應變數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%87%AA%E5%8F%98%E9%87%8F%E5%92%8C%E5%9B%A0%E5%8F%98%E9%87%8F" title="自变量和因变量 – Chinese" lang="zh" hreflang="zh" data-title="自变量和因变量" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q320723#sitelinks-wikipedia" title="Edit interlanguage 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class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Independent_variable&redirect=no" class="mw-redirect" title="Independent variable">Independent variable</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For dependent and independent random variables, see <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence (probability theory)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Concept in mathematical modeling, statistical modeling and experimental sciences</div> <p>A variable is considered <b>dependent</b> if it depends on an <b>independent variable</b>. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a <a href="/wiki/Mathematical_function" class="mw-redirect" title="Mathematical function">mathematical function</a>), on the values of other variables. Independent variables, in turn, are not seen as depending on any other variable in the scope of the experiment in question.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> In this sense, some common independent variables are <a href="/wiki/Time" title="Time">time</a>, <a href="/wiki/Space" title="Space">space</a>, <a href="/wiki/Density" title="Density">density</a>, <a href="/wiki/Mass" title="Mass">mass</a>, <a href="/wiki/Volumetric_flow_rate" title="Volumetric flow rate">fluid flow rate</a>,<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and previous values of some observed value of interest (e.g. human population size) to predict future values (the dependent variable).<sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Of the two, it is always the dependent variable whose <a href="https://en.wiktionary.org/wiki/variation" class="extiw" title="wikt:variation">variation</a> is being studied, by altering inputs, also known as <b>regressors</b> in a <a href="/wiki/Statistics" title="Statistics">statistical</a> context. In an experiment, any variable that can be attributed a value without attributing a value to any other variable is called an independent variable. <a href="/wiki/Conceptual_model" title="Conceptual model">Models</a> and <a href="/wiki/Experiment" title="Experiment">experiments</a> test the effects that the independent variables have on the dependent variables. Sometimes, even if their influence is not of direct interest, independent variables may be included for other reasons, such as to account for their potential <a href="/wiki/Confounding" title="Confounding">confounding</a> effect. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Polynomialdeg2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/220px-Polynomialdeg2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/330px-Polynomialdeg2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/440px-Polynomialdeg2.svg.png 2x" data-file-width="320" data-file-height="320" /></a><figcaption>In single variable <a href="/wiki/Calculus" title="Calculus">calculus</a>, a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> is typically <a href="/wiki/Graph_of_a_function" title="Graph of a function">graphed</a> with the <a href="/wiki/Horizontal_axis" class="mw-redirect" title="Horizontal axis">horizontal axis</a> representing the independent variable and the <a href="/wiki/Vertical_axis" class="mw-redirect" title="Vertical axis">vertical axis</a> representing the dependent variable.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In this function, <i>y</i> is the dependent variable and <i>x</i> is the independent variable.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="In_pure_mathematics">In pure mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dependent_and_independent_variables&action=edit&section=1" title="Edit section: In pure mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In mathematics, a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> is a rule for taking an input (in the simplest case, a number or set of numbers)<sup id="cite_ref-carlson_6-0" class="reference"><a href="#cite_note-carlson-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and providing an output (which may also be a number).<sup id="cite_ref-carlson_6-1" class="reference"><a href="#cite_note-carlson-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A symbol that stands for an arbitrary input is called an <b>independent variable</b>, while a symbol that stands for an arbitrary output is called a <b>dependent variable</b>.<sup id="cite_ref-stewart_7-0" class="reference"><a href="#cite_note-stewart-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The most common symbol for the input is <span class="texhtml"><i>x</i></span>, and the most common symbol for the output is <span class="texhtml"><i>y</i></span>; the function itself is commonly written <span class="texhtml"><i>y</i> = <i>f</i>(<i>x</i>)</span>.<sup id="cite_ref-stewart_7-1" class="reference"><a href="#cite_note-stewart-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>It is possible to have multiple independent variables or multiple dependent variables. For instance, in <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a>, one often encounters functions of the form <span class="texhtml"><i>z</i> = <i>f</i>(<i>x</i>,<i>y</i>)</span>, where <span class="texhtml"><i>z</i></span> is a dependent variable and <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> are independent variables.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Functions with multiple outputs are often referred to as <a href="/wiki/Vector-valued_functions" class="mw-redirect" title="Vector-valued functions">vector-valued functions</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_modeling_and_statistics">In modeling and statistics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dependent_and_independent_variables&action=edit&section=2" title="Edit section: In modeling and statistics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Mathematical_modeling" class="mw-redirect" title="Mathematical modeling">mathematical modeling</a>, the relationship between the set of dependent variables and set of independent variables is studied.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2024)">citation needed</span></a></i>]</sup> </p><p>In the simple <a href="/wiki/Stochastic" title="Stochastic">stochastic</a> <a href="/wiki/Linear_model" title="Linear model">linear model</a> <span class="texhtml"><i>y</i><sub><i>i</i></sub> = a + b<i>x</i><sub><i>i</i></sub> + <i>e</i><sub><i>i</i></sub></span> the term <span class="texhtml"><i>y</i><sub><i>i</i></sub></span> is the <span class="texhtml mvar" style="font-style:italic;">i</span>th value of the dependent variable and <span class="texhtml"><i>x</i><sub><i>i</i></sub></span> is the <span class="texhtml mvar" style="font-style:italic;">i</span>th value of the independent variable. The term <span class="texhtml"><i>e</i><sub><i>i</i></sub></span> is known as the "error" and contains the variability of the dependent variable not explained by the independent variable.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2024)">citation needed</span></a></i>]</sup> </p><p>With multiple independent variables, the model is <span class="texhtml"><i>y</i><sub><i>i</i></sub> = a + b<i>x</i><sub><i>i</i>,1</sub> + b<i>x</i><sub><i>i</i>,2</sub> + ... + b<i>x</i><sub><i>i,n</i></sub> + <i>e</i><sub>i</sub></span>, where <span class="texhtml"><i>n</i></span> is the number of independent variables.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2019)">citation needed</span></a></i>]</sup> </p><p>In statistics, more specifically in <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a>, a <a href="/wiki/Scatter_plot" title="Scatter plot">scatter plot</a> of data is generated with <span class="texhtml mvar" style="font-style:italic;">X</span> as the independent variable and <span class="texhtml mvar" style="font-style:italic;">Y</span> as the dependent variable. This is also called a bivariate dataset, <span class="texhtml">(<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>)(<i>x</i><sub>2</sub>, <i>y</i><sub>2</sub>) ...(<i>x</i><sub><i>i</i></sub>, <i>y</i><sub><i>i</i></sub>)</span>. The simple linear regression model takes the form of <span class="texhtml"><i>Y</i><sub><i>i</i></sub> = a + B<i>x</i><sub><i>i</i></sub> + <i>U</i><sub><i>i</i></sub></span>, for <span class="texhtml"><i>i</i> = 1, 2, ... , <i>n</i></span>. In this case, <span class="texhtml"><i>U</i><sub><i>i</i></sub>, ... ,<i>U</i><sub><i>n</i></sub></span> are independent random variables. This occurs when the measurements do not influence each other. Through propagation of independence, the independence of <span class="texhtml"><i>U</i><sub><i>i</i></sub></span> implies independence of <span class="texhtml"><i>Y</i><sub><i>i</i></sub></span>, even though each <span class="texhtml"><i>Y</i><sub><i>i</i></sub></span> has a different expectation value. Each <span class="texhtml"><i>U</i><sub><i>i</i></sub></span> has an expectation value of 0 and a variance of <span class="texhtml">σ<sup>2</sup></span>.<sup id="cite_ref-Dekking_10-0" class="reference"><a href="#cite_note-Dekking-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Expectation of <span class="texhtml"><i>Y</i><sub><i>i</i></sub></span> Proof:<sup id="cite_ref-Dekking_10-1" class="reference"><a href="#cite_note-Dekking-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></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 E[Y_{i}]=E[\alpha +\beta x_{i}+U_{i}]=\alpha +\beta x_{i}+E[U_{i}]=\alpha +\beta x_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">[</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[Y_{i}]=E[\alpha +\beta x_{i}+U_{i}]=\alpha +\beta x_{i}+E[U_{i}]=\alpha +\beta x_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/060c6d2c55947cceb008cbe330e1b07a2c119d3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.124ex; height:2.843ex;" alt="{\displaystyle E[Y_{i}]=E[\alpha +\beta x_{i}+U_{i}]=\alpha +\beta x_{i}+E[U_{i}]=\alpha +\beta x_{i}.}"></span></dd></dl> <p>The line of best fit for the <a href="/wiki/Bivariate_data" title="Bivariate data">bivariate dataset</a> takes the form <span class="texhtml"><i>y</i> = <i>α</i> + <i>βx</i></span> and is called the regression line. <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> correspond to the intercept and slope, respectively.<sup id="cite_ref-Dekking_10-2" class="reference"><a href="#cite_note-Dekking-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>In an <a href="/wiki/Design_of_experiments" title="Design of experiments">experiment</a>, the variable manipulated by an experimenter is something that is proven to work, called an independent variable.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The dependent variable is the event expected to change when the independent variable is manipulated.<sup id="cite_ref-random_house_12-0" class="reference"><a href="#cite_note-random_house-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Data_mining" title="Data mining">data mining</a> tools (for <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">multivariate statistics</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>), the dependent variable is assigned a <i>role</i> as <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="target_variable"></span><span class="vanchor-text">target variable</span></span></b> (or in some tools as <i>label attribute</i>), while an independent variable may be assigned a role as <i>regular variable</i> <sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> or feature variable. Known values for the target variable are provided for the training data set and <a href="/wiki/Test_data" title="Test data">test data</a> set, but should be predicted for other data. The target variable is used in <a href="/wiki/Supervised_learning" title="Supervised learning">supervised learning</a> algorithms but not in unsupervised learning. </p> <div class="mw-heading mw-heading2"><h2 id="Synonyms">Synonyms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dependent_and_independent_variables&action=edit&section=3" title="Edit section: Synonyms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Depending on the context, an independent variable is sometimes called a "predictor variable", "regressor", "covariate", "manipulated variable", "explanatory variable", "exposure variable" (see <a href="/wiki/Reliability_theory" class="mw-redirect" title="Reliability theory">reliability theory</a>), "<a href="/wiki/Risk_factor" title="Risk factor">risk factor</a>" (see <a href="/wiki/Medical_statistics" title="Medical statistics">medical statistics</a>), "<a href="/wiki/Feature_(machine_learning)" title="Feature (machine learning)">feature</a>" (in <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> and <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a>) or "input variable".<sup id="cite_ref-Dodgeindepvar_14-0" class="reference"><a href="#cite_note-Dodgeindepvar-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Dodgeregression_15-0" class="reference"><a href="#cite_note-Dodgeregression-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, the term "control variable" is usually used instead of "covariate".<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>"Explanatory variable"<span class="anchor" id="explanatory_variable"></span> is preferred by some authors over "independent variable" when the quantities treated as independent variables may not be statistically independent or independently manipulable by the researcher.<sup id="cite_ref-Everitt1_21-0" class="reference"><a href="#cite_note-Everitt1-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Dodge1_22-0" class="reference"><a href="#cite_note-Dodge1-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> If the independent variable is referred to as an "explanatory variable" then the term "response variable"<span class="anchor" id="response_variable"></span> is preferred by some authors for the dependent variable.<sup id="cite_ref-Dodgeregression_15-1" class="reference"><a href="#cite_note-Dodgeregression-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Everitt1_21-1" class="reference"><a href="#cite_note-Everitt1-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Dodge1_22-1" class="reference"><a href="#cite_note-Dodge1-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Depending on the context, a dependent variable is sometimes called a "response variable", "regressand", "criterion", "predicted variable", "measured variable", "explained variable", "experimental variable", "responding variable", "outcome variable", "output variable", "target" or "label".<sup id="cite_ref-Dodgeregression_15-2" class="reference"><a href="#cite_note-Dodgeregression-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In economics endogenous variables are usually referencing the target. </p><p>"Explained variable"<span class="anchor" id="explained_variable"></span> is preferred by some authors over "dependent variable" when the quantities treated as "dependent variables" may not be statistically dependent.<sup id="cite_ref-DAUME_23-0" class="reference"><a href="#cite_note-DAUME-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> If the dependent variable is referred to as an "explained variable" then the term "predictor variable"<span class="anchor" id="predictor_variable"></span> is preferred by some authors for the independent variable.<sup id="cite_ref-DAUME_23-1" class="reference"><a href="#cite_note-DAUME-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>An example is provided by the analysis of trend in sea level by <a href="#CITEREFWoodworth1987">Woodworth (1987)</a>. Here the dependent variable (and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available. The primary independent variable was time. Use was made of a covariate consisting of yearly values of annual mean atmospheric pressure at sea level. The results showed that inclusion of the covariate allowed improved estimates of the trend against time to be obtained, compared to analyses which omitted the covariate. </p> <table class="wikitable"> <caption>Antonym pairs </caption> <tbody><tr> <td>independent</td> <td>dependent </td></tr> <tr> <td>input</td> <td>output </td></tr> <tr> <td>regressor</td> <td>regressand </td></tr> <tr> <td>predictor</td> <td>predicted </td></tr> <tr> <td>explanatory</td> <td>explained </td></tr> <tr> <td>exogenous</td> <td>endogenous </td></tr> <tr> <td>manipulated</td> <td>measured </td></tr> <tr> <td>exposure</td> <td>outcome </td></tr> <tr> <td>feature </td> <td>label or target </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Other_variables">Other variables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dependent_and_independent_variables&action=edit&section=4" title="Edit section: Other variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A variable may be thought to alter the dependent or independent variables, but may not actually be the focus of the experiment. So that the variable will be kept constant or monitored to try to minimize its effect on the experiment. Such variables may be designated as either a "controlled variable", "<a href="/wiki/Control_variable" title="Control variable">control variable</a>", or "fixed variable". </p><p>Extraneous variables, if included in a <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a> as independent variables, may aid a researcher with accurate response parameter estimation, <a href="/wiki/Prediction" title="Prediction">prediction</a>, and <a href="/wiki/Goodness_of_fit" title="Goodness of fit">goodness of fit</a>, but are not of substantive interest to the <a href="/wiki/Hypothesis" title="Hypothesis">hypothesis</a> under examination. For example, in a study examining the effect of post-secondary education on lifetime earnings, some extraneous variables might be gender, ethnicity, social class, genetics, intelligence, age, and so forth. A variable is extraneous only when it can be assumed (or shown) to influence the <a href="/wiki/Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variable</a>. If included in a regression, it can improve the <a href="/wiki/Model_fitting" class="mw-redirect" title="Model fitting">fit of the model</a>. If it is excluded from the regression and if it has a non-zero <a href="/wiki/Covariance" title="Covariance">covariance</a> with one or more of the independent variables of interest, its omission will <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">bias</a> the regression's result for the effect of that independent variable of interest. This effect is called <a href="/wiki/Confounding" title="Confounding">confounding</a> or <a href="/wiki/Omitted_variable_bias" class="mw-redirect" title="Omitted variable bias">omitted variable bias</a>; in these situations, design changes and/or controlling for a variable statistical control is necessary. </p><p>Extraneous variables are often classified into three types: </p> <ol><li>Subject variables, which are the characteristics of the individuals being studied that might affect their actions. These variables include age, gender, health status, mood, background, etc.</li> <li>Blocking variables or experimental variables are characteristics of the persons conducting the experiment which might influence how a person behaves. Gender, the presence of racial discrimination, language, or other factors may qualify as such variables.</li> <li>Situational variables are features of the environment in which the study or research was conducted, which have a bearing on the outcome of the experiment in a negative way. Included are the air temperature, level of activity, lighting, and time of day.</li></ol> <p>In modelling, variability that is not covered by the independent variable is designated by <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 e_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8f6a77f9502357c64944f69d25b9148162e822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.144ex; height:2.009ex;" alt="{\displaystyle e_{I}}"></span> and is known as the "<a href="/wiki/Errors_and_residuals" title="Errors and residuals">residual</a>", "side effect", "<a href="/wiki/Errors_and_residuals" title="Errors and residuals">error</a>", "unexplained share", "residual variable", "disturbance", or "tolerance". </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dependent_and_independent_variables&action=edit&section=5" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Effect of fertilizer on plant growths:</li></ul> <dl><dd>In a study measuring the influence of different quantities of fertilizer on plant growth, the independent variable would be the amount of fertilizer used. The dependent variable would be the growth in height or mass of the plant. The controlled variables would be the type of plant, the type of fertilizer, the amount of sunlight the plant gets, the size of the pots, etc.</dd></dl> <ul><li>Effect of drug dosage on symptom severity:</li></ul> <dl><dd>In a study of how different doses of a drug affect the severity of symptoms, a researcher could compare the frequency and intensity of symptoms when different doses are administered. Here the independent variable is the dose and the dependent variable is the frequency/intensity of symptoms.</dd></dl> <ul><li>Effect of temperature on pigmentation:</li></ul> <dl><dd>In measuring the amount of color removed from beetroot samples at different temperatures, temperature is the independent variable and amount of pigment removed is the dependent variable.</dd></dl> <ul><li>Effect of sugar added in a coffee:</li></ul> <dl><dd>The taste varies with the amount of sugar added in the coffee. Here, the sugar is the independent variable, while the taste is the dependent variable.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dependent_and_independent_variables&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Abscissa_and_ordinate" title="Abscissa and ordinate">Abscissa and ordinate</a></li> <li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking (statistics)</a></li> <li><a href="/wiki/Latent_and_observable_variables" title="Latent and observable variables">Latent and observable variables</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dependent_and_independent_variables&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Even if the existing dependency is invertible (e.g., by finding the <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> when it exists), the nomenclature is kept if the inverse dependency is not the object of study in the experiment.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dependent_and_independent_variables&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:1-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-:1_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output 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CRC Press, 2006. p.183</span> </li> <li id="cite_note-stewart-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-stewart_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-stewart_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Stewart, James. Calculus. Cengage Learning, 2011. Section 1.1</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Anton, Howard, Irl C. Bivens, and Stephen Davis. Calculus Single Variable. John Wiley & Sons, 2012. Section 0.1</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Larson, Ron, and Bruce Edwards. Calculus. Cengage Learning, 2009. Section 13.1</span> </li> <li id="cite_note-Dekking-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dekking_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dekking_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Dekking_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDekking2005" class="citation cs2">Dekking, Frederik Michel (2005), <i>A modern introduction to probability and statistics: understanding why and how</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-85233-896-2" title="Special:BookSources/1-85233-896-2"><bdi>1-85233-896-2</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/783259968">783259968</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+modern+introduction+to+probability+and+statistics%3A+understanding+why+and+how&rft.pub=Springer&rft.date=2005&rft_id=info%3Aoclcnum%2F783259968&rft.isbn=1-85233-896-2&rft.aulast=Dekking&rft.aufirst=Frederik+Michel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADependent+and+independent+variables" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://onlinestatbook.com/2/introduction/variables.html">"Variables"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Variables&rft_id=http%3A%2F%2Fonlinestatbook.com%2F2%2Fintroduction%2Fvariables.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADependent+and+independent+variables" class="Z3988"></span></span> </li> <li id="cite_note-random_house-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-random_house_12-0">^</a></b></span> <span class="reference-text"><i> Random House Webster's Unabridged Dictionary.</i> Random House, Inc. 2001. Page 534, 971. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-375-42566-7" title="Special:BookSources/0-375-42566-7">0-375-42566-7</a>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://1xltkxylmzx3z8gd647akcdvov.wpengine.netdna-cdn.com/wp-content/uploads/2013/10/rapidminer-5.0-manual-english_v1.0.pdf">English Manual version 1.0</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140210002634/http://1xltkxylmzx3z8gd647akcdvov.wpengine.netdna-cdn.com/wp-content/uploads/2013/10/rapidminer-5.0-manual-english_v1.0.pdf">Archived</a> 2014-02-10 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> for <a href="/wiki/RapidMiner" title="RapidMiner">RapidMiner</a> 5.0, October 2013.</span> </li> <li id="cite_note-Dodgeindepvar-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dodgeindepvar_14-0">^</a></b></span> <span class="reference-text">Dodge, Y. 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(2003) <i>The Oxford Dictionary of Statistical Terms</i>, OUP. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-920613-9" title="Special:BookSources/0-19-920613-9">0-19-920613-9</a> (entry for "regression")</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGujaratiPorter2009" class="citation book cs1">Gujarati, Damodar N.; <a href="/wiki/Dawn_C._Porter" title="Dawn C. Porter">Porter, Dawn C.</a> (2009). "Terminology and Notation". <i>Basic Econometrics</i> (Fifth international ed.). New York: McGraw-Hill. p. 21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-007-127625-2" title="Special:BookSources/978-007-127625-2"><bdi>978-007-127625-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Terminology+and+Notation&rft.btitle=Basic+Econometrics&rft.place=New+York&rft.pages=21&rft.edition=Fifth+international&rft.pub=McGraw-Hill&rft.date=2009&rft.isbn=978-007-127625-2&rft.aulast=Gujarati&rft.aufirst=Damodar+N.&rft.au=Porter%2C+Dawn+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADependent+and+independent+variables" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWooldridge2012" class="citation book cs1">Wooldridge, Jeffrey (2012). <i>Introductory Econometrics: A Modern Approach</i> (Fifth ed.). Mason, OH: South-Western Cengage Learning. pp. 22–23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-111-53104-1" title="Special:BookSources/978-1-111-53104-1"><bdi>978-1-111-53104-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+Econometrics%3A+A+Modern+Approach&rft.place=Mason%2C+OH&rft.pages=22-23&rft.edition=Fifth&rft.pub=South-Western+Cengage+Learning&rft.date=2012&rft.isbn=978-1-111-53104-1&rft.aulast=Wooldridge&rft.aufirst=Jeffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADependent+and+independent+variables" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLast2001" class="citation book cs1">Last, John M., ed. (2001). <i>A Dictionary of Epidemiology</i> (Fourth ed.). Oxford UP. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-514168-7" title="Special:BookSources/0-19-514168-7"><bdi>0-19-514168-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Dictionary+of+Epidemiology&rft.edition=Fourth&rft.pub=Oxford+UP&rft.date=2001&rft.isbn=0-19-514168-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADependent+and+independent+variables" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEveritt2002" class="citation book cs1">Everitt, B. S. (2002). <i>The Cambridge Dictionary of Statistics</i> (2nd ed.). Cambridge UP. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-81099-X" title="Special:BookSources/0-521-81099-X"><bdi>0-521-81099-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Cambridge+Dictionary+of+Statistics&rft.edition=2nd&rft.pub=Cambridge+UP&rft.date=2002&rft.isbn=0-521-81099-X&rft.aulast=Everitt&rft.aufirst=B.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADependent+and+independent+variables" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoodworth1987" class="citation journal cs1">Woodworth, P. L. (1987). "Trends in U.K. mean sea level". <i>Marine Geodesy</i>. <b>11</b> (1): 57–87. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987MarGe..11...57W">1987MarGe..11...57W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F15210608709379549">10.1080/15210608709379549</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Marine+Geodesy&rft.atitle=Trends+in+U.K.+mean+sea+level&rft.volume=11&rft.issue=1&rft.pages=57-87&rft.date=1987&rft_id=info%3Adoi%2F10.1080%2F15210608709379549&rft_id=info%3Abibcode%2F1987MarGe..11...57W&rft.aulast=Woodworth&rft.aufirst=P.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADependent+and+independent+variables" class="Z3988"></span></span> </li> <li id="cite_note-Everitt1-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-Everitt1_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Everitt1_21-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Everitt, B.S. (2002) Cambridge Dictionary of Statistics, CUP. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-81099-X" title="Special:BookSources/0-521-81099-X">0-521-81099-X</a></span> </li> <li id="cite_note-Dodge1-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dodge1_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dodge1_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Dodge, Y. (2003) <i>The Oxford Dictionary of Statistical Terms</i>, OUP. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-920613-9" title="Special:BookSources/0-19-920613-9">0-19-920613-9</a></span> </li> <li id="cite_note-DAUME-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-DAUME_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DAUME_23-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Ash Narayan Sah (2009) Data Analysis Using Microsoft Excel, New Delhi. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7446-716-4" title="Special:BookSources/978-81-7446-716-4">978-81-7446-716-4</a></span> </li> </ol></div></div> <style 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Independent_variable" class="extiw" title="v:Independent variable">Independent variable</a></b></i></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Dependent_variable" class="extiw" title="v:Dependent variable">Dependent variable</a></b></i></div></div> </div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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href="/wiki/Template:Differential_equations_topics" title="Template:Differential equations topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations_topics" title="Template talk:Differential equations topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations_topics" title="Special:EditPage/Template:Differential equations topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Differential_equations" style="font-size:114%;margin:0 4em"><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classification</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Operations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation for differentiation</a></li> <li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li> <li><a href="/wiki/Holonomic_function" title="Holonomic function">Holonomic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Attributes of variables</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Dependent and independent variables</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a></li> <li><a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Coupled</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Decoupled</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Order</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Degree</a></li> <li><a href="/wiki/Autonomous_system_(mathematics)" title="Autonomous system (mathematics)">Autonomous</a></li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact differential equation</a></li> <li><a href="/wiki/Jet_bundle#Partial_differential_equations" title="Jet bundle">On jet bundles</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Relation to processes</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference</a> (discrete analogue)</li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solutions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Existence/uniqueness</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Phase_space" title="Phase space">Phase space</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li> <li><a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic stability</a></li> <li><a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series solutions</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> solutions</li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_mathematical_jargon#Proof_techniques" class="mw-redirect" title="List of mathematical jargon">Inspection</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a></li> <li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Method of undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Euler_method" title="Euler method">Euler method</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference method</a></li> <li><a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson method</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume method</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></li> <li><a href="/wiki/List_of_linear_ordinary_differential_equations" title="List of linear ordinary differential equations">List of linear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_ordinary_differential_equations" title="List of nonlinear ordinary differential equations">List of nonlinear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_partial_differential_equations" title="List of nonlinear partial differential equations">List of nonlinear partial differential equations</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Ernst_Leonard_Lindel%C3%B6f" title="Ernst Leonard Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Sofya_Kovalevskaya" title="Sofya Kovalevskaya">Sofya Kovalevskaya</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Dependent_and_independent_variables&oldid=1256050811">https://en.wikipedia.org/w/index.php?title=Dependent_and_independent_variables&oldid=1256050811</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Design_of_experiments" title="Category:Design of 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