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class="vector-toc-link" href="#Probabilitateen_kalkulua_eta_zorizko_aldagai_normalen_estandarizazioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Probabilitateen kalkulua eta zorizko aldagai normalen estandarizazioa</span> </div> </a> <ul id="toc-Probabilitateen_kalkulua_eta_zorizko_aldagai_normalen_estandarizazioa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ikus,_gainera" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ikus,_gainera"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ikus, gainera</span> </div> </a> <ul id="toc-Ikus,_gainera-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kanpo_estekak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kanpo_estekak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Kanpo estekak</span> </div> </a> <ul id="toc-Kanpo_estekak-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="Edukiak" 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="Eduki taularen ikusgarritasuna aldatu" > <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">Eduki taularen ikusgarritasuna aldatu</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">Banaketa normal</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="Joan beste hizkuntza batean idatzitako artikulu batera. 73 hizkuntzatan eskuragarri." > <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-73" 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">73 hizkuntza</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Normalverteilung" title="Normalverteilung – Suitzako alemana" lang="gsw" hreflang="gsw" data-title="Normalverteilung" data-language-autonym="Alemannisch" data-language-local-name="Suitzako alemana" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84%D9%8A_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A" title="توزيع احتمالي طبيعي – arabiera" lang="ar" hreflang="ar" data-title="توزيع احتمالي طبيعي" data-language-autonym="العربية" data-language-local-name="arabiera" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal – asturiera" lang="ast" hreflang="ast" data-title="Distribución normal" data-language-autonym="Asturianu" data-language-local-name="asturiera" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Normal_paylanma" title="Normal paylanma – azerbaijanera" lang="az" hreflang="az" data-title="Normal paylanma" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijanera" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%86%D9%88%D8%B1%D9%85%D8%A7%D9%84_%D8%AF%D8%A7%D8%BA%DB%8C%D9%84%DB%8C%D9%85" title="نورمال داغیلیم – South Azerbaijani" lang="azb" hreflang="azb" data-title="نورمال داغیلیم" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B0%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D0%B5_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Нармальнае размеркаванне – bielorrusiera" lang="be" hreflang="be" data-title="Нармальнае размеркаванне" data-language-autonym="Беларуская" data-language-local-name="bielorrusiera" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%BE_%D1%80%D0%B0%D0%B7%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Нормално разпределение – bulgariera" lang="bg" hreflang="bg" data-title="Нормално разпределение" data-language-autonym="Български" data-language-local-name="bulgariera" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Normalna_raspodjela" title="Normalna raspodjela – bosniera" lang="bs" hreflang="bs" data-title="Normalna raspodjela" data-language-autonym="Bosanski" data-language-local-name="bosniera" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_normal" title="Distribució normal – katalana" lang="ca" hreflang="ca" data-title="Distribució normal" data-language-autonym="Català" data-language-local-name="katalana" 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/Norm%C3%A1ln%C3%AD_rozd%C4%9Blen%C3%AD" title="Normální rozdělení – txekiera" lang="cs" hreflang="cs" data-title="Normální rozdělení" data-language-autonym="Čeština" data-language-local-name="txekiera" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D0%B2%D0%B0%D0%BB%D0%B5%C3%A7%C4%95%D0%B2%C4%95" title="Гаусс валеçĕвĕ – txuvaxera" lang="cv" hreflang="cv" data-title="Гаусс валеçĕвĕ" data-language-autonym="Чӑвашла" data-language-local-name="txuvaxera" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dosraniad_normal" title="Dosraniad normal – galesa" lang="cy" hreflang="cy" data-title="Dosraniad normal" data-language-autonym="Cymraeg" data-language-local-name="galesa" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling – daniera" lang="da" hreflang="da" data-title="Normalfordeling" data-language-autonym="Dansk" data-language-local-name="daniera" 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/Normalverteilung" title="Normalverteilung – alemana" lang="de" hreflang="de" data-title="Normalverteilung" data-language-autonym="Deutsch" data-language-local-name="alemana" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B1%CE%BD%CE%BF%CE%BD%CE%B9%CE%BA%CE%AE_%CE%BA%CE%B1%CF%84%CE%B1%CE%BD%CE%BF%CE%BC%CE%AE" title="Κανονική κατανομή – greziera" lang="el" hreflang="el" data-title="Κανονική κατανομή" data-language-autonym="Ελληνικά" data-language-local-name="greziera" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Normal_distribution" title="Normal distribution – ingelesa" lang="en" hreflang="en" data-title="Normal distribution" data-language-autonym="English" data-language-local-name="ingelesa" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Normala_distribuo" title="Normala distribuo – esperantoa" lang="eo" hreflang="eo" data-title="Normala distribuo" data-language-autonym="Esperanto" data-language-local-name="esperantoa" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal – gaztelania" lang="es" hreflang="es" data-title="Distribución normal" data-language-autonym="Español" data-language-local-name="gaztelania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Normaaljaotus" title="Normaaljaotus – estoniera" lang="et" hreflang="et" data-title="Normaaljaotus" data-language-autonym="Eesti" data-language-local-name="estoniera" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%D9%86%D8%B1%D9%85%D8%A7%D9%84" title="توزیع نرمال – persiera" lang="fa" hreflang="fa" data-title="توزیع نرمال" data-language-autonym="فارسی" data-language-local-name="persiera" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Normaalijakauma" title="Normaalijakauma – finlandiera" lang="fi" hreflang="fi" data-title="Normaalijakauma" data-language-autonym="Suomi" data-language-local-name="finlandiera" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://fr.wikipedia.org/wiki/Loi_normale" title="Loi normale – frantsesa" lang="fr" hreflang="fr" data-title="Loi normale" data-language-autonym="Français" data-language-local-name="frantsesa" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Normoolferdialang" title="Normoolferdialang – iparraldeko frisiera" lang="frr" hreflang="frr" data-title="Normoolferdialang" data-language-autonym="Nordfriisk" data-language-local-name="iparraldeko frisiera" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/D%C3%A1ileadh_normalach" title="Dáileadh normalach – irlandera" lang="ga" hreflang="ga" data-title="Dáileadh normalach" data-language-autonym="Gaeilge" data-language-local-name="irlandera" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal – galiziera" lang="gl" hreflang="gl" data-title="Distribución normal" data-language-autonym="Galego" data-language-local-name="galiziera" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" title="התפלגות נורמלית – hebreera" lang="he" hreflang="he" data-title="התפלגות נורמלית" data-language-autonym="עברית" data-language-local-name="hebreera" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%E0%A4%AC%E0%A4%82%E0%A4%9F%E0%A4%A8" title="प्रसामान्य बंटन – hindia" lang="hi" hreflang="hi" data-title="प्रसामान्य बंटन" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Normalna_raspodjela" title="Normalna raspodjela – kroaziera" lang="hr" hreflang="hr" data-title="Normalna raspodjela" data-language-autonym="Hrvatski" data-language-local-name="kroaziera" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Norm%C3%A1lis_eloszl%C3%A1s" title="Normális eloszlás – hungariera" lang="hu" hreflang="hu" data-title="Normális eloszlás" data-language-autonym="Magyar" data-language-local-name="hungariera" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%B8%D6%80%D5%B4%D5%A1%D5%AC_%D5%A2%D5%A1%D5%B7%D5%AD%D5%B8%D6%82%D5%B4" title="Նորմալ բաշխում – armeniera" lang="hy" hreflang="hy" data-title="Նորմալ բաշխում" data-language-autonym="Հայերեն" data-language-local-name="armeniera" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Distribusi_normal" title="Distribusi normal – indonesiera" lang="id" hreflang="id" data-title="Distribusi normal" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiera" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Normaldreifing" title="Normaldreifing – islandiera" lang="is" hreflang="is" data-title="Normaldreifing" data-language-autonym="Íslenska" data-language-local-name="islandiera" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_normale" title="Distribuzione normale – italiera" lang="it" hreflang="it" data-title="Distribuzione normale" data-language-autonym="Italiano" data-language-local-name="italiera" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83" title="正規分布 – japoniera" lang="ja" hreflang="ja" data-title="正規分布" data-language-autonym="日本語" data-language-local-name="japoniera" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%9D%E1%83%A0%E1%83%9B%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%92%E1%83%90%E1%83%9C%E1%83%90%E1%83%AC%E1%83%98%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="ნორმალური განაწილება – georgiera" lang="ka" hreflang="ka" data-title="ნორმალური განაწილება" data-language-autonym="ქართული" data-language-local-name="georgiera" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%9A%D0%B0%D0%BB%D1%8B%D0%BF%D1%82%D1%8B_%D0%B4%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F" title="Қалыпты дисперсия – kazakhera" lang="kk" hreflang="kk" data-title="Қалыпты дисперсия" data-language-autonym="Қазақша" data-language-local-name="kazakhera" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EA%B7%9C_%EB%B6%84%ED%8F%AC" title="정규 분포 – koreera" lang="ko" hreflang="ko" data-title="정규 분포" data-language-autonym="한국어" data-language-local-name="koreera" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Distributio_normalis" title="Distributio normalis – latina" lang="la" hreflang="la" data-title="Distributio normalis" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Distribuzzion_normala" title="Distribuzzion normala – Lombard" lang="lmo" hreflang="lmo" data-title="Distribuzzion normala" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Normalusis_skirstinys" title="Normalusis skirstinys – lituaniera" lang="lt" hreflang="lt" data-title="Normalusis skirstinys" data-language-autonym="Lietuvių" data-language-local-name="lituaniera" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Norm%C4%81lais_sadal%C4%ABjums" title="Normālais sadalījums – letoniera" lang="lv" hreflang="lv" data-title="Normālais sadalījums" data-language-autonym="Latviešu" data-language-local-name="letoniera" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B1%D0%B0" title="Нормална распределба – mazedoniera" lang="mk" hreflang="mk" data-title="Нормална распределба" data-language-autonym="Македонски" data-language-local-name="mazedoniera" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%E0%A4%B5%E0%A4%BF%E0%A4%A4%E0%A4%B0%E0%A4%A3" title="सामान्य वितरण – marathera" lang="mr" hreflang="mr" data-title="सामान्य वितरण" data-language-autonym="मराठी" data-language-local-name="marathera" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Taburan_normal" title="Taburan normal – malaysiera" lang="ms" hreflang="ms" data-title="Taburan normal" data-language-autonym="Bahasa Melayu" data-language-local-name="malaysiera" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Normale_verdeling" title="Normale verdeling – nederlandera" lang="nl" hreflang="nl" data-title="Normale verdeling" data-language-autonym="Nederlands" data-language-local-name="nederlandera" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling – nynorsk (norvegiera)" lang="nn" hreflang="nn" data-title="Normalfordeling" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk (norvegiera)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Normalfordeling" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål (norvegiera)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_normalny" title="Rozkład normalny – poloniera" lang="pl" hreflang="pl" data-title="Rozkład normalny" data-language-autonym="Polski" data-language-local-name="poloniera" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Distribussion_%C3%ABd_Gauss" title="Distribussion ëd Gauss – Piedmontese" lang="pms" hreflang="pms" data-title="Distribussion ëd Gauss" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_normal" title="Distribuição normal – portugesa" lang="pt" hreflang="pt" data-title="Distribuição normal" data-language-autonym="Português" data-language-local-name="portugesa" 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/Distribu%C8%9Bia_Gauss" title="Distribuția Gauss – errumaniera" lang="ro" hreflang="ro" data-title="Distribuția Gauss" data-language-autonym="Română" data-language-local-name="errumaniera" 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%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Нормальное распределение – errusiera" lang="ru" hreflang="ru" data-title="Нормальное распределение" data-language-autonym="Русский" data-language-local-name="errusiera" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Normalna_raspodela" title="Normalna raspodela – serbokroaziera" lang="sh" hreflang="sh" data-title="Normalna raspodela" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaziera" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Normal_distribution" title="Normal distribution – Simple English" lang="en-simple" hreflang="en-simple" data-title="Normal distribution" 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/Norm%C3%A1lne_rozdelenie" title="Normálne rozdelenie – eslovakiera" lang="sk" hreflang="sk" data-title="Normálne rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="eslovakiera" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Normalna_porazdelitev" title="Normalna porazdelitev – esloveniera" lang="sl" hreflang="sl" data-title="Normalna porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="esloveniera" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shp%C3%ABrndarja_normale" title="Shpërndarja normale – albaniera" lang="sq" hreflang="sq" data-title="Shpërndarja normale" data-language-autonym="Shqip" data-language-local-name="albaniera" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D1%81%D0%BF%D0%BE%D0%B4%D0%B5%D0%BB%D0%B0" title="Нормална расподела – serbiera" lang="sr" hreflang="sr" data-title="Нормална расподела" data-language-autonym="Српски / srpski" data-language-local-name="serbiera" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Sebaran_normal" title="Sebaran normal – sundanera" lang="su" hreflang="su" data-title="Sebaran normal" data-language-autonym="Sunda" data-language-local-name="sundanera" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Normalf%C3%B6rdelning" title="Normalfördelning – suediera" lang="sv" hreflang="sv" data-title="Normalfördelning" data-language-autonym="Svenska" data-language-local-name="suediera" 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%87%E0%AE%AF%E0%AE%B2%E0%AF%8D%E0%AE%A8%E0%AE%BF%E0%AE%B2%E0%AF%88%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%B0%E0%AE%B5%E0%AE%B2%E0%AF%8D" title="இயல்நிலைப் பரவல் – tamilera" lang="ta" hreflang="ta" data-title="இயல்நிலைப் பரவல்" data-language-autonym="தமிழ்" data-language-local-name="tamilera" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%88%E0%B8%81%E0%B9%81%E0%B8%88%E0%B8%87%E0%B8%9B%E0%B8%A3%E0%B8%81%E0%B8%95%E0%B8%B4" title="การแจกแจงปรกติ – thailandiera" lang="th" hreflang="th" data-title="การแจกแจงปรกติ" data-language-autonym="ไทย" data-language-local-name="thailandiera" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distribusyong_normal" title="Distribusyong normal – tagaloa" lang="tl" hreflang="tl" data-title="Distribusyong normal" data-language-autonym="Tagalog" data-language-local-name="tagaloa" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Normal_da%C4%9F%C4%B1l%C4%B1m" title="Normal dağılım – turkiera" lang="tr" hreflang="tr" data-title="Normal dağılım" data-language-autonym="Türkçe" data-language-local-name="turkiera" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D0%B1%D2%AF%D0%BB%D0%B5%D0%BD%D0%B5%D1%88%D0%B5" title="Гаусс бүленеше – tatarera" lang="tt" hreflang="tt" data-title="Гаусс бүленеше" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarera" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D1%80%D0%BE%D0%B7%D0%BF%D0%BE%D0%B4%D1%96%D0%BB" title="Нормальний розподіл – ukrainera" lang="uk" hreflang="uk" data-title="Нормальний розподіл" data-language-autonym="Українська" data-language-local-name="ukrainera" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%A7%D8%B1%D9%85%D9%84_%D8%AA%D9%82%D8%B3%DB%8C%D9%85" title="نارمل تقسیم – urdua" lang="ur" hreflang="ur" data-title="نارمل تقسیم" data-language-autonym="اردو" data-language-local-name="urdua" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_ph%E1%BB%91i_chu%E1%BA%A9n" title="Phân phối chuẩn – vietnamera" lang="vi" hreflang="vi" data-title="Phân phối chuẩn" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamera" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83" title="正态分布 – wu txinera" lang="wuu" hreflang="wuu" data-title="正态分布" data-language-autonym="吴语" data-language-local-name="wu txinera" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A0%D7%90%D7%A8%D7%9E%D7%90%D7%9C%D7%A2_%D7%A4%D7%90%D7%A8%D7%98%D7%99%D7%99%D7%9C%D7%95%D7%A0%D7%92" title="נארמאלע פארטיילונג – yiddisha" lang="yi" hreflang="yi" data-title="נארמאלע פארטיילונג" data-language-autonym="ייִדיש" data-language-local-name="yiddisha" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83" title="正态分布 – txinera" lang="zh" hreflang="zh" data-title="正态分布" data-language-autonym="中文" data-language-local-name="txinera" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Si%C3%B4ng-th%C3%A0i_hun-p%C3%B2%CD%98" title="Siông-thài hun-pò͘ – Minnan" lang="nan" hreflang="nan" data-title="Siông-thài hun-pò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B8%B8%E6%85%8B%E5%88%86%E4%BD%88" title="常態分佈 – kantonera" lang="yue" hreflang="yue" data-title="常態分佈" data-language-autonym="粵語" data-language-local-name="kantonera" class="interlanguage-link-target"><span>粵語</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/Q133871#sitelinks-wikipedia" title="Aldatu hizkuntzen arteko loturak" class="wbc-editpage">Aldatu loturak</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Izen-tarteak"> <div 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<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="eu" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Normal_distribution_pdf.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Normal_distribution_pdf.png/250px-Normal_distribution_pdf.png" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Normal_distribution_pdf.png/375px-Normal_distribution_pdf.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Normal_distribution_pdf.png/500px-Normal_distribution_pdf.png 2x" data-file-width="1300" data-file-height="975" /></a><figcaption><a href="/wiki/Parametro_(estatistika)" title="Parametro (estatistika)">Parametro</a> ezberdinak dituzten lau banaketa normalen trinkotasun funtzioak. Kolore berdez irudikatzen dena N(0,1) banaketa normal estandarra da.</figcaption></figure> <p><a href="/wiki/Probabilitate_teoria" title="Probabilitate teoria">Probabilitate teorian</a> eta <a href="/wiki/Estatistika" title="Estatistika">estatistikan</a>, <a href="/wiki/Zorizko_aldagai" title="Zorizko aldagai">zorizko aldagai</a> bat <b>banaketa normalari</b> jarraitzen diola, <b>zorizko aldagai gausstarra</b> dela, edo laburrago normal banatzen dela esaten da, zorizko aldagaiaren <a href="/w/index.php?title=Trinkotasun_funtzio&action=edit&redlink=1" class="new" title="Trinkotasun funtzio (sortu gabe)">trinkotasun funtzioa</a> honelakoa denean: </p><p><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 f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\mathrm {e} ^{\displaystyle -{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}};-\infty <x<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> </msup> <mo>;</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\mathrm {e} ^{\displaystyle -{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}};-\infty <x<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90f7fbfa8b5099a908a7e464ac02668e3b0817c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:45.083ex; height:9.343ex;" alt="{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\mathrm {e} ^{\displaystyle -{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}};-\infty <x<\infty }"></span> </p><p><a href="/wiki/Estatistika" title="Estatistika">Estatistikan</a> gehien erabiltzen den <a href="/wiki/Probabilitate-banaketa" title="Probabilitate-banaketa">probabilitate banaketa</a> da, bere ezaugarri bereziengatik. Zorizko aldagaiak har ditzakeen balioei buruz inongo murrizketarik jartzen ez duela (bere balio posibleak <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 -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span>-tik <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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>-ra baitoaz), bere trinkotasun funtzioak kanpai itxura erakusten du beti (eta horregatik <b>Gaussen kanpaia</b> ere deitzen zaio), datuen <a href="/wiki/Histograma" title="Histograma">histograma</a> irudikatuz gero errealitateko aldagai asko bezalaxe. Hori dela eta, aldagai askoren <a href="/wiki/Eredu_(argipena)" title="Eredu (argipena)">eredu</a> gisa aukeratzen da, hortik datorkio <i>normal</i> izena. Bestalde, oso propietate matematiko interesgarriak ditu: <a href="/wiki/Probabilitate-banaketa" title="Probabilitate-banaketa">probabilitate banaketa</a> anitzen limitea da eta <a href="/wiki/Inferentzia" class="mw-disambig" title="Inferentzia">inferentzian</a> zenbatesle askoren banaketa izanik, <a href="/wiki/Hipotesi-froga" class="mw-redirect" title="Hipotesi-froga">hipotesi kontraste</a> eta <a href="/wiki/Konfiantza-tarte" title="Konfiantza-tarte">konfiantza tarte</a> askotarako erabiltzen da. <a href="/wiki/Limitearen_teorema_zentral" title="Limitearen teorema zentral">Limitearen teorema zentralari</a> esker, banaketa normala zorizko aldagaia faktore anitzen ekarpenen <a href="/wiki/Batuketa" title="Batuketa">batura</a> denerako ere da baliozkoa. </p><p>Banakuntza normala bi <a href="/wiki/Parametro_(estatistika)" title="Parametro (estatistika)">parametroren</a> araberakoa da: μ eta σ, <a href="/wiki/Batezbesteko" title="Batezbesteko">batez bestekoa</a> edo <a href="/wiki/Itxaropen_matematiko" title="Itxaropen matematiko">itxaropen matematikoa</a> eta <a href="/wiki/Desbideratze_estandar" title="Desbideratze estandar">desbideratze estandarra</a> hurrenez hurren. Horrela, X aldagai bat banaketa normalari jarraitzen diola honela adierazten da: </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\sim N{\big (}\mu ,\ \sigma {\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>μ</mi> <mo>,</mo> <mtext> </mtext> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim N{\big (}\mu ,\ \sigma {\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e6b29b20486c85a7b3ed8852ed27de4d77e150" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.618ex; height:3.176ex;" alt="{\displaystyle X\sim N{\big (}\mu ,\ \sigma {\big )}}"></span></dd></dl> <p><b>Banakuntza normal estandarra</b> μ=0 eta σ=1 parametroak dituen banaketa normala da eta beste banaketa normaletako <a href="/wiki/Probabilitate" title="Probabilitate">probabilitateak</a> kalkulatzeko oinarri gisa erabiltzen da. Banakuntza normal estandarra honela irudikatzen da: </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 Z\sim N{\big (}0,\ 1{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>∼</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\sim N{\big (}0,\ 1{\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629e725df84f7a73b82097cd73daec8b1a684269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.912ex; height:3.176ex;" alt="{\displaystyle Z\sim N{\big (}0,\ 1{\big )}}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Propietateak">Propietateak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Banaketa_normal&veaction=edit&section=1" title="Aldatu atal hau: «Propietateak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Banaketa_normal&action=edit&section=1" title="Edit section's source code: Propietateak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Banakuntza normalaren <a href="/wiki/Itxaropen_matematiko" title="Itxaropen matematiko">itxaropen matematikoa</a> μ da. <a href="/wiki/Desbideratze_estandar" title="Desbideratze estandar">Desbideratze estandarra</a> σ da.</li> <li>Banakuntza normalaren trinkotasun funtzioa <a href="/wiki/Simetria" title="Simetria">simetrikoa</a> da, <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=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20a0e3d1103b2704e577d15b7319bf0870e5d97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.83ex; height:2.176ex;" alt="{\displaystyle x=\mu }"></span> ardatzaren inguruan.</li> <li><a href="/wiki/Mediana" title="Mediana">Mediana</a> eta <a href="/wiki/Moda" title="Moda">moda</a> bat datoz μ itxaropen matematikoarekin.</li> <li>Itxaropen matematikoaren inguruko probabilitateak hauek ditugu: <ul><li>[<i>μ</i> - <i>σ</i>, <i>μ</i> + <i>σ</i>] tarteko <a href="/wiki/Probabilitate" title="Probabilitate">probabilitatea</a> %68,26 da.</li> <li>[<i>μ</i> - 2<i>σ</i>, <i>μ</i> + 2<i>σ</i>] tarteko probabilitatea %95,44 da.</li> <li>[<i>μ</i> -3<i>σ</i>, <i>μ</i> + 3<i>σ</i>] tarteko probabilitatea %99,74 da.</li></ul></li> <li>Banakuntza normal estandarrean <a href="/wiki/Inferentzia" class="mw-disambig" title="Inferentzia">inferentzia</a> estatistikoan maiz erabiltzen diren <a href="/wiki/Pertzentil" title="Pertzentil">pertzentil</a> jakingarri hauek daude: <ul><li><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 P(Z<3.29053)=0.9995}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>3.29053</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.9995</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Z<3.29053)=0.9995}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153f426cbd224816b49c6e3028a69933b6b4038d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.513ex; height:2.843ex;" alt="{\displaystyle P(Z<3.29053)=0.9995}"></span>, hau da, banaketa normal estandar batean bere azpitik 0.9995eko probabilitatea uzten duen balioa 3.29053 da.</li> <li><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 P(Z<3.09023)=0.999}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>3.09023</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.999</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Z<3.09023)=0.999}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97e37a394e56b9d614cc03e07ac5ac85f1104240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.35ex; height:2.843ex;" alt="{\displaystyle P(Z<3.09023)=0.999}"></span></li> <li><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 P(Z<2.57583)=0.995}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>2.57583</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.995</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Z<2.57583)=0.995}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8138c8c44617361416970bbece5e2b58315617a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.35ex; height:2.843ex;" alt="{\displaystyle P(Z<2.57583)=0.995}"></span></li> <li><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 P(Z<2.32635)=0.99}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>2.32635</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.99</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Z<2.32635)=0.99}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df65958d7c3c6d31243b72ae32444758d31d552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.188ex; height:2.843ex;" alt="{\displaystyle P(Z<2.32635)=0.99}"></span></li> <li><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 P(Z<2.05375)=0.98}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>2.05375</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.98</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Z<2.05375)=0.98}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b46471fc42a4ebbea0af353d160e3584535d414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.188ex; height:2.843ex;" alt="{\displaystyle P(Z<2.05375)=0.98}"></span></li> <li><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 P(Z<1.95996)=0.975}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>1.95996</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.975</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Z<1.95996)=0.975}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c80c9fb5d119ffa9e8945841ee5ea7054e3eebfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.35ex; height:2.843ex;" alt="{\displaystyle P(Z<1.95996)=0.975}"></span></li> <li><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 P(Z<1.64485)=0.95}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>1.64485</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.95</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Z<1.64485)=0.95}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b562599042db4e081f39bfd0173170922a3d889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.188ex; height:2.843ex;" alt="{\displaystyle P(Z<1.64485)=0.95}"></span></li> <li><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 P(Z<1.28155)=0.90}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>1.28155</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.90</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Z<1.28155)=0.90}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63cb809abaa6afbf744ed9ae238396bd68df8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.188ex; height:2.843ex;" alt="{\displaystyle P(Z<1.28155)=0.90}"></span></li></ul></li> <li><i>X</i> aldagai normal bati aldagai aldaketa lineala egiten bazaio, <i>X</i> ~ N(<i>μ</i>, <i>σ</i>) eta <i>a</i>, <i>b</i> <a href="/wiki/Zenbaki_erreal" title="Zenbaki erreal">zenbaki errealak</a> izanik, orduan (<i>aX</i> + <i>b</i>) ~ N(<i>aμ</i>+<i>b</i>, <i>|a|σ</i>).</li> <li><i>X</i> ~ N(<i>μ<sub>x</sub></i>, <i>σ<sub>x</sub></i>) eta <i>Y</i> ~ N(<i>μ<sub>y</sub></i>, <i>σ<sub>y</sub></i>) izanik, non <i>X¡</i> eta <i>Y</i> elkarrekiko independenteak diren, orduan: <ul><li><i>U</i> = <i>X</i> + <i>Y</i> ~ N(<i>μ<sub>x</sub></i> + <i>μ<sub>y</sub></i>, <i>σ<sub>x</sub></i><sup>2</sup> + <i>σ<sub>y</sub></i><sup>2</sup>)</li> <li>Arestikoaren alderantziz, independenteak diren bi zorizko aldagaien batura normal banatzen bada, bi zorizko aldagaiak normalak izan behar dira, Cramerren teoremaren arabera.</li> <li><i>V</i> = <i>X</i> - <i>Y</i> ~ N(<i>μ<sub>x</sub></i> - <i>μ<sub>y</sub></i>, <i>σ<sub>x</sub></i><sup>2</sup> + <i>σ<sub>y</sub></i><sup>2</sup>)</li> <li><i>X</i> eta <i>Y</i> zorizko aldagaiek bariantza berdinak badituzte, <i>U</i> eta <i>V</i> elkarrekiko independenteak dira.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Probabilitateen_kalkulua_eta_zorizko_aldagai_normalen_estandarizazioa">Probabilitateen kalkulua eta zorizko aldagai normalen estandarizazioa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Banaketa_normal&veaction=edit&section=2" title="Aldatu atal hau: «Probabilitateen kalkulua eta zorizko aldagai normalen estandarizazioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Banaketa_normal&action=edit&section=2" title="Edit section's source code: Probabilitateen kalkulua eta zorizko aldagai normalen estandarizazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Probabilitate" title="Probabilitate">Probabilitateak</a> <a href="/wiki/Banaketa-funtzio" title="Banaketa-funtzio">banaketa funtzioa</a> erabiliz kalkulatzen dira, μ eta σ zehaztu ondoren: </p><p><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 F_{X}(x)=P(X\leq x)=\int _{-\infty }^{x}f_{X}(x)dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,dx,\quad x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=P(X\leq x)=\int _{-\infty }^{x}f_{X}(x)dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,dx,\quad x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957e65c58bc9ccd711c03d38ce227f4088a33731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:70.644ex; height:7.176ex;" alt="{\displaystyle F_{X}(x)=P(X\leq x)=\int _{-\infty }^{x}f_{X}(x)dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,dx,\quad x\in \mathbb {R} }"></span> </p><p>Aurreko kalkulua <i>eskuz</i> oso konplexua denez, probabilitateak <b>estandarizazioa</b> delako prozesu batez kalkula daitezke errazago. Estandarizazioaz banaketa normal guztiak banaketa normal estandarrera bihurtzen dira. Bihurketa hau egin eta gero, banaketa normal estandarreko probabilitateak taula batez aise kalkulatzen dira (taula ikusteko, <a href="/wiki/Eranskina:_Banaketa_normalaren_banaketa_funtziorako_taula" title="Eranskina: Banaketa normalaren banaketa funtziorako taula">eranskinera</a> jo). <i>X</i> zorizko aldagai normal bat estandarizatzeko, μ <a href="/wiki/Itxaropen_matematiko" title="Itxaropen matematiko">itxaropena</a> kendu eta emaitza σ <a href="/wiki/Desbideratze_estandar" title="Desbideratze estandar">desbideratze estandarraz</a> zatitzen da: </p><p><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 P(X<x)=P{\Big (}{\frac {X-\mu }{\sigma }}<{\frac {x-\mu }{\sigma }}{\Big )}=P{\Big (}Z<{\frac {x-\mu }{\sigma }}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo><</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>X</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X<x)=P{\Big (}{\frac {X-\mu }{\sigma }}<{\frac {x-\mu }{\sigma }}{\Big )}=P{\Big (}Z<{\frac {x-\mu }{\sigma }}{\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246ecf243b1db811be78daeee5db17348f4d5a1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.953ex; height:5.343ex;" alt="{\displaystyle P(X<x)=P{\Big (}{\frac {X-\mu }{\sigma }}<{\frac {x-\mu }{\sigma }}{\Big )}=P{\Big (}Z<{\frac {x-\mu }{\sigma }}{\Big )}}"></span> </p><p>Adibidez, <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\sim N{\big (}\mu =8,\ \sigma =2{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>μ</mi> <mo>=</mo> <mn>8</mn> <mo>,</mo> <mtext> </mtext> <mi>σ</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim N{\big (}\mu =8,\ \sigma =2{\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f97c5dee4f931820ddd790881fb5050cd76e510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.14ex; height:3.176ex;" alt="{\displaystyle X\sim N{\big (}\mu =8,\ \sigma =2{\big )}}"></span> dugularik, zenbatekoa da <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 P{\big (}X<10.2{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>X</mi> <mo><</mo> <mn>10.2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P{\big (}X<10.2{\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8404aba9f8d8ed2b5e453cab6adc8c6e347acbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.088ex; height:3.176ex;" alt="{\displaystyle P{\big (}X<10.2{\big )}}"></span> probabilitatea? </p><p><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 P{\big (}X<10.2{\big )}=P{\Big (}Z<{\frac {10.2-8}{2}}{\Big )}=P(Z<1.1)=0.8643}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>X</mi> <mo><</mo> <mn>10.2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>Z</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>10.2</mn> <mo>−</mo> <mn>8</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo><</mo> <mn>1.1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.8643</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P{\big (}X<10.2{\big )}=P{\Big (}Z<{\frac {10.2-8}{2}}{\Big )}=P(Z<1.1)=0.8643}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d7e5ad773e1d5d996a259adbc46cf11e61c5e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.421ex; height:5.176ex;" alt="{\displaystyle P{\big (}X<10.2{\big )}=P{\Big (}Z<{\frac {10.2-8}{2}}{\Big )}=P(Z<1.1)=0.8643}"></span> </p><p>Balio batetik gorako probabilitateak eta balio negatiboei dagozkien probabilitateak banaketa normalaren <a href="/wiki/Simetria" title="Simetria">simetria</a> eta probabilitateen osagarritasuna erabiliz kalkulatzen dira. <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 z\geq 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>≥</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\geq 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912d67a3b5dff3b836720bfd4101eb3c9e82324a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.349ex; height:2.343ex;" alt="{\displaystyle z\geq 4}"></span> balioetarako probabilitateak ia 1 direla esan daiteke, taulan agertzen den zehaztasunerako. </p><p>Taularik ezean, banaketa funtzioaren probabilitateak <a href="/wiki/Errore_funtzio" title="Errore funtzio">errore funtzioa</a>, <b>erf</b> ikurrez irudikatzen dena, eta bere Taylorren garapena erabiliz kalkula daitezke. Banakuntza normal estandarraren banaketa funtzioa honela lotzen da errore funtzioarekin: </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 \Phi (z)={\frac {1}{2}}{\Bigl [}1+\operatorname {erf} {\Bigl (}{\frac {z}{\sqrt {2}}}{\Bigr )}{\Bigr ]},\quad x\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mi>erf</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (z)={\frac {1}{2}}{\Bigl [}1+\operatorname {erf} {\Bigl (}{\frac {z}{\sqrt {2}}}{\Bigr )}{\Bigr ]},\quad x\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36751a284a015993eaa134753db993cb65bd1792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.475ex; height:6.176ex;" alt="{\displaystyle \Phi (z)={\frac {1}{2}}{\Bigl [}1+\operatorname {erf} {\Bigl (}{\frac {z}{\sqrt {2}}}{\Bigr )}{\Bigr ]},\quad x\in \mathbb {R} .}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ikus,_gainera"><span id="Ikus.2C_gainera"></span>Ikus, gainera</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Banaketa_normal&veaction=edit&section=3" title="Aldatu atal hau: «Ikus, gainera»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Banaketa_normal&action=edit&section=3" title="Edit section's source code: Ikus, gainera"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Eranskina:_Banaketa_normalaren_banaketa_funtziorako_taula" title="Eranskina: Banaketa normalaren banaketa funtziorako taula">Eranskina: Banaketa normalaren banaketa funtziorako taula</a></li> 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src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datuak:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q133871" class="extiw" title="wikidata:Q133871">Q133871</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, 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