Standart sapma - Vikipedi

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bir varyasyon ölçüsü </div> </div> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"><a role="button" href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#p-lang" data-mw="interface" data-event-name="menu.languages" title="Dil" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> <span class="minerva-icon minerva-icon--language"></span> <span>Dil</span> </a></li> <li id="page-actions-watch" class="page-actions-menu__list-item"><a role="button" id="ca-watch" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=%C3%96zel:Kullan%C4%B1c%C4%B1OturumuA%C3%A7ma&returnto=Standart+sapma&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> <span class="minerva-icon minerva-icon--star"></span> <span>İzle</span> </a></li> <li id="page-actions-edit" class="page-actions-menu__list-item"><a role="button" id="ca-edit" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Değiştir</span> </a></li> </ul> </nav> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="tr" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <table class="box-Düzenle plainlinks metadata ambox ambox-style ambox-Wikify" role="presentation"> <tbody> <tr> <td class="mbox-text"> <div class="mbox-text-span"> <b>Bu madde, <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Vikipedi:Bi%C3%A7em_el_kitab%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vikipedi:Biçem el kitabı">Vikipedi biçem el kitabına</a> uygun değildir</b>.<span class="hide-when-compact"> Maddeyi, Vikipedi standartlarına uygun biçimde düzenleyerek Vikipedi'ye katkıda bulunabilirsiniz. <small>Gerekli düzenleme yapılmadan bu şablon kaldırılmamalıdır.</small></span> <small class="date-container"><i>(<span class="date">Şubat 2018</span>)</i></small> </div></td> </tr> </tbody> </table> <table class="box-Kaynaksız plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"> <tbody> <tr> <td class="mbox-text"> <div class="mbox-text-span"> Bu madde <b>hiçbir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Vikipedi:Kaynak_g%C3%B6sterme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vikipedi:Kaynak gösterme">kaynak</a> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Vikipedi:Do%C4%9Frulanabilirlik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vikipedi:Doğrulanabilirlik">içermemektedir</a>.</b><span class="hide-when-compact"> Lütfen <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Yard%C4%B1m:Dipnotlar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Yardım:Dipnotlar">güvenilir kaynaklar ekleyerek</a> <a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/w/index.php?title%3DStandart_sapma%26action%3Dedit">madde içeriğinin geliştirilmesine</a> yardımcı olun. Kaynaksız içerik itiraz konusu olabilir ve <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Vikipedi:Do%C4%9Frulanabilirlik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Kan%C4%B1t_sorumlulu%C4%9Fu" title="Vikipedi:Doğrulanabilirlik">kaldırılabilir</a>.<br><small><span class="plainlinks"><i>Kaynak ara:</i> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.google.com/search?as_eq%3Dwikipedia%26q%3D%2522Standart%2Bsapma%2522">"Standart sapma"</a> – <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.google.com/search?tbm%3Dnws%26q%3D%2522Standart%2Bsapma%2522%2B-wikipedia">haber</a> · <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.google.com/search?%26q%3D%2522Standart%2Bsapma%2522%2Bsite:news.google.com/newspapers%26source%3Dnewspapers">gazete</a> · <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.google.com/search?tbs%3Dbks:1%26q%3D%2522Standart%2Bsapma%2522%2B-wikipedia">kitap</a> · <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://scholar.google.com/scholar?q%3D%2522Standart%2Bsapma%2522">akademik</a> · <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/action/doBasicSearch?Query%3D%2522Standart%2Bsapma%2522%26acc%3Don%26wc%3Don">JSTOR</a></span></small></span> <small class="date-container"><i>(<span class="date">Eylül 2022</span>)</i></small><small class="hide-when-compact"><i> (<a href="https://tr-m-wikipedia-org.translate.goog/wiki/Yard%C4%B1m:Bak%C4%B1m_%C5%9Fablonunu_kald%C4%B1rmak?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Yardım:Bakım şablonunu kaldırmak">Bu şablonun nasıl ve ne zaman kaldırılması gerektiğini öğrenin</a>)</i></small> </div></td> </tr> </tbody> </table> <p><b>Standart sapma</b>, <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Olas%C4%B1l%C4%B1k_kuram%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Olasılık kuramı">Olasılık kuramı</a> ve <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C4%B0statistik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="İstatistik">istatistik</a> bilim dallarında, bir <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=%C4%B0statistiksel_anak%C3%BCtle&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="İstatistiksel anakütle (sayfa mevcut değil)">anakütle</a>, bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%96rneklem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect mw-disambig" title="Örneklem">örneklem</a>, bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Olas%C4%B1l%C4%B1k_da%C4%9F%C4%B1l%C4%B1m%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Olasılık dağılımı">olasılık dağılımı</a> veya bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Rassal_de%C4%9Fi%C5%9Fken?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rassal değişken">rassal değişken</a>, veri değerlerinin yayılımının özetlenmesi için kullanılan bir ölçüdür. <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Matematik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matematik">Matematik</a> notasyonunda genel olarak, bir anakütle veya bir rassal değişken veya bir olasılık dağılımı için standart sapma σ (eski Yunan harfi olan küçük sigma) ile ifade edilir; örneklem verileri için standart sapma için ise s veya s' (anakütle σ değeri için yansız kestirim kullanılır.)</p> <p>Standart sapma <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Varyans?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Varyans">varyansın</a> kareköküdür. Daha matematiksel bir ifade ile standart sapma veri değerlerinin <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Aritmetik_ortalama?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aritmetik ortalama">aritmetik ortalamadan</a> farklarının karelerinin toplamının veri sayısı -1'e bölümünün kareköküdür, yani verilerin ortalamadan sapmalarının kareler ortalamasının karekökü olarak tanımlanır. Standart sapma kavramının yayılma ölçüsü olarak kullanılmasını anlamak için ölçüm birimine bakmak gerekir. Diğer yayılma ölçüsü olan <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Varyans?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Varyans">varyans</a> verilerin ortalamadan farklarının karelerinin ortalaması olarak tanımlanır. Böylece varyans ölçüsü için veri birimlerinin karesi alınması gerekir ve varyansın birimi veri biriminin karesidir. Bu durum pratikte istenmeyen sonuçlar yaratabilir (Örneğin veriler birimi kilogram ise varyans birimi kilogram kare olur). Bundan kaçınmak için standart sapma için varyansın karekökü alınarak standart sapma birim veri birimi olması sağlanır ve verinin yayılımı böylece veri birimleri ile ölçülür.</p> <p>Standart sapma genel olarak niceliksel ölçekli sayılar için en çok kullanılan verilerin ortalamaya göre yayılmasını gösteren bir istatistiksel ölçüdür. Eğer birçok veri ortalamaya yakın ise, standart sapma değeri küçüktür; eğer birçok veri ortalamadan uzakta yayılmışlarsa standart sapma değeri büyük olur. Eğer bütün veri değerleri tıpatıp ayni ise standart sapma değeri sıfırdır</p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Dosya:Standard_deviation.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Standard_deviation.svg/220px-Standard_deviation.svg.png" decoding="async" width="220" height="177" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Standard_deviation.svg/330px-Standard_deviation.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Standard_deviation.svg/440px-Standard_deviation.svg.png 2x" data-file-width="512" data-file-height="411"></a> <figcaption> Mavi olarak gösterilen bir rassal değişken dağılımı için standart sapma değeri σ rassal değişken değerlerinin ortalama μ değeri etrafında yayılmasını gösterir. </figcaption> </figure> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="tr" dir="ltr"> <h2 id="mw-toc-heading">İçindekiler</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Tan%C4%B1mlama_ve_hesaplama"><span class="tocnumber">1</span> <span class="toctext">Tanımlama ve hesaplama</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Rassal_de%C4%9Fi%C5%9Fken_i%C3%A7in_standart_sapma"><span class="tocnumber">1.1</span> <span class="toctext">Rassal değişken için standart sapma</span></a></li> <li class="toclevel-2 tocsection-3"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Anak%C3%BCtle_standart_sapma_de%C4%9Ferinin_%C3%B6rneklem_standart_sapma_kullan%C4%B1larak_kestirimi"><span class="tocnumber">1.2</span> <span class="toctext">Anakütle standart sapma değerinin örneklem standart sapma kullanılarak kestirimi</span></a></li> <li class="toclevel-2 tocsection-4"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Bir_s%C3%BCrekli_rassal_de%C4%9Fi%C5%9Fken_i%C3%A7in_standart_sapma"><span class="tocnumber">1.3</span> <span class="toctext">Bir sürekli rassal değişken için standart sapma</span></a></li> </ul></li> <li class="toclevel-1 tocsection-5"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%C3%96rne%C4%9Fin"><span class="tocnumber">2</span> <span class="toctext">Örneğin</span></a></li> <li class="toclevel-1 tocsection-6"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#A%C3%A7%C4%B1klama_ve_uygulama"><span class="tocnumber">3</span> <span class="toctext">Açıklama ve uygulama</span></a> <ul> <li class="toclevel-2 tocsection-7"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Normal_da%C4%9F%C4%B1l%C4%B1m_g%C3%B6steren_veriler_i%C3%A7in_kurallar"><span class="tocnumber">3.1</span> <span class="toctext">Normal dağılım gösteren veriler için kurallar</span></a></li> <li class="toclevel-2 tocsection-8"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%C3%87ebi%C5%9Fev'in_e%C5%9Fitsizli%C4%9Fi"><span class="tocnumber">3.2</span> <span class="toctext">Çebişev'in eşitsizliği</span></a></li> </ul></li> <li class="toclevel-1 tocsection-9"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Standart_sapma_ve_ortalama_aras%C4%B1ndaki_ili%C5%9Fki"><span class="tocnumber">4</span> <span class="toctext">Standart sapma ve ortalama arasındaki ilişki</span></a></li> <li class="toclevel-1 tocsection-10"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ayr%C4%B1ca_bak%C4%B1n%C4%B1z"><span class="tocnumber">5</span> <span class="toctext">Ayrıca bakınız</span></a></li> <li class="toclevel-1 tocsection-11"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Kaynak%C3%A7a"><span class="tocnumber">6</span> <span class="toctext">Kaynakça</span></a></li> <li class="toclevel-1 tocsection-12"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#D%C4%B1%C5%9F_kaynaklar"><span class="tocnumber">7</span> <span class="toctext">Dış kaynaklar</span></a></li> <li class="toclevel-1 tocsection-13"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_sapma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#D%C4%B1%C5%9F_ba%C4%9Flant%C4%B1lar"><span class="tocnumber">8</span> <span class="toctext">Dış bağlantılar</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Tanımlama_ve_hesaplama"><span id="Tan.C4.B1mlama_ve_hesaplama"></span>Tanımlama ve hesaplama</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Tanımlama ve hesaplama" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <div class="mw-heading mw-heading3"> <h3 id="Rassal_değişken_için_standart_sapma"><span id="Rassal_de.C4.9Fi.C5.9Fken_i.C3.A7in_standart_sapma"></span>Rassal değişken için standart sapma</h3><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Rassal değişken için standart sapma" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <p>Bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Rassal_de%C4%9Fi%C5%9Fken?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rassal değişken">rassal değişken</a> olan <i>X</i> için standart sapma şöyle tanımlanır:</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 {\begin{array}{lcl}\sigma &=&{\sqrt {\operatorname {E} ((X-\operatorname {E} (X))^{2})}}={\sqrt {\operatorname {E} (X^{2})-(\operatorname {E} (X))^{2}}}\\&=&{\sqrt {\operatorname {Var} (X)}}\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> σ </mi> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal"> E </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mi> X </mi> <mo> − </mo> <mi mathvariant="normal"> E </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> X </mi> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </msqrt> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal"> E </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msup> <mi> X </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> E </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> X </mi> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> Var </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> X </mi> <mo stretchy="false"> ) </mo> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{lcl}\sigma &=&{\sqrt {\operatorname {E} ((X-\operatorname {E} (X))^{2})}}={\sqrt {\operatorname {E} (X^{2})-(\operatorname {E} (X))^{2}}}\\&=&{\sqrt {\operatorname {Var} (X)}}\end{array}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102be82c387781da351c2298d29ecbe16b9808f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.248ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{lcl}\sigma &=&{\sqrt {\operatorname {E} ((X-\operatorname {E} (X))^{2})}}={\sqrt {\operatorname {E} (X^{2})-(\operatorname {E} (X))^{2}}}\\&=&{\sqrt {\operatorname {Var} (X)}}\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 50.248ex;height: 7.176ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102be82c387781da351c2298d29ecbe16b9808f5" data-alt="{\displaystyle {\begin{array}{lcl}\sigma &=&{\sqrt {\operatorname {E} ((X-\operatorname {E} (X))^{2})}}={\sqrt {\operatorname {E} (X^{2})-(\operatorname {E} (X))^{2}}}\\&=&{\sqrt {\operatorname {Var} (X)}}\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Burada E(<i>X</i>) <i>X</i> için <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Beklenen_de%C4%9Fer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Beklenen değer">beklenen değer</a> yani <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Ortalama?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ortalama">ortalama</a> ve Var(<i>X</i>) <i>X</i> için varyans değeridir.</p> <p><i>Her</i> rassal değişken dağılım tipi için bir standart değer var olması gerekli değildir. Çünkü bazı dağılımlar için <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Beklenen_de%C4%9Fer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Beklenen değer">beklenen değer</a> bulunamaz. Örneğin, <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Cauchy_da%C4%9F%C4%B1l%C4%B1m%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cauchy dağılımı">Cauchy dağılımı</a> gösteren bir rassal değişken <i>X</i> için bir standart sapma yoktur; çünkü E(<i>X</i>) tanımlanamaz.</p> <p>Eğer bir rassal değişken <i>X</i> (<a href="https://tr-m-wikipedia-org.translate.goog/wiki/Reel_say%C4%B1lar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Reel sayılar">reel sayılar</a> olan) <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 \scriptstyle x_{1},\dots ,x_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle x_{1},\dots ,x_{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594e7d1418c6dcc36247acfdf5e310cc9c82e406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.517ex; height:1.509ex;" alt="{\displaystyle \scriptstyle x_{1},\dots ,x_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.517ex;height: 1.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594e7d1418c6dcc36247acfdf5e310cc9c82e406" data-alt="{\displaystyle \scriptstyle x_{1},\dots ,x_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> değerlerini eşit olasılıkla alırsa, o rassal değişken için standart sapma şöyle hesaplanır:</p> <p>Önce, <i>X</i> için <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Ortalama?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ortalama">ortalama</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.445ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" data-alt="{\displaystyle {\overline {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, şu toplam olarak tanımlanır:</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 {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mo> ⋯ </mo> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mrow> <mi> n </mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e3ccab7d7fb59582bc656289083aa1c07d053f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.528ex; height:6.843ex;" alt="{\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}}"> </noscript><span class="lazy-image-placeholder" style="width: 35.528ex;height: 6.843ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e3ccab7d7fb59582bc656289083aa1c07d053f" data-alt="{\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Burada <i>N</i> alınan örneklem büyüklüğü sayısıdır.</p> <p>Sonra, standart sapma ifadesi şöyle basitleştirilir:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f828739d3a59a36fea27d5d76b779fbbbcecc83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.65ex; height:7.509ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 22.65ex;height: 7.509ex;vertical-align: -3.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f828739d3a59a36fea27d5d76b779fbbbcecc83" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Yani, bir aralıklı tekdüze dağılım gösteren rassal değişken <i>X</i> için standart sapma şöyle hesaplanır:</p> <ol> <li>Her <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_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{i}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.129ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" data-alt="{\displaystyle x_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> değeri için <i>x</i><sub><i>i</i></sub> le ortalama değer olan <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 \scriptstyle {\overline {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle {\overline {x}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d61e55efaaade217e62e2c2e93e578f66842784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.034ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\overline {x}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.034ex;height: 1.843ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d61e55efaaade217e62e2c2e93e578f66842784" data-alt="{\displaystyle \scriptstyle {\overline {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> arasında olan farklar <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 \scriptstyle x_{i}-{\overline {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle x_{i}-{\overline {x}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d67ffecf3df83a049116f253a102635e4494412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.877ex; height:2.009ex;" alt="{\displaystyle \scriptstyle x_{i}-{\overline {x}}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.877ex;height: 2.009ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d67ffecf3df83a049116f253a102635e4494412" data-alt="{\displaystyle \scriptstyle x_{i}-{\overline {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> olarak bulunur.</li> <li>Bu farkların kareleri hesaplanır.</li> <li>Bu farkların karelerinin ortalaması bulunur. Bu değer <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Varyans?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Varyans">varyans</a>, yani σ<sup>2</sup>, olur.</li> <li>Bu varyans değerinin kare kökü alınır.</li> </ol> <p>Ancak hesapları elle veya el hesap makinesi ile yapmak için genellikle daha uygun bir formül kullanılır:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\left(\sum _{i=1}^{n}x_{i}^{2}-n{\overline {x}}^{2}\right)}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> − </mo> <mi> n </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\left(\sum _{i=1}^{n}x_{i}^{2}-n{\overline {x}}^{2}\right)}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ec43c877dfd1992cbbbc1c502126689633201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.688ex; height:8.176ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\left(\sum _{i=1}^{n}x_{i}^{2}-n{\overline {x}}^{2}\right)}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 26.688ex;height: 8.176ex;vertical-align: -3.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ec43c877dfd1992cbbbc1c502126689633201" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\left(\sum _{i=1}^{n}x_{i}^{2}-n{\overline {x}}^{2}\right)}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Bu iki formülün birbire eşitliği biraz <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Cebir?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cebir">cebir</a> kullanılarak gösterilebilir:</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 {\begin{aligned}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}&={}\sum _{i=1}^{n}(x_{i}^{2}-2x_{i}{\overline {x}}+{\overline {x}}^{2})\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-\left(2{\overline {x}}\sum _{i=1}^{n}x_{i}\right)+n{\overline {x}}^{2}\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-2{\overline {x}}(n{\overline {x}})+n{\overline {x}}^{2}\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-n{\overline {x}}^{2}.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> − </mo> <mn> 2 </mn> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> + </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> − </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> n </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> − </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> n </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> − </mo> <mi> n </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}&={}\sum _{i=1}^{n}(x_{i}^{2}-2x_{i}{\overline {x}}+{\overline {x}}^{2})\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-\left(2{\overline {x}}\sum _{i=1}^{n}x_{i}\right)+n{\overline {x}}^{2}\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-2{\overline {x}}(n{\overline {x}})+n{\overline {x}}^{2}\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-n{\overline {x}}^{2}.\end{aligned}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5026d1235156776d0bb1060732440a00ef88a159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:48.411ex; height:29.843ex;" alt="{\displaystyle {\begin{aligned}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}&={}\sum _{i=1}^{n}(x_{i}^{2}-2x_{i}{\overline {x}}+{\overline {x}}^{2})\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-\left(2{\overline {x}}\sum _{i=1}^{n}x_{i}\right)+n{\overline {x}}^{2}\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-2{\overline {x}}(n{\overline {x}})+n{\overline {x}}^{2}\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-n{\overline {x}}^{2}.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 48.411ex;height: 29.843ex;vertical-align: -14.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5026d1235156776d0bb1060732440a00ef88a159" data-alt="{\displaystyle {\begin{aligned}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}&={}\sum _{i=1}^{n}(x_{i}^{2}-2x_{i}{\overline {x}}+{\overline {x}}^{2})\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-\left(2{\overline {x}}\sum _{i=1}^{n}x_{i}\right)+n{\overline {x}}^{2}\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-2{\overline {x}}(n{\overline {x}})+n{\overline {x}}^{2}\\&{}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)-n{\overline {x}}^{2}.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Anakütle_standart_sapma_değerinin_örneklem_standart_sapma_kullanılarak_kestirimi"><span id="Anak.C3.BCtle_standart_sapma_de.C4.9Ferinin_.C3.B6rneklem_standart_sapma_kullan.C4.B1larak_kestirimi"></span>Anakütle standart sapma değerinin örneklem standart sapma kullanılarak kestirimi</h3><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Anakütle standart sapma değerinin örneklem standart sapma kullanılarak kestirimi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <p>Pratik hayatta, her bir anakütle elemanın ölçülmesini gerektiren bir <i>anakütle standart sapma</i> değeri bulmak, bazı çok nadir haller dışında (örnegin <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_hale_getirilmi%C5%9F_mekanik_test_etme&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Standart hale getirilmiş mekanik test etme (sayfa mevcut değil)">standart hale getirilmiş mekanik test etme</a>), hiç realistik değildir. Nerede ise her halde, anakütleden bir rastgele örneklem alınır ve bu örneklemden anakütle standart sapması için bir kestirim değer bulunur. Bu kestirim, çok kere <i>örneklem standart sapma</i>sını anakütle standard sapmasının aynı olan bir formülü kullanmak suretiyle yapılır:</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 s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}}\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> s </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mi> N </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}}\,,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f9a59b127439fa7925e9d000b14603e087c1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.599ex; height:8.009ex;" alt="{\displaystyle s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}}\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 27.599ex;height: 8.009ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f9a59b127439fa7925e9d000b14603e087c1a7" data-alt="{\displaystyle s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}}\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Burada <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 \scriptstyle \{x_{1},\,x_{2},\,\ldots ,\,x_{n}\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mspace width="thinmathspace"></mspace> <mo> … </mo> <mo> , </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle \{x_{1},\,x_{2},\,\ldots ,\,x_{n}\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a5250b12c29dce1aebbcabc679fd32d4b59721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.552ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \{x_{1},\,x_{2},\,\ldots ,\,x_{n}\}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.552ex;height: 2.176ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a5250b12c29dce1aebbcabc679fd32d4b59721" data-alt="{\displaystyle \scriptstyle \{x_{1},\,x_{2},\,\ldots ,\,x_{n}\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> örneklem değerleri ve <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 \scriptstyle {\overline {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle {\overline {x}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d61e55efaaade217e62e2c2e93e578f66842784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.034ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\overline {x}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.034ex;height: 1.843ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d61e55efaaade217e62e2c2e93e578f66842784" data-alt="{\displaystyle \scriptstyle {\overline {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> örneklem ortalamasıdır. Bölen değer olan <i>n</i> − 1</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 \scriptstyle (x_{1}-{\overline {x}},\,\dots ,\,x_{N}-{\overline {x}})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> , </mo> <mspace width="thinmathspace"></mspace> <mo> … </mo> <mo> , </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle (x_{1}-{\overline {x}},\,\dots ,\,x_{N}-{\overline {x}})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad825e2e42881c24e8a9262e51fb41a2ad8f07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.579ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (x_{1}-{\overline {x}},\,\dots ,\,x_{N}-{\overline {x}})}"> </noscript><span class="lazy-image-placeholder" style="width: 13.579ex;height: 2.176ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad825e2e42881c24e8a9262e51fb41a2ad8f07d" data-alt="{\displaystyle \scriptstyle (x_{1}-{\overline {x}},\,\dots ,\,x_{N}-{\overline {x}})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </dd> </dl> <p>vektörü içinde bulunan <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Serbestik_derecesi&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Serbestik derecesi (sayfa mevcut değil)">serbestik derecesi</a> olur.</p> <p>Bu belki bir bakıma uygundur; çünkü eğer bir anakütle varyansının kavramsal olarak var olduğu biliniyorsa ve örneklem için anakütleden her eleman çekiminden sonra bu eleman geri konulursa, bilinmektedir ki örneklem varyansı (yani <i>s</i><sup>2</sup>) anakütle varyansı (yani σ<sup>2</sup>) için bir <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Yans%C4%B1z_kestirim&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Yansız kestirim (sayfa mevcut değil)">yansız kestirim</a> olur. Ancak bu standart sapmalar için doğru değildir ; yani yukaridaki gibi bulunan örneklem standart sapması (<i>s</i>) anakütle standart sapması (σ) için yansız kestirim değeri değildir ve <i>s</i> ile anakütle standart sapması biraz daha küçükçe tahmin edilir. Eğer rassal değişken <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Normal_da%C4%9F%C4%B1l%C4%B1m?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Normal dağılım">normal dağılım</a> gösteriyorsa, bu yansız olan kestirim pratikte çok kolay olmayan bir dönüşüm ile elde edilebilmektedir. Ayrıca zaten bir kestirim için yansız olmak karakteri her zaman çok istenir bir özellik değildir.</p> <p>Çok kullanılan diğer bir kestrim ise benzer bir ifade ile şöyle verilir:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4048555f54b9cd7ce802768d330bee5e83d0b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.996ex; height:7.509ex;" alt="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 18.996ex;height: 7.509ex;vertical-align: -3.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4048555f54b9cd7ce802768d330bee5e83d0b6e" data-alt="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>olur. Eğer anakütle normal dağılım gösteriyorsa, bu şekildeki kestirim yansız kestirimden her zaman biraz daha küçük <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Ortalama_hata_karesi&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Ortalama hata karesi (sayfa mevcut değil)">ortalama hata karesi</a> gösterir ve bu nedenle normal için <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Maksimum_olabilirlik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Maksimum olabilirlik">maksimum olabilirlik kestirimi</a> olur.</p> <div class="mw-heading mw-heading3"> <h3 id="Bir_sürekli_rassal_değişken_için_standart_sapma"><span id="Bir_s.C3.BCrekli_rassal_de.C4.9Fi.C5.9Fken_i.C3.A7in_standart_sapma"></span>Bir sürekli rassal değişken için standart sapma</h3><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Bir sürekli rassal değişken için standart sapma" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <p><a href="https://tr-m-wikipedia-org.translate.goog/wiki/S%C3%BCrekli_olas%C4%B1l%C4%B1k_da%C4%9F%C4%B1l%C4%B1mlar%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sürekli olasılık dağılımları">Sürekli olasılık dağılımları</a> için genellikle standart sapma değerinin dağılıma özel olan parametreleri kullanılarak hesaplanması için formül vardır. Genel olarak ise, <i>p</i>(<i>x</i>) <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Olas%C4%B1l%C4%B1k_yo%C4%9Funluk_fonksiyonu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Olasılık yoğunluk fonksiyonu">olasılık yoğunluk fonksiyonu</a> olan bir sürekli rassal değişken olan <i>X</i> için standart sapma şöyle verilir:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {\int (x-\mu )^{2}\,p(x)\,dx}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo> ∫ </mo> <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> <mspace width="thinmathspace"></mspace> <mi> p </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {\int (x-\mu )^{2}\,p(x)\,dx}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75dc3ebea441fe7cdc07227b0d75cc75653352ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.009ex; height:7.509ex;" alt="{\displaystyle \sigma ={\sqrt {\int (x-\mu )^{2}\,p(x)\,dx}}}"> </noscript><span class="lazy-image-placeholder" style="width: 25.009ex;height: 7.509ex;vertical-align: -3.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75dc3ebea441fe7cdc07227b0d75cc75653352ff" data-alt="{\displaystyle \sigma ={\sqrt {\int (x-\mu )^{2}\,p(x)\,dx}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Burada</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 \mu =\int x\,p(x)\,dx}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> μ </mi> <mo> = </mo> <mo> ∫ </mo> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mi> p </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mu =\int x\,p(x)\,dx} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc81a779a41a646514f59c045e2d0068778e83e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.039ex; height:5.676ex;" alt="{\displaystyle \mu =\int x\,p(x)\,dx}"> </noscript><span class="lazy-image-placeholder" style="width: 16.039ex;height: 5.676ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc81a779a41a646514f59c045e2d0068778e83e6" data-alt="{\displaystyle \mu =\int x\,p(x)\,dx}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Örneğin"><span id=".C3.96rne.C4.9Fin"></span>Örneğin</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Örneğin" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Burada önce çok ufak bir anakütle veri serisi için standart sapma hesaplaması gösterilmektedir. Bu seri bir inşaat firmasının yabancılara yaptığı aylık daire satış sayılarını göstermektedir ve veri serisi şudur: <b>{ 5, 2, 11, 12, 3, 6 }</b>.</p> <p>1. Önce bir aritmetik ortalama <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 {\overline {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.445ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" data-alt="{\displaystyle {\overline {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> şöyle hesaplanır:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4048555f54b9cd7ce802768d330bee5e83d0b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.996ex; height:7.509ex;" alt="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 18.996ex;height: 7.509ex;vertical-align: -3.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4048555f54b9cd7ce802768d330bee5e83d0b6e" data-alt="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </dd> </dl> <p>Burada i her veriye verilen sıra numarasıdır yani i=1,2,3,...,6. Yani</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_{1}=5\,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 5 </mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{1}=5\,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e40b744482a62598ea702b8acdc9eafe88a8173d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:7.032ex; height:2.509ex;" alt="{\displaystyle x_{1}=5\,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 7.032ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e40b744482a62598ea702b8acdc9eafe88a8173d" data-alt="{\displaystyle x_{1}=5\,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}=2\,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{2}=2\,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090a407a0b3d33f371f5dccca9d22e51f7f63f63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:7.032ex; height:2.509ex;" alt="{\displaystyle x_{2}=2\,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 7.032ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090a407a0b3d33f371f5dccca9d22e51f7f63f63" data-alt="{\displaystyle x_{2}=2\,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3}=11\,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> = </mo> <mn> 11 </mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{3}=11\,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d261817b40ddfbed8be3a7ae71043eb82d8fa2b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:8.194ex; height:2.509ex;" alt="{\displaystyle x_{3}=11\,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 8.194ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d261817b40ddfbed8be3a7ae71043eb82d8fa2b1" data-alt="{\displaystyle x_{3}=11\,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{4}=12\,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <mo> = </mo> <mn> 12 </mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{4}=12\,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b38b588bef85e9673b6907e9b4cf78820092390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:8.194ex; height:2.509ex;" alt="{\displaystyle x_{4}=12\,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 8.194ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b38b588bef85e9673b6907e9b4cf78820092390" data-alt="{\displaystyle x_{4}=12\,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{5}=3\,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msub> <mo> = </mo> <mn> 3 </mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{5}=3\,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b5a6e435fc208c37aca994e01635094d300f3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:7.032ex; height:2.509ex;" alt="{\displaystyle x_{5}=3\,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 7.032ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b5a6e435fc208c37aca994e01635094d300f3b4" data-alt="{\displaystyle x_{5}=3\,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{6}=6\,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 6 </mn> </mrow> </msub> <mo> = </mo> <mn> 6 </mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{6}=6\,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9754a3586323ff352db3665dd8b37c415a08dfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:7.032ex; height:2.509ex;" alt="{\displaystyle x_{6}=6\,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 7.032ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9754a3586323ff352db3665dd8b37c415a08dfc" data-alt="{\displaystyle x_{6}=6\,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Bu halde <i>N</i> = 6 olup veri büyüklüğü veya anakütle hacmidir.</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 {\overline {x}}={\frac {1}{6}}\sum _{i=1}^{6}x_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 6 </mn> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}={\frac {1}{6}}\sum _{i=1}^{6}x_{i}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/705453c18d6f0a8047a47ceb0204068fae6602ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.8ex; height:7.343ex;" alt="{\displaystyle {\overline {x}}={\frac {1}{6}}\sum _{i=1}^{6}x_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.8ex;height: 7.343ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/705453c18d6f0a8047a47ceb0204068fae6602ec" data-alt="{\displaystyle {\overline {x}}={\frac {1}{6}}\sum _{i=1}^{6}x_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>        <i>N</i> yerine 6 </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}={\frac {1}{6}}\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 6 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}={\frac {1}{6}}\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37ac4674d541339040351992d8493fe29055abe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.244ex; height:5.176ex;" alt="{\displaystyle {\overline {x}}={\frac {1}{6}}\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 37.244ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37ac4674d541339040351992d8493fe29055abe" data-alt="{\displaystyle {\overline {x}}={\frac {1}{6}}\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <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 {\overline {x}}={\frac {1}{6}}\left(5+2+11+12+3+6\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> + </mo> <mn> 2 </mn> <mo> + </mo> <mn> 11 </mn> <mo> + </mo> <mn> 12 </mn> <mo> + </mo> <mn> 3 </mn> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}={\frac {1}{6}}\left(5+2+11+12+3+6\right)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94144f700ab69a56dbd924a4f75e79fb231ce22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.24ex; height:5.176ex;" alt="{\displaystyle {\overline {x}}={\frac {1}{6}}\left(5+2+11+12+3+6\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 32.24ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94144f700ab69a56dbd924a4f75e79fb231ce22" data-alt="{\displaystyle {\overline {x}}={\frac {1}{6}}\left(5+2+11+12+3+6\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <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 {\overline {x}}=6.5}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mn> 6.5 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}=6.5} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d8e92a1a0553711d79bab1f71b9eba544e4bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.515ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}=6.5}"> </noscript><span class="lazy-image-placeholder" style="width: 7.515ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d8e92a1a0553711d79bab1f71b9eba544e4bbb" data-alt="{\displaystyle {\overline {x}}=6.5}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>    Bu aritmetik ortalamadır. </dd> </dl> <p>2. Standart sapma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma \,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b645815a7c06785b9cc44600737d59624e4c75f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle \sigma \,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 1.717ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b645815a7c06785b9cc44600737d59624e4c75f7" data-alt="{\displaystyle \sigma \,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> değerini bulma:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4048555f54b9cd7ce802768d330bee5e83d0b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.996ex; height:7.509ex;" alt="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 18.996ex;height: 7.509ex;vertical-align: -3.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4048555f54b9cd7ce802768d330bee5e83d0b6e" data-alt="{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}\,\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\sum _{i=1}^{6}(x_{i}-{\overline {x}})^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 6 </mn> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\sum _{i=1}^{6}(x_{i}-{\overline {x}})^{2}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e09e72b22de88b203af96ebd029b6390db1446d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.901ex; height:8.009ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\sum _{i=1}^{6}(x_{i}-{\overline {x}})^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 21.901ex;height: 8.009ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e09e72b22de88b203af96ebd029b6390db1446d" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\sum _{i=1}^{6}(x_{i}-{\overline {x}})^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>        <i>N</i> yerine 6 </dd> </dl> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\sum _{i=1}^{6}(x_{i}-6.5)^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 6 </mn> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mn> 6.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\sum _{i=1}^{6}(x_{i}-6.5)^{2}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d9299a24bd70809de138a8bccb55a3dfbb4aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.428ex; height:8.009ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\sum _{i=1}^{6}(x_{i}-6.5)^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 23.428ex;height: 8.009ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d9299a24bd70809de138a8bccb55a3dfbb4aad" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\sum _{i=1}^{6}(x_{i}-6.5)^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>       <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 {\overline {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.445ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" data-alt="{\displaystyle {\overline {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> yerine 6.5 </dd> </dl> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left[(5-6.5)^{2}+(2-6.5)^{2}+(11-6.5)^{2}+(12-6.5)^{2}+(3-6.5)^{2}+(6-6.5)^{2}\right]}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mrow> <mo> [ </mo> <mrow> <mo stretchy="false"> ( </mo> <mn> 5 </mn> <mo> − </mo> <mn> 6.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo> − </mo> <mn> 6.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 11 </mn> <mo> − </mo> <mn> 6.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 12 </mn> <mo> − </mo> <mn> 6.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 3 </mn> <mo> − </mo> <mn> 6.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 6 </mn> <mo> − </mo> <mn> 6.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ] </mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left[(5-6.5)^{2}+(2-6.5)^{2}+(11-6.5)^{2}+(12-6.5)^{2}+(3-6.5)^{2}+(6-6.5)^{2}\right]}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ebea7b51a954c6816c62562891fd11b4d643ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:86.633ex; height:6.176ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left[(5-6.5)^{2}+(2-6.5)^{2}+(11-6.5)^{2}+(12-6.5)^{2}+(3-6.5)^{2}+(6-6.5)^{2}\right]}}}"> </noscript><span class="lazy-image-placeholder" style="width: 86.633ex;height: 6.176ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ebea7b51a954c6816c62562891fd11b4d643ce" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left[(5-6.5)^{2}+(2-6.5)^{2}+(11-6.5)^{2}+(12-6.5)^{2}+(3-6.5)^{2}+(6-6.5)^{2}\right]}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left((-1.5)^{2}+(-4.5)^{2}+(4.5)^{2}+(5.5)^{2}+(-3.5)^{2}+(-0.5)^{2}\right)}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 4.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 4.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 5.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 3.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 0.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left((-1.5)^{2}+(-4.5)^{2}+(4.5)^{2}+(5.5)^{2}+(-3.5)^{2}+(-0.5)^{2}\right)}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480aaf101cb10accce6c278c46a9072440753bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:67.714ex; height:6.176ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left((-1.5)^{2}+(-4.5)^{2}+(4.5)^{2}+(5.5)^{2}+(-3.5)^{2}+(-0.5)^{2}\right)}}}"> </noscript><span class="lazy-image-placeholder" style="width: 67.714ex;height: 6.176ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480aaf101cb10accce6c278c46a9072440753bb9" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left((-1.5)^{2}+(-4.5)^{2}+(4.5)^{2}+(5.5)^{2}+(-3.5)^{2}+(-0.5)^{2}\right)}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left(2.25+20.25+20.25+30.25+12.25+0.25\right)}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2.25 </mn> <mo> + </mo> <mn> 20.25 </mn> <mo> + </mo> <mn> 20.25 </mn> <mo> + </mo> <mn> 30.25 </mn> <mo> + </mo> <mn> 12.25 </mn> <mo> + </mo> <mn> 0.25 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left(2.25+20.25+20.25+30.25+12.25+0.25\right)}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8133eea3ce5086e3ed9307db9d046bfc8ac1b20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.604ex; height:6.176ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left(2.25+20.25+20.25+30.25+12.25+0.25\right)}}}"> </noscript><span class="lazy-image-placeholder" style="width: 54.604ex;height: 6.176ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8133eea3ce5086e3ed9307db9d046bfc8ac1b20c" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left(2.25+20.25+20.25+30.25+12.25+0.25\right)}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {\frac {85.5}{6}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn> 85.5 </mn> <mn> 6 </mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {\frac {85.5}{6}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa34a6d4a35e5d876dd206379db875fff43cb3ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.722ex; height:6.176ex;" alt="{\displaystyle \sigma ={\sqrt {\frac {85.5}{6}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.722ex;height: 6.176ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa34a6d4a35e5d876dd206379db875fff43cb3ba" data-alt="{\displaystyle \sigma ={\sqrt {\frac {85.5}{6}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {14.25}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 14.25 </mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {14.25}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6295d74807078d1382bb851b9fd4481720dabac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.661ex; height:3.009ex;" alt="{\displaystyle \sigma ={\sqrt {14.25}}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.661ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6295d74807078d1382bb851b9fd4481720dabac4" data-alt="{\displaystyle \sigma ={\sqrt {14.25}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =3.77\,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mn> 3.77 </mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma =3.77\,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d82bba27db54c17ce5a7dfe06c3ceac272ce9a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:8.949ex; height:2.176ex;" alt="{\displaystyle \sigma =3.77\,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 8.949ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d82bba27db54c17ce5a7dfe06c3ceac272ce9a3" data-alt="{\displaystyle \sigma =3.77\,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>   Bu standart sapma değeri olur. </dd> </dl> <p>Bu sonucun dikkati çekecek bir yanı verilerin <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Tam_say%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tam sayı">tam sayı</a> olmasına rağmen standart sapmanın (ve ayni şekilde aritmetik ortalamanın) kesirli olmasıdır.</p> <p>Bu hesaplamayı daha kolaylaştırmak için şu formül kullanılabilir:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\left(\sum _{i=1}^{n}x_{i}^{2}-n{\overline {x}}^{2}\right)}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> − </mo> <mi> n </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\left(\sum _{i=1}^{n}x_{i}^{2}-n{\overline {x}}^{2}\right)}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ec43c877dfd1992cbbbc1c502126689633201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.688ex; height:8.176ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\left(\sum _{i=1}^{n}x_{i}^{2}-n{\overline {x}}^{2}\right)}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 26.688ex;height: 8.176ex;vertical-align: -3.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ec43c877dfd1992cbbbc1c502126689633201" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{n}}\left(\sum _{i=1}^{n}x_{i}^{2}-n{\overline {x}}^{2}\right)}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>1. Önce bir aritmetik ortalama <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 {\overline {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.445ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" data-alt="{\displaystyle {\overline {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> hesaplanır:</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 {\overline {x}}={\frac {1}{N}}\sum _{i=1}^{N}x_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> N </mi> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}={\frac {1}{N}}\sum _{i=1}^{N}x_{i}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/269c0c2d794f2caa3e7c35c8de0a91d5d555e7c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.701ex; height:7.343ex;" alt="{\displaystyle {\overline {x}}={\frac {1}{N}}\sum _{i=1}^{N}x_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.701ex;height: 7.343ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/269c0c2d794f2caa3e7c35c8de0a91d5d555e7c1" data-alt="{\displaystyle {\overline {x}}={\frac {1}{N}}\sum _{i=1}^{N}x_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}={\frac {1}{6}}\left(5+2+11+12+3+6\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> + </mo> <mn> 2 </mn> <mo> + </mo> <mn> 11 </mn> <mo> + </mo> <mn> 12 </mn> <mo> + </mo> <mn> 3 </mn> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}={\frac {1}{6}}\left(5+2+11+12+3+6\right)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94144f700ab69a56dbd924a4f75e79fb231ce22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.24ex; height:5.176ex;" alt="{\displaystyle {\overline {x}}={\frac {1}{6}}\left(5+2+11+12+3+6\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 32.24ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94144f700ab69a56dbd924a4f75e79fb231ce22" data-alt="{\displaystyle {\overline {x}}={\frac {1}{6}}\left(5+2+11+12+3+6\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}=6.5}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mn> 6.5 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}=6.5} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d8e92a1a0553711d79bab1f71b9eba544e4bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.515ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}=6.5}"> </noscript><span class="lazy-image-placeholder" style="width: 7.515ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d8e92a1a0553711d79bab1f71b9eba544e4bbb" data-alt="{\displaystyle {\overline {x}}=6.5}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>    Bu aritmetik ortalamadır. </dd> </dl> <p>2. Sonra toplam kareler bulunur:</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 \sum {(x_{i})^{2}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum {(x_{i})^{2}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe11b71d6806dadb55ae91dce5ba99dfa089a5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.735ex; height:3.843ex;" alt="{\displaystyle \sum {(x_{i})^{2}}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.735ex;height: 3.843ex;vertical-align: -1.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe11b71d6806dadb55ae91dce5ba99dfa089a5b8" data-alt="{\displaystyle \sum {(x_{i})^{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> = 5<sup>2</sup> + 2<sup>2</sup> + 11<sup>2</sup> + 12<sup>2</sup> + 3<sup>2</sup> + 6 <sup>2</sup> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum {(x_{i})^{2}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum {(x_{i})^{2}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe11b71d6806dadb55ae91dce5ba99dfa089a5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.735ex; height:3.843ex;" alt="{\displaystyle \sum {(x_{i})^{2}}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.735ex;height: 3.843ex;vertical-align: -1.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe11b71d6806dadb55ae91dce5ba99dfa089a5b8" data-alt="{\displaystyle \sum {(x_{i})^{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> = 25+4+121+144+9+36 </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum {(x_{i})^{2}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum {(x_{i})^{2}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe11b71d6806dadb55ae91dce5ba99dfa089a5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.735ex; height:3.843ex;" alt="{\displaystyle \sum {(x_{i})^{2}}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.735ex;height: 3.843ex;vertical-align: -1.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe11b71d6806dadb55ae91dce5ba99dfa089a5b8" data-alt="{\displaystyle \sum {(x_{i})^{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> = 339 </dd> </dl> <p>3. Bunlar formüle konulur:</p> <p>Yani <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 \sum {(x_{i})^{2}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum {(x_{i})^{2}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe11b71d6806dadb55ae91dce5ba99dfa089a5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.735ex; height:3.843ex;" alt="{\displaystyle \sum {(x_{i})^{2}}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.735ex;height: 3.843ex;vertical-align: -1.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe11b71d6806dadb55ae91dce5ba99dfa089a5b8" data-alt="{\displaystyle \sum {(x_{i})^{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> = 339     <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 {\overline {x}}=6.5}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mn> 6.5 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {x}}=6.5} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d8e92a1a0553711d79bab1f71b9eba544e4bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.515ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}=6.5}"> </noscript><span class="lazy-image-placeholder" style="width: 7.515ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d8e92a1a0553711d79bab1f71b9eba544e4bbb" data-alt="{\displaystyle {\overline {x}}=6.5}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> = </mo> <mn> 6 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n=6} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=6}"> </noscript><span class="lazy-image-placeholder" style="width: 5.656ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f" data-alt="{\displaystyle n=6}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>     formüle girer:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left(339-6\times {6.5}^{2}\right)}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mn> 339 </mn> <mo> − </mo> <mn> 6 </mn> <mo> × </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn> 6.5 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left(339-6\times {6.5}^{2}\right)}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ebf30a60f88ac51421b0012afd363b5018c46c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.624ex; height:6.176ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left(339-6\times {6.5}^{2}\right)}}}"> </noscript><span class="lazy-image-placeholder" style="width: 25.624ex;height: 6.176ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ebf30a60f88ac51421b0012afd363b5018c46c" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\left(339-6\times {6.5}^{2}\right)}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\ (339-253.5)}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mtext>   </mtext> <mo stretchy="false"> ( </mo> <mn> 339 </mn> <mo> − </mo> <mn> 253.5 </mn> <mo stretchy="false"> ) </mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\ (339-253.5)}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0504ab73fd93c97deeb1ac01e2ae419e2c7cf5b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.765ex; height:6.176ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\ (339-253.5)}}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.765ex;height: 6.176ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0504ab73fd93c97deeb1ac01e2ae419e2c7cf5b3" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\ (339-253.5)}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\ (85.5)}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mtext>   </mtext> <mo stretchy="false"> ( </mo> <mn> 85.5 </mn> <mo stretchy="false"> ) </mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\ (85.5)}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa5d973d0b7ddbb3938cc4cbca17c13e5c33360e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.275ex; height:6.176ex;" alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\ (85.5)}}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.275ex;height: 6.176ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa5d973d0b7ddbb3938cc4cbca17c13e5c33360e" data-alt="{\displaystyle \sigma ={\sqrt {{\frac {1}{6}}\ (85.5)}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {14.25}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 14.25 </mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma ={\sqrt {14.25}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6295d74807078d1382bb851b9fd4481720dabac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.661ex; height:3.009ex;" alt="{\displaystyle \sigma ={\sqrt {14.25}}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.661ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6295d74807078d1382bb851b9fd4481720dabac4" data-alt="{\displaystyle \sigma ={\sqrt {14.25}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =3.77\,\!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo> = </mo> <mn> 3.77 </mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma =3.77\,\!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d82bba27db54c17ce5a7dfe06c3ceac272ce9a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:8.949ex; height:2.176ex;" alt="{\displaystyle \sigma =3.77\,\!}"> </noscript><span class="lazy-image-placeholder" style="width: 8.949ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d82bba27db54c17ce5a7dfe06c3ceac272ce9a3" data-alt="{\displaystyle \sigma =3.77\,\!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>   Bu standart sapma değeridir. </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Açıklama_ve_uygulama"><span id="A.C3.A7.C4.B1klama_ve_uygulama"></span>Açıklama ve uygulama</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Açıklama ve uygulama" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>Belli bir seri sayı için standart sapma değerini bilmek ve bu kavramı anlamak demek bir ortalama etrafında bu serinin ne kadar yayılım gösterdiğini anlamaktır. Standart sapmanın büyük olması veri noktalarının ortalamadan daha uzak yayıldıklarını; küçük bir standart sapma ise ortalama etrafında daha çok yakın gruplaştıklarını gösterir.</p> <p>Standart sapma belirsizliğin bir ölçüsü olarak hizmet edebilir. Fiziksel bilimlerde, tekrar tekrar yapılan deneyler ve deneylerde alınan ölçüler ise gösterilen standart sapma olgusu bu deneyin ölçülmesindeki kesinlik ve doğruluğunu gösterir. Ölçümlerin teoriye dayanan bir tahmin ile karşılaştırıp birbirine uygunluk gösterip göstermediğine karar vermede ölçümlerin standart sapması önemli rol oynar. Eğer ölçümlerin standart sapması teorik tahminden çok daha uzaksa, sınanan teorinin değiştirilmesi gerekir. İşte bu uzaklık standart sapmalarla belirlenir.</p> <p>Finansmanda, standart sapma verilmiş bir menkul (hisse seneti, tahvil, emlak vb.) için rizikonun veya bir menkuller portföyü için rizikoları temsil eder. Bir yatırım portföyünün etkin olarak idare edilmesini tayin eden en önemli faktörlerden birisi rizikodur. Çünkü her tek bir menkulün veya bir menkuller portföyünün getirisindeki mümkün yayılımını riziko tanımlar ve rizikonun standart sapma ile tanımlanması ise yatırım kararları için bir matematiksel temel sağlar. En geniş kavramla, yatırım rizikosu arttıkça menkul veya menkuller portföyünün beklenen getirisi da artış gösterir. Buna neden yatırımcıların menkul getirileri için riziko primlerini artırmaları olarak açıklanır. Diğer bir deyişle, eğer bir yatırım daha yüksek riziko seviyesi taşıyorsa, yatırımcılar o yatırımından daha yüksek bir getiri beklemeleri gereklidir.</p> <p>Uzunca bir zaman içinde herhangi bir menkul için yıllık getirilerinin ortalamasını bulmakla o menkul için beklenen getiri değerini vermektedir. Her yıl için elde edilen getiriden bu beklenen getiri farkı bulunursa buna finansmancılar ve muhasebeciler tarafından varyans adı verilir (Dikkat edilirse bu istatistiksel varyans kavramından farklıdır). Her bir yıl için varyansın karesini bulmak ve bu varyans karelerinin ortalamasının kare kökü o menkulün standart sapmasını yani rizikosunu gösterir. İşte bu rizikolar yani varyansların karelerinin toplamının ortalamasının kare kökü, standart sapmadır ve rizikoyu ölçer. Menkullerin karşılaştırılımı için temel çalışma işte bu ölçü ile yapılır.</p> <p>Standart sapmalar için pratik uygulamalar daha değişik alanlarda da verilebilir; fakat burada bu ufak sayıda uygulamalar bile standart sapmanın uygun bir şekilde önemini ortaya çıkartmaktadır.</p> <div class="mw-heading mw-heading3"> <h3 id="Normal_dağılım_gösteren_veriler_için_kurallar"><span id="Normal_da.C4.9F.C4.B1l.C4.B1m_g.C3.B6steren_veriler_i.C3.A7in_kurallar"></span>Normal dağılım gösteren veriler için kurallar</h3><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Normal dağılım gösteren veriler için kurallar" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Dosya:Standard_deviation_diagram.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/350px-Standard_deviation_diagram.svg.png" decoding="async" width="350" height="175" class="mw-file-element" data-file-width="400" data-file-height="200"> </noscript><span class="lazy-image-placeholder" style="width: 350px;height: 175px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/350px-Standard_deviation_diagram.svg.png" data-width="350" data-height="175" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/525px-Standard_deviation_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/700px-Standard_deviation_diagram.svg.png 2x" data-class="mw-file-element"> </span></a> <figcaption> Koyu mavi ortalamadan bir standart sapmadan daha düşük değerleri gösterir. <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Normal_da%C4%9F%C4%B1l%C4%B1m?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Normal dağılım">Normal dağılım</a> için bu %68,27 olur; (orta ile koyu mavi) ortalamadan iki standart sapma için %95,45; (açık, orta ve koyu mavi için) ortalamadan üç standart sapma %99,73 olur. </figcaption> </figure> <p>Pratikte, çok zaman verilerin yaklaşık olarak bir normal dağılım gösteren anakütleden geldiği varsayılır. Bu varsayıma neden olarak merkezsel <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Limit?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Limit">limit</a> teoreminin geçerliliği iddiası olur. Merkezsel limit teoremine göre birçok birbirinden bağımsız ve hepsi aynı dağılım gösteren rassal değişkenlerin toplamı limitte bir normal dağılıma göre eğilim gösterirler. Eğer bu varsayım geçerli ise, değerler yaklaşık %68,27 olasılıkla ortalamadan eksi ve artı bir standart sapma noktalarının arasında bulunur; ortalamadan artı ve eksi 2 standart sapma noktaları arasında %95,45 olasılıkla ve ortalamadan artı ve eksi 3 standart sapma noktaları arasında %99,73 olasılıkla bulunur. Bu <i><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=68-95-99.7_kural%C4%B1&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="68-95-99.7 kuralı (sayfa mevcut değil)">68-95-99.7 kuralı</a></i> veya bir <i>emprik kural</i> olarak bilinir.</p> <p><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=G%C3%BCvenlik_aral%C4%B1k&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Güvenlik aralık (sayfa mevcut değil)">Güvenlik aralıkları</a> şöyle gösterilebilir:</p> <table border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"> <tbody> <tr> <td align="center">σ</td> <td>%68,26894921371</td> </tr> <tr> <td>2σ</td> <td>%95,44997361036</td> </tr> <tr> <td>3σ</td> <td>%99,73002039367</td> </tr> <tr> <td>4σ</td> <td>%99,99366575163</td> </tr> <tr> <td>5σ</td> <td>%99,99994266969</td> </tr> <tr> <td>6σ</td> <td>%99,99999980268</td> </tr> <tr> <td>7σ</td> <td>%99,99999999974</td> </tr> </tbody> </table> <p>Normal dağılımlar için ortalamadan bir standart sapma uzaklıktaki eğri üzerindeki noktalar bir enfeksiyon noktası da olurlar.</p> <div class="mw-heading mw-heading3"> <h3 id="Çebişev'in_eşitsizliği"><span id=".C3.87ebi.C5.9Fev.27in_e.C5.9Fitsizli.C4.9Fi"></span>Çebişev'in eşitsizliği</h3><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Çebişev'in eşitsizliği" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <p><i>Yakınlık</i> standart sapma birimlerinde ifade edilirse, herhangi bir veri serisi için, <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%87ebi%C5%9Fev%27in_e%C5%9Fitsizli%C4%9Fi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Çebişev'in eşitsizliği">Çebişev'in eşitsizliği</a> ile ispat edilmiştir ki veri değerlerin çok büyük çoğunluğu ortalama değere <i>yakın</i>dır. <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%87ebi%C5%9Fev%27in_e%C5%9Fitsizli%C4%9Fi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Çebişev'in eşitsizliği">Çebişev'in eşitsizliği</a> sadece normal dağılım gösteren seriler için değil, bütün rastgele dağılım gösteren veri serileri için geçerlidir. Buna göre, şu zayıf sınırlar ve bu sınırlar içinde bulunan veri yüzdesi şöyle verilebilir:</p> <dl> <dd> Ortalamadan √2 standart sapma uzaklıkları arasında değerlerin en aşağı %50si bulunur. </dd> <dd> Ortalamadan 2 standart sapma uzaklıkları arasında değerlerin en aşağı %75i bulunur. </dd> <dd> Ortalamadan 3 standart sapma uzaklıkları arasında değerlerin en aşağı %89u bulunur. </dd> <dd> Ortalamadan 4 standart sapma uzaklıkları arasında değerlerin en aşağı %94ü bulunur. </dd> <dd> Ortalamadan 5 standart sapma uzaklıkları arasında değerlerin en aşağı %96sı bulunur. </dd> <dd> Ortalamadan 6 standart sapma uzaklıkları arasında değerlerin en aşağı %97si bulunur. </dd> <dd> Ortalamadan 7 standart sapma uzaklıkları arasında değerlerin en aşağı %98i bulunur. </dd> </dl> <p>Genel olarak:</p> <dl> <dd> ortalamadan <i>k</i> standart sapma uzaklıkları arasında değerlerin en aşağı %(1 − 1/<i>k</i><sup>2</sup>) × 100 si bulunur. </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Standart_sapma_ve_ortalama_arasındaki_ilişki"><span id="Standart_sapma_ve_ortalama_aras.C4.B1ndaki_ili.C5.9Fki"></span>Standart sapma ve ortalama arasındaki ilişki</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Standart sapma ve ortalama arasındaki ilişki" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <p>Çok kere bir veri serisinin özetlenmesinde ortalama ve standart sapma birlikte bildirilmektedir. Bir anlamda, eğer ortalama verilerinin merkezi olarak kullanılan ölçü ise, standart sapma veri yayılımının <i>doğal</i> ölçüsüdür. Buna neden ortalama noktasından standart sapmanın, verinin herhangi bir noktasından standarize edilmiş sapmadan daha küçük olduğudur. Bu matematiksel ifade ile şöyle gösterilebilir: <i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub> reel sayılar olsun ve şu fonksiyon tanımlansın:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (r)={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-r)^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> σ </mi> <mo stretchy="false"> ( </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mi> r </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma (r)={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-r)^{2}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a390379896280ef3194c3c41db017c9f28fae8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.465ex; height:7.509ex;" alt="{\displaystyle \sigma (r)={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-r)^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 24.465ex;height: 7.509ex;vertical-align: -3.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a390379896280ef3194c3c41db017c9f28fae8af" data-alt="{\displaystyle \sigma (r)={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-r)^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Ya birinci <a href="https://tr-m-wikipedia-org.translate.goog/wiki/T%C3%BCrev?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Türev">türev</a> alınıp sıfıra eşit yaparak veya daha kolay bir cebirsel yol olan <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Kare_tamamlamas%C4%B1&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Kare tamamlaması (sayfa mevcut değil)">kare tamamlaması</a> kullanarak σ(<i>r</i>) nın tek ve sadece tek bir minimum noktasının aritmetik ortalama olduğu; yani</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 r={\overline {x}}.\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> . </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r={\overline {x}}.\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76f600dd5a4d354fffe87b5a2cf79a8a715a42fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.626ex; height:2.343ex;" alt="{\displaystyle r={\overline {x}}.\,}"> </noscript><span class="lazy-image-placeholder" style="width: 6.626ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76f600dd5a4d354fffe87b5a2cf79a8a715a42fa" data-alt="{\displaystyle r={\overline {x}}.\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>gösterilebilir.</p> <p>Standart sapma ile ortalama arasındaki diğer bir ilişki ise yayılım özelliğine dayanan veri karşılaştırılmaları için kullanılan <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Varyasyon_katsay%C4%B1s%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Varyasyon katsayısı">varyasyon katsayısıdır</a>. Bir veri serisi için varyasyon katsayısı standart sapma ile ortalama arasındaki orandır. Böylece, standart sapma (ve ortalama) veri birimleri ile boyutlu iken (örneğin veri TL ile ise standart sapma ve ortalama TL birimlerindedir); varyasyon katsayısı boyutsuz sırf bir sayıdır. Bu nedenle değişik birimlerde olan verilerin yayılımlarının karşılaştırılması için kullanılabilir.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Ayrıca_bakınız"><span id="Ayr.C4.B1ca_bak.C4.B1n.C4.B1z"></span>Ayrıca bakınız</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Ayrıca bakınız" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <div style="-moz-column-count:2; column-count:2;"> <ul> <li><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Varyans_hesaplanmas%C4%B1_i%C3%A7in_algoritmalar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Varyans hesaplanması için algoritmalar">Varyans hesaplanması için algoritmalar</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%87ebi%C5%9Fev%27in_e%C5%9Fitsizli%C4%9Fi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Çebişev'in eşitsizliği">Çebişev'in eşitsizliği</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=G%C3%BCvenlik_aral%C4%B1%C4%9F%C4%B1&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Güvenlik aralığı (sayfa mevcut değil)">Güvenlik aralığı</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Kumulant&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Kumulant (sayfa mevcut değil)">Kumulant</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Geometrik_standard_sapma&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Geometrik standard sapma (sayfa mevcut değil)">Geometrik standard sapma</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Bas%C4%B1kl%C4%B1k?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Basıklık">Basıklık</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Ortalama_mutlak_hata&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Ortalama mutlak hata (sayfa mevcut değil)">Ortalama mutlak hata</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Ortalama?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ortalama">Ortalama</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Ortalama_karesi_karek%C3%B6k%C3%BC&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Ortalama karesi karekökü (sayfa mevcut değil)">Ortalama karesi karekökü</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=%C3%96rneklem_b%C3%BCy%C3%BCkl%C3%BC%C4%9F%C3%BC&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Örneklem büyüklüğü (sayfa mevcut değil)">Örneklem büyüklüğü</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%87arp%C4%B1kl%C4%B1k?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Çarpıklık">Çarpıklık</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Standart_hata_(istatistik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Standart hata (istatistik)">Standart hata</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_puanlama&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Standart puanlama (sayfa mevcut değil)">Standart puanlama</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Varyans?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Varyans">Varyans</a></li> <li><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Volatilite_(finansman)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Volatilite (finansman) (sayfa mevcut değil)">Volatilite</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Kaynakça"><span id="Kaynak.C3.A7a"></span>Kaynakça</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Kaynakça" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <style data-mw-deduplicate="TemplateStyles:r32805677">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-count:2}.mw-parser-output .reflist-columns-3{column-count:3}.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"> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Dış_kaynaklar"><span id="D.C4.B1.C5.9F_kaynaklar"></span>Dış kaynaklar</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Dış kaynaklar" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul> <li>Spiegel, Murray R ve Stephens, Larry J. (Tr.Çev.: Çelebioğlu, Salih) (2013) <i>İstatistik</i>, İstanbul: Nobel Akademik Yayıncılık <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%96zel:KitapKaynaklar%C4%B1/9786051337043?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 9786051337043</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Dış_bağlantılar"><span id="D.C4.B1.C5.9F_ba.C4.9Flant.C4.B1lar"></span>Dış bağlantılar</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Dış bağlantılar" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>değiştir</span> </a> </span> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20141110093543/http://hesabet.com/Hesaplamalar/Mod_Medyan_Ortalama_Standart_Sapma/">Mod, medyan, ortalama ve stantart sapma hesaplayabileceğiniz Türkçe bir sayfa.</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20131022135203/http://invsee.asu.edu/srinivas/stdev.html">Standart sapma hesaplayıcısı</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20100420134759/http://www.stats4students.com/Essentials/Measures-Of-Spread/Overview_3.php">Standart sapmayı anlamak ve hesaplamak için bir kılavuz</a> <span style="font-size:0.95em; font-weight:bold; color:inherit;">(İngilizce)</span></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.techbookreport.com/tutorials/stddev-30-secs.html">Standart sapma - matematik kullanmadan bir açıklama</a>10 Ocak 2010 tarihinde <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Wayback_Machine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine">Wayback Machine</a> sitesinde <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20100110105637/http://www.techbookreport.com/tutorials/stddev-30-secs.html">arşivlendi</a>. <span style="font-size:0.95em; font-weight:bold; color:inherit;">(İngilizce)</span></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://davidmlane.com/hyperstat/A16252.html">Standart sapma, bir basit giriş</a>8 Şubat 2010 tarihinde <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Wayback_Machine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine">Wayback Machine</a> sitesinde <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20100208075158/http://davidmlane.com/hyperstat/A16252.html">arşivlendi</a>. <span style="font-size:0.95em; font-weight:bold; color:inherit;">(İngilizce)</span></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.robertniles.com/stats/stdev.shtml">Standart sapma: dergi ve gazete yazarları için basitce bir açıklama</a>8 Şubat 2010 tarihinde <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Wayback_Machine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine">Wayback Machine</a> sitesinde <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20100208035722/http://www.robertniles.com/stats/stdev.shtml">arşivlendi</a>. <span style="font-size:0.95em; font-weight:bold; color:inherit;">(İngilizce)</span></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.stat.tamu.edu/~jhardin/applets/">Texas A&M standart sapma ve güven aralığı hesaplayıcıları</a>12 Ocak 2010 tarihinde <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Wayback_Machine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine">Wayback Machine</a> sitesinde <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20100112125614/http://www.stat.tamu.edu/~jhardin/applets/">arşivlendi</a>. <span style="font-size:0.95em; font-weight:bold; color:inherit;">(İngilizce)</span></li> </ul> <p><br></p> </section> </div> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=mobile&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> "<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/w/index.php?title%3DStandart_sapma%26oldid%3D33556535">https://tr.wikipedia.org/w/index.php?title=Standart_sapma&oldid=33556535</a>" sayfasından alınmıştır </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Standart_sapma&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"><span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Kararınca" data-user-gender="male" data-timestamp="1721971686"> <span>Son düzenleme 26 Temmuz 2024, 05.28 tarihinde yapıldı</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Diller</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://af.wikipedia.org/wiki/Standaardafwyking" title="Standaardafwyking - Afrikaanca" lang="af" hreflang="af" data-title="Standaardafwyking" data-language-autonym="Afrikaans" data-language-local-name="Afrikaanca" class="interlanguage-link-target"><span>Afrikaans</span></a></li> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D8%25A7%25D9%2586%25D8%25AD%25D8%25B1%25D8%25A7%25D9%2581_%25D9%2585%25D8%25B9%25D9%258A%25D8%25A7%25D8%25B1%25D9%258A" title="انحراف معياري - Arapça" lang="ar" hreflang="ar" data-title="انحراف معياري" data-language-autonym="العربية" data-language-local-name="Arapça" class="interlanguage-link-target"><span>العربية</span></a></li> <li class="interlanguage-link interwiki-as mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://as.wikipedia.org/wiki/%25E0%25A6%25AE%25E0%25A6%25BE%25E0%25A6%25A8%25E0%25A6%2595_%25E0%25A6%25AC%25E0%25A6%25BF%25E0%25A6%259A%25E0%25A6%25B2%25E0%25A6%25A8" title="মানক বিচলন - Assamca" lang="as" hreflang="as" data-title="মানক বিচলন" data-language-autonym="অসমীয়া" data-language-local-name="Assamca" class="interlanguage-link-target"><span>অসমীয়া</span></a></li> <li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Esviaci%25C3%25B3n_t%25C3%25ADpica" title="Esviación típica - Asturyasça" lang="ast" hreflang="ast" data-title="Esviación típica" data-language-autonym="Asturianu" data-language-local-name="Asturyasça" class="interlanguage-link-target"><span>Asturianu</span></a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/Orta_kvadratik_meyl" title="Orta kvadratik meyl - Azerbaycan dili" lang="az" hreflang="az" data-title="Orta kvadratik meyl" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaycan dili" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%25A1%25D1%258F%25D1%2580%25D1%258D%25D0%25B4%25D0%25BD%25D1%258F%25D0%25B5_%25D0%25BA%25D0%25B2%25D0%25B0%25D0%25B4%25D1%2580%25D0%25B0%25D1%2582%25D0%25BE%25D0%25B2%25D0%25B0%25D0%25B5_%25D0%25B0%25D0%25B4%25D1%2585%25D1%2596%25D0%25BB%25D0%25B5%25D0%25BD%25D0%25BD%25D0%25B5" title="Сярэдняе квадратовае адхіленне - Belarusça" lang="be" hreflang="be" data-title="Сярэдняе квадратовае адхіленне" data-language-autonym="Беларуская" data-language-local-name="Belarusça" class="interlanguage-link-target"><span>Беларуская</span></a></li> <li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be-tarask.wikipedia.org/wiki/%25D0%25A1%25D1%2582%25D0%25B0%25D0%25BD%25D0%25B4%25D0%25B0%25D1%2580%25D1%2582%25D0%25BD%25D0%25B0%25D0%25B5_%25D0%25B0%25D0%25B4%25D1%2585%25D1%2596%25D0%25BB%25D0%25B5%25D0%25BD%25D1%258C%25D0%25BD%25D0%25B5" title="Стандартнае адхіленьне - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Стандартнае адхіленьне" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li> <li class="interlanguage-link interwiki-bew mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bew.wikipedia.org/wiki/Pel%25C3%25A8ncongan_pakem" title="Pelèncongan pakem - Betawi" lang="bew" hreflang="bew" data-title="Pelèncongan pakem" data-language-autonym="Betawi" data-language-local-name="Betawi" class="interlanguage-link-target"><span>Betawi</span></a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%25A1%25D1%2582%25D0%25B0%25D0%25BD%25D0%25B4%25D0%25B0%25D1%2580%25D1%2582%25D0%25BD%25D0%25BE_%25D0%25BE%25D1%2582%25D0%25BA%25D0%25BB%25D0%25BE%25D0%25BD%25D0%25B5%25D0%25BD%25D0%25B8%25D0%25B5" title="Стандартно отклонение - Bulgarca" lang="bg" hreflang="bg" data-title="Стандартно отклонение" data-language-autonym="Български" data-language-local-name="Bulgarca" class="interlanguage-link-target"><span>Български</span></a></li> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bn.wikipedia.org/wiki/%25E0%25A6%25B8%25E0%25A6%25AE%25E0%25A6%2595_%25E0%25A6%25AC%25E0%25A6%25BF%25E0%25A6%259A%25E0%25A7%258D%25E0%25A6%25AF%25E0%25A7%2581%25E0%25A6%25A4%25E0%25A6%25BF" title="সমক বিচ্যুতি - Bengalce" lang="bn" hreflang="bn" data-title="সমক বিচ্যুতি" data-language-autonym="বাংলা" data-language-local-name="Bengalce" class="interlanguage-link-target"><span>বাংলা</span></a></li> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bs.wikipedia.org/wiki/Standardna_devijacija" title="Standardna devijacija - Boşnakça" lang="bs" hreflang="bs" data-title="Standardna devijacija" data-language-autonym="Bosanski" data-language-local-name="Boşnakça" class="interlanguage-link-target"><span>Bosanski</span></a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Desviaci%25C3%25B3_tipus" title="Desviació tipus - Katalanca" lang="ca" hreflang="ca" data-title="Desviació tipus" data-language-autonym="Català" data-language-local-name="Katalanca" class="interlanguage-link-target"><span>Català</span></a></li> <li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cbk-zam.wikipedia.org/wiki/Tipico_desviacion" title="Tipico desviacion - Chavacano" lang="cbk" hreflang="cbk" data-title="Tipico desviacion" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target"><span>Chavacano de Zamboanga</span></a></li> <li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ckb.wikipedia.org/wiki/%25D9%2584%25D8%25A7%25D8%25AF%25D8%25A7%25D9%2586%25DB%258C_%25D9%25BE%25DB%258E%25D9%2588%25D8%25A7%25D9%2586%25DB%2595%25DB%258C%25DB%258C" title="لادانی پێوانەیی - Orta Kürtçe" lang="ckb" hreflang="ckb" data-title="لادانی پێوانەیی" data-language-autonym="کوردی" data-language-local-name="Orta Kürtçe" class="interlanguage-link-target"><span>کوردی</span></a></li> <li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://crh.wikipedia.org/wiki/Ortalama_kvadratik_tayp%25C4%25B1n%25C4%25B1%25C5%259F" title="Ortalama kvadratik taypınış - Kırım Tatarcası" lang="crh" hreflang="crh" data-title="Ortalama kvadratik taypınış" data-language-autonym="Qırımtatarca" data-language-local-name="Kırım Tatarcası" class="interlanguage-link-target"><span>Qırımtatarca</span></a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Sm%25C4%259Brodatn%25C3%25A1_odchylka" title="Směrodatná odchylka - Çekçe" lang="cs" hreflang="cs" data-title="Směrodatná odchylka" data-language-autonym="Čeština" data-language-local-name="Çekçe" class="interlanguage-link-target"><span>Čeština</span></a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25D0%25A2%25C4%2583%25D0%25B2%25D0%25B0%25D1%2582%25D0%25BA%25D0%25B0%25D0%25BB%25D0%25BB%25D0%25B0_%25D0%25B2%25C4%2583%25D1%2582%25D0%25B0%25D0%25BC_%25D0%25BF%25C4%2583%25D1%2580%25C4%2583%25D0%25BD%25D0%25B0%25D0%25B2" title="Тăваткалла вăтам пăрăнав - Çuvaşça" lang="cv" hreflang="cv" data-title="Тăваткалла вăтам пăрăнав" data-language-autonym="Чӑвашла" data-language-local-name="Çuvaşça" class="interlanguage-link-target"><span>Чӑвашла</span></a></li> <li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cy.wikipedia.org/wiki/Gwyriad_safonol" title="Gwyriad safonol - Galce" lang="cy" hreflang="cy" data-title="Gwyriad safonol" data-language-autonym="Cymraeg" data-language-local-name="Galce" class="interlanguage-link-target"><span>Cymraeg</span></a></li> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://da.wikipedia.org/wiki/Standardafvigelse" title="Standardafvigelse - Danca" lang="da" hreflang="da" data-title="Standardafvigelse" data-language-autonym="Dansk" data-language-local-name="Danca" class="interlanguage-link-target"><span>Dansk</span></a></li> <li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Standardabweichung_(Stochastik)" title="Standardabweichung (Stochastik) - Almanca" lang="de" hreflang="de" data-title="Standardabweichung (Stochastik)" data-language-autonym="Deutsch" data-language-local-name="Almanca" class="interlanguage-link-target"><span>Deutsch</span></a></li> <li class="interlanguage-link interwiki-el mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://el.wikipedia.org/wiki/%25CE%25A4%25CF%2585%25CF%2580%25CE%25B9%25CE%25BA%25CE%25AE_%25CE%25B1%25CF%2580%25CF%258C%25CE%25BA%25CE%25BB%25CE%25B9%25CF%2583%25CE%25B7" title="Τυπική απόκλιση - Yunanca" lang="el" hreflang="el" data-title="Τυπική απόκλιση" data-language-autonym="Ελληνικά" data-language-local-name="Yunanca" class="interlanguage-link-target"><span>Ελληνικά</span></a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Standard_deviation" title="Standard deviation - İngilizce" lang="en" hreflang="en" data-title="Standard deviation" data-language-autonym="English" data-language-local-name="İngilizce" class="interlanguage-link-target"><span>English</span></a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Norma_devio" title="Norma devio - Esperanto" lang="eo" hreflang="eo" data-title="Norma devio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Desviaci%25C3%25B3n_t%25C3%25ADpica" title="Desviación típica - İspanyolca" lang="es" hreflang="es" data-title="Desviación típica" data-language-autonym="Español" data-language-local-name="İspanyolca" class="interlanguage-link-target"><span>Español</span></a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Standardh%25C3%25A4lve" title="Standardhälve - Estonca" lang="et" hreflang="et" data-title="Standardhälve" data-language-autonym="Eesti" data-language-local-name="Estonca" class="interlanguage-link-target"><span>Eesti</span></a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Desbideratze_estandar" title="Desbideratze estandar - Baskça" lang="eu" hreflang="eu" data-title="Desbideratze estandar" data-language-autonym="Euskara" data-language-local-name="Baskça" class="interlanguage-link-target"><span>Euskara</span></a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D8%25A7%25D9%2586%25D8%25AD%25D8%25B1%25D8%25A7%25D9%2581_%25D9%2585%25D8%25B9%25DB%258C%25D8%25A7%25D8%25B1" title="انحراف معیار - Farsça" lang="fa" hreflang="fa" data-title="انحراف معیار" data-language-autonym="فارسی" data-language-local-name="Farsça" class="interlanguage-link-target"><span>فارسی</span></a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Keskihajonta" title="Keskihajonta - Fince" lang="fi" hreflang="fi" data-title="Keskihajonta" data-language-autonym="Suomi" data-language-local-name="Fince" class="interlanguage-link-target"><span>Suomi</span></a></li> <li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="kaliteli madde"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/%25C3%2589cart_type" title="Écart type - Fransızca" lang="fr" hreflang="fr" data-title="Écart type" data-language-autonym="Français" data-language-local-name="Fransızca" class="interlanguage-link-target"><span>Français</span></a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Desviaci%25C3%25B3n_t%25C3%25ADpica" title="Desviación típica - Galiçyaca" lang="gl" hreflang="gl" data-title="Desviación típica" data-language-autonym="Galego" data-language-local-name="Galiçyaca" class="interlanguage-link-target"><span>Galego</span></a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%25A1%25D7%2598%25D7%2599%25D7%2599%25D7%25AA_%25D7%25AA%25D7%25A7%25D7%259F" title="סטיית תקן - İbranice" lang="he" hreflang="he" data-title="סטיית תקן" data-language-autonym="עברית" data-language-local-name="İbranice" class="interlanguage-link-target"><span>עברית</span></a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25AE%25E0%25A4%25BE%25E0%25A4%25A8%25E0%25A4%2595_%25E0%25A4%25B5%25E0%25A4%25BF%25E0%25A4%259A%25E0%25A4%25B2%25E0%25A4%25A8" title="मानक विचलन - Hintçe" lang="hi" hreflang="hi" data-title="मानक विचलन" data-language-autonym="हिन्दी" data-language-local-name="Hintçe" class="interlanguage-link-target"><span>हिन्दी</span></a></li> <li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hr.wikipedia.org/wiki/Standardna_devijacija" title="Standardna devijacija - Hırvatça" lang="hr" hreflang="hr" data-title="Standardna devijacija" data-language-autonym="Hrvatski" data-language-local-name="Hırvatça" class="interlanguage-link-target"><span>Hrvatski</span></a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Sz%25C3%25B3r%25C3%25A1s_(val%25C3%25B3sz%25C3%25ADn%25C5%25B1s%25C3%25A9gsz%25C3%25A1m%25C3%25ADt%25C3%25A1s)" title="Szórás (valószínűségszámítás) - Macarca" lang="hu" hreflang="hu" data-title="Szórás (valószínűségszámítás)" data-language-autonym="Magyar" data-language-local-name="Macarca" class="interlanguage-link-target"><span>Magyar</span></a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Simpangan_baku" title="Simpangan baku - Endonezce" lang="id" hreflang="id" data-title="Simpangan baku" data-language-autonym="Bahasa Indonesia" data-language-local-name="Endonezce" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li> <li class="interlanguage-link interwiki-is mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/Sta%25C3%25B0alfr%25C3%25A1vik" title="Staðalfrávik - İzlandaca" lang="is" hreflang="is" data-title="Staðalfrávik" data-language-autonym="Íslenska" data-language-local-name="İzlandaca" class="interlanguage-link-target"><span>Íslenska</span></a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Scarto_quadratico_medio" title="Scarto quadratico medio - İtalyanca" lang="it" hreflang="it" data-title="Scarto quadratico medio" data-language-autonym="İtaliano" data-language-local-name="İtalyanca" class="interlanguage-link-target"><span>İtaliano</span></a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E6%25A8%2599%25E6%25BA%2596%25E5%2581%258F%25E5%25B7%25AE" title="標準偏差 - Japonca" lang="ja" hreflang="ja" data-title="標準偏差" data-language-autonym="日本語" data-language-local-name="Japonca" class="interlanguage-link-target"><span>日本語</span></a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%259A%25D0%25B2%25D0%25B0%25D0%25B4%25D1%2580%25D0%25B0%25D1%2582%25D1%2582%25D1%258B%25D2%259B_%25D0%25B0%25D1%2583%25D1%258B%25D1%2582%25D2%259B%25D1%2583" title="Квадраттық ауытқу - Kazakça" lang="kk" hreflang="kk" data-title="Квадраттық ауытқу" data-language-autonym="Қазақша" data-language-local-name="Kazakça" class="interlanguage-link-target"><span>Қазақша</span></a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25ED%2591%259C%25EC%25A4%2580_%25ED%258E%25B8%25EC%25B0%25A8" title="표준 편차 - Korece" lang="ko" hreflang="ko" data-title="표준 편차" data-language-autonym="한국어" data-language-local-name="Korece" class="interlanguage-link-target"><span>한국어</span></a></li> <li class="interlanguage-link interwiki-la mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://la.wikipedia.org/wiki/Deviatio_canonica" title="Deviatio canonica - Latince" lang="la" hreflang="la" data-title="Deviatio canonica" data-language-autonym="Latina" data-language-local-name="Latince" class="interlanguage-link-target"><span>Latina</span></a></li> <li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Standartinis_nuokrypis" title="Standartinis nuokrypis - Litvanca" lang="lt" hreflang="lt" data-title="Standartinis nuokrypis" data-language-autonym="Lietuvių" data-language-local-name="Litvanca" class="interlanguage-link-target"><span>Lietuvių</span></a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/Standartnovirze" title="Standartnovirze - Letonca" lang="lv" hreflang="lv" data-title="Standartnovirze" data-language-autonym="Latviešu" data-language-local-name="Letonca" class="interlanguage-link-target"><span>Latviešu</span></a></li> <li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mk.wikipedia.org/wiki/%25D0%25A1%25D1%2582%25D0%25B0%25D0%25BD%25D0%25B4%25D0%25B0%25D1%2580%25D0%25B4%25D0%25BD%25D0%25BE_%25D0%25BE%25D1%2582%25D1%2581%25D1%2582%25D0%25B0%25D0%25BF%25D1%2583%25D0%25B2%25D0%25B0%25D1%259A%25D0%25B5" title="Стандардно отстапување - Makedonca" lang="mk" hreflang="mk" data-title="Стандардно отстапување" data-language-autonym="Македонски" data-language-local-name="Makedonca" class="interlanguage-link-target"><span>Македонски</span></a></li> <li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mn.wikipedia.org/wiki/%25D0%25A1%25D1%2582%25D0%25B0%25D0%25BD%25D0%25B4%25D0%25B0%25D1%2580%25D1%2582_%25D1%2585%25D0%25B0%25D0%25B7%25D0%25B0%25D0%25B9%25D0%25BB%25D1%2582" title="Стандарт хазайлт - Moğolca" lang="mn" hreflang="mn" data-title="Стандарт хазайлт" data-language-autonym="Монгол" data-language-local-name="Moğolca" class="interlanguage-link-target"><span>Монгол</span></a></li> <li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ms.wikipedia.org/wiki/Sisihan_piawai" title="Sisihan piawai - Malayca" lang="ms" hreflang="ms" data-title="Sisihan piawai" data-language-autonym="Bahasa Melayu" data-language-local-name="Malayca" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li> <li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ne.wikipedia.org/wiki/%25E0%25A4%25B8%25E0%25A5%258D%25E0%25A4%25A4%25E0%25A4%25B0%25E0%25A5%2580%25E0%25A4%25AF_%25E0%25A4%25AD%25E0%25A4%25BF%25E0%25A4%25A8%25E0%25A5%258D%25E0%25A4%25A8%25E0%25A4%25A4%25E0%25A4%25BE" title="स्तरीय भिन्नता - Nepalce" lang="ne" hreflang="ne" data-title="स्तरीय भिन्नता" data-language-autonym="नेपाली" data-language-local-name="Nepalce" class="interlanguage-link-target"><span>नेपाली</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Standaardafwijking" title="Standaardafwijking - Felemenkçe" lang="nl" hreflang="nl" data-title="Standaardafwijking" data-language-autonym="Nederlands" data-language-local-name="Felemenkçe" class="interlanguage-link-target"><span>Nederlands</span></a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Standardavvik" title="Standardavvik - Norveççe Nynorsk" lang="nn" hreflang="nn" data-title="Standardavvik" data-language-autonym="Norsk nynorsk" data-language-local-name="Norveççe Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Standardavvik" title="Standardavvik - Norveççe Bokmål" lang="nb" hreflang="nb" data-title="Standardavvik" data-language-autonym="Norsk bokmål" data-language-local-name="Norveççe Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li> <li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oc.wikipedia.org/wiki/Desviacion_tipica" title="Desviacion tipica - Oksitan dili" lang="oc" hreflang="oc" data-title="Desviacion tipica" data-language-autonym="Occitan" data-language-local-name="Oksitan dili" class="interlanguage-link-target"><span>Occitan</span></a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Odchylenie_standardowe" title="Odchylenie standardowe - Lehçe" lang="pl" hreflang="pl" data-title="Odchylenie standardowe" data-language-autonym="Polski" data-language-local-name="Lehçe" class="interlanguage-link-target"><span>Polski</span></a></li> <li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="seçkin madde"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Desvio_padr%25C3%25A3o" title="Desvio padrão - Portekizce" lang="pt" hreflang="pt" data-title="Desvio padrão" data-language-autonym="Português" data-language-local-name="Portekizce" class="interlanguage-link-target"><span>Português</span></a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Abatere_standard" title="Abatere standard - Rumence" lang="ro" hreflang="ro" data-title="Abatere standard" data-language-autonym="Română" data-language-local-name="Rumence" class="interlanguage-link-target"><span>Română</span></a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%25A1%25D1%2580%25D0%25B5%25D0%25B4%25D0%25BD%25D0%25B5%25D0%25BA%25D0%25B2%25D0%25B0%25D0%25B4%25D1%2580%25D0%25B0%25D1%2582%25D0%25B8%25D1%2587%25D0%25B5%25D1%2581%25D0%25BA%25D0%25BE%25D0%25B5_%25D0%25BE%25D1%2582%25D0%25BA%25D0%25BB%25D0%25BE%25D0%25BD%25D0%25B5%25D0%25BD%25D0%25B8%25D0%25B5" title="Среднеквадратическое отклонение - Rusça" lang="ru" hreflang="ru" data-title="Среднеквадратическое отклонение" data-language-autonym="Русский" data-language-local-name="Rusça" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://scn.wikipedia.org/wiki/Diviazzioni_standard" title="Diviazzioni standard - Sicilyaca" lang="scn" hreflang="scn" data-title="Diviazzioni standard" data-language-autonym="Sicilianu" data-language-local-name="Sicilyaca" class="interlanguage-link-target"><span>Sicilianu</span></a></li> <li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sd.wikipedia.org/wiki/%25D9%2585%25D8%25B9%25D9%258A%25D8%25A7%25D8%25B1%25D9%258A_%25D8%25A7%25D9%2586%25D8%25AD%25D8%25B1%25D8%25A7%25D9%2581" title="معياري انحراف - Sindhi dili" lang="sd" hreflang="sd" data-title="معياري انحراف" data-language-autonym="سنڌي" data-language-local-name="Sindhi dili" class="interlanguage-link-target"><span>سنڌي</span></a></li> <li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sh.wikipedia.org/wiki/Standardna_devijacija" title="Standardna devijacija - Sırp-Hırvat Dili" lang="sh" hreflang="sh" data-title="Standardna devijacija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Sırp-Hırvat Dili" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li> <li class="interlanguage-link interwiki-si mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://si.wikipedia.org/wiki/%25E0%25B7%2583%25E0%25B6%25B8%25E0%25B7%258A%25E0%25B6%25B8%25E0%25B6%25AD_%25E0%25B6%2585%25E0%25B6%25B4%25E0%25B6%259C%25E0%25B6%25B8%25E0%25B6%25B1%25E0%25B6%25BA" title="සම්මත අපගමනය - Sinhali dili" lang="si" hreflang="si" data-title="සම්මත අපගමනය" data-language-autonym="සිංහල" data-language-local-name="Sinhali dili" class="interlanguage-link-target"><span>සිංහල</span></a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Standard_deviation" title="Standard deviation - Simple English" lang="en-simple" hreflang="en-simple" data-title="Standard deviation" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Smerodajn%25C3%25A1_odch%25C3%25BDlka" title="Smerodajná odchýlka - Slovakça" lang="sk" hreflang="sk" data-title="Smerodajná odchýlka" data-language-autonym="Slovenčina" data-language-local-name="Slovakça" class="interlanguage-link-target"><span>Slovenčina</span></a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Standardni_odklon" title="Standardni odklon - Slovence" lang="sl" hreflang="sl" data-title="Standardni odklon" data-language-autonym="Slovenščina" data-language-local-name="Slovence" class="interlanguage-link-target"><span>Slovenščina</span></a></li> <li class="interlanguage-link interwiki-sm mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sm.wikipedia.org/wiki/Fa%2527ailoga_masani" title="Fa'ailoga masani - Samoa dili" lang="sm" hreflang="sm" data-title="Fa'ailoga masani" data-language-autonym="Gagana Samoa" data-language-local-name="Samoa dili" class="interlanguage-link-target"><span>Gagana Samoa</span></a></li> <li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sq.wikipedia.org/wiki/Devijimi_standard" title="Devijimi standard - Arnavutça" lang="sq" hreflang="sq" data-title="Devijimi standard" data-language-autonym="Shqip" data-language-local-name="Arnavutça" class="interlanguage-link-target"><span>Shqip</span></a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%25A1%25D1%2582%25D0%25B0%25D0%25BD%25D0%25B4%25D0%25B0%25D1%2580%25D0%25B4%25D0%25BD%25D0%25B0_%25D0%25B4%25D0%25B5%25D0%25B2%25D0%25B8%25D1%2598%25D0%25B0%25D1%2586%25D0%25B8%25D1%2598%25D0%25B0" title="Стандардна девијација - Sırpça" lang="sr" hreflang="sr" data-title="Стандардна девијација" data-language-autonym="Српски / srpski" data-language-local-name="Sırpça" class="interlanguage-link-target"><span>Српски / srpski</span></a></li> <li class="interlanguage-link interwiki-su mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://su.wikipedia.org/wiki/Simpangan_baku" title="Simpangan baku - Sunda dili" lang="su" hreflang="su" data-title="Simpangan baku" data-language-autonym="Sunda" data-language-local-name="Sunda dili" class="interlanguage-link-target"><span>Sunda</span></a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Standardavvikelse" title="Standardavvikelse - İsveççe" lang="sv" hreflang="sv" data-title="Standardavvikelse" data-language-autonym="Svenska" data-language-local-name="İsveççe" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sw.wikipedia.org/wiki/Mkengeuko_wastani" title="Mkengeuko wastani - Svahili dili" lang="sw" hreflang="sw" data-title="Mkengeuko wastani" data-language-autonym="Kiswahili" data-language-local-name="Svahili dili" class="interlanguage-link-target"><span>Kiswahili</span></a></li> <li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ta.wikipedia.org/wiki/%25E0%25AE%25A8%25E0%25AE%25BF%25E0%25AE%25AF%25E0%25AE%25AE%25E0%25AE%25B5%25E0%25AE%25BF%25E0%25AE%25B2%25E0%25AE%2595%25E0%25AE%25B2%25E0%25AF%258D" title="நியமவிலகல் - Tamilce" lang="ta" hreflang="ta" data-title="நியமவிலகல்" data-language-autonym="தமிழ்" data-language-local-name="Tamilce" class="interlanguage-link-target"><span>தமிழ்</span></a></li> <li class="interlanguage-link interwiki-th mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://th.wikipedia.org/wiki/%25E0%25B8%2584%25E0%25B9%2588%25E0%25B8%25B2%25E0%25B9%2580%25E0%25B8%259A%25E0%25B8%25B5%25E0%25B9%2588%25E0%25B8%25A2%25E0%25B8%2587%25E0%25B9%2580%25E0%25B8%259A%25E0%25B8%2599%25E0%25B8%25A1%25E0%25B8%25B2%25E0%25B8%2595%25E0%25B8%25A3%25E0%25B8%2590%25E0%25B8%25B2%25E0%25B8%2599" title="ค่าเบี่ยงเบนมาตรฐาน - Tayca" lang="th" hreflang="th" data-title="ค่าเบี่ยงเบนมาตรฐาน" data-language-autonym="ไทย" data-language-local-name="Tayca" class="interlanguage-link-target"><span>ไทย</span></a></li> <li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tl.wikipedia.org/wiki/Standard_deviation" title="Standard deviation - Tagalogca" lang="tl" hreflang="tl" data-title="Standard deviation" data-language-autonym="Tagalog" data-language-local-name="Tagalogca" class="interlanguage-link-target"><span>Tagalog</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%25A1%25D1%2582%25D0%25B0%25D0%25BD%25D0%25B4%25D0%25B0%25D1%2580%25D1%2582%25D0%25BD%25D0%25B5_%25D0%25B2%25D1%2596%25D0%25B4%25D1%2585%25D0%25B8%25D0%25BB%25D0%25B5%25D0%25BD%25D0%25BD%25D1%258F" title="Стандартне відхилення - Ukraynaca" lang="uk" hreflang="uk" data-title="Стандартне відхилення" data-language-autonym="Українська" data-language-local-name="Ukraynaca" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ur.wikipedia.org/wiki/%25D9%2585%25D8%25B9%25DB%258C%25D8%25A7%25D8%25B1%25DB%258C_%25D8%25A7%25D9%2586%25D8%25AD%25D8%25B1%25D8%25A7%25D9%2581" title="معیاری انحراف - Urduca" lang="ur" hreflang="ur" data-title="معیاری انحراف" data-language-autonym="اردو" data-language-local-name="Urduca" class="interlanguage-link-target"><span>اردو</span></a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/%25C4%2590%25E1%25BB%2599_l%25E1%25BB%2587ch_chu%25E1%25BA%25A9n" title="Độ lệch chuẩn - Vietnamca" lang="vi" hreflang="vi" data-title="Độ lệch chuẩn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamca" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> <li class="interlanguage-link interwiki-war mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://war.wikipedia.org/wiki/Standard_deviation" title="Standard deviation - Varay" lang="war" hreflang="war" data-title="Standard deviation" data-language-autonym="Winaray" data-language-local-name="Varay" class="interlanguage-link-target"><span>Winaray</span></a></li> <li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://wuu.wikipedia.org/wiki/%25E6%25A0%2587%25E5%2587%2586%25E5%25B7%25AE" title="标准差 - Wu Çincesi" lang="wuu" hreflang="wuu" data-title="标准差" data-language-autonym="吴语" data-language-local-name="Wu Çincesi" class="interlanguage-link-target"><span>吴语</span></a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E6%25A8%2599%25E6%25BA%2596%25E5%25B7%25AE" title="標準差 - Çince" lang="zh" hreflang="zh" data-title="標準差" data-language-autonym="中文" data-language-local-name="Çince" class="interlanguage-link-target"><span>中文</span></a></li> <li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-min-nan.wikipedia.org/wiki/Piau-ch%25C3%25BAn_phian-chha" title="Piau-chún phian-chha - Min Nan Çincesi" lang="nan" hreflang="nan" data-title="Piau-chún phian-chha" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Min Nan Çincesi" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E6%25A8%2599%25E6%25BA%2596%25E5%25B7%25AE" title="標準差 - Kantonca" lang="yue" hreflang="yue" data-title="標準差" data-language-autonym="粵語" data-language-local-name="Kantonca" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-tr.svg" alt="Vikipedi" width="107" height="18" style="width: 6.6875em; height: 1.125em;"> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">Sayfa en son 05.28, 26 Temmuz 2024 tarihinde değiştirildi.</li> <li id="footer-info-copyright">Aksi belirtilmedikçe içeriğin kullanımı <a class="external" rel="nofollow" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://creativecommons.org/licenses/by-sa/4.0/deed.tr">CC BY-SA 4.0</a> lisansı kapsamında uygundur.</li> </ul> <ul 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Standart sapma - Vikipedi