{"id":7162,"date":"2011-11-12T18:05:54","date_gmt":"2011-11-12T18:05:54","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7162"},"modified":"2026-06-05T00:25:06","modified_gmt":"2026-06-04T23:25:06","slug":"distribuicao-normal","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7162","title":{"rendered":"Distribui\u00e7\u00e3o normal"},"content":{"rendered":"<p><ul id='GTTabs_ul_7162' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7162' class='GTTabs_curr'><a  id=\"7162_0\" onMouseOver=\"GTTabsShowLinks('Distribui\u00e7\u00e3o normal'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Distribui\u00e7\u00e3o normal<\/a><\/li>\n<li id='GTTabs_li_1_7162' ><a  id=\"7162_1\" onMouseOver=\"GTTabsShowLinks('Folha de c\u00e1lculo'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Folha de c\u00e1lculo<\/a><\/li>\n<li id='GTTabs_li_2_7162' ><a  id=\"7162_2\" onMouseOver=\"GTTabsShowLinks('GeoGebra'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>GeoGebra<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7162'>\n<span class='GTTabs_titles'><b>Distribui\u00e7\u00e3o normal<\/b><\/span><\/p>\n<div id=\"attachment_7164\" style=\"width: 245px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Abraham_de_moivre2.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7164\" data-attachment-id=\"7164\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7164\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Abraham_de_moivre2.jpg\" data-orig-size=\"471,600\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Abraham De Moivre\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Abraham De Moivre&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Abraham_de_moivre2.jpg\" class=\"   wp-image-7164 size-medium\" title=\"Abraham De Moivre\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Abraham_de_moivre2-235x300.jpg\" alt=\"\" width=\"235\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Abraham_de_moivre2-235x300.jpg 235w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Abraham_de_moivre2-117x150.jpg 117w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Abraham_de_moivre2-400x509.jpg 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Abraham_de_moivre2.jpg 471w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/><\/a><p id=\"caption-attachment-7164\" class=\"wp-caption-text\">Abraham De Moivre<\/p><\/div>\n<p>A distribui\u00e7\u00e3o normal \u00e9 uma das mais importantes distribui\u00e7\u00f5es da estat\u00edstica, conhecida tamb\u00e9m como Distribui\u00e7\u00e3o de Gauss ou Gaussiana. Foi primeiramente introduzida pelo matem\u00e1tico <a href=\"https:\/\/en.wikipedia.org\/wiki\/Moivre\" target=\"_blank\" rel=\"noopener\">Abraham de Moivre<\/a>.<\/p>\n<p>Al\u00e9m de descrever uma s\u00e9rie de fen\u00f3menos f\u00edsicos e financeiros, possui grande uso na estat\u00edstica inferencial. \u00c9 inteiramente descrita por seus par\u00e2metros de m\u00e9dia e desvio padr\u00e3o, ou seja, conhecendo-se estes consegue-se determinar qualquer probabilidade em uma distribui\u00e7\u00e3o Normal.<\/p>\n<p>Um interessante uso da Distribui\u00e7\u00e3o Normal \u00e9 que ela serve de aproxima\u00e7\u00e3o para o c\u00e1lculo de outras distribui\u00e7\u00f5es quando o n\u00famero de observa\u00e7\u00f5es fica grande. Essa importante propriedade prov\u00e9m do Teorema Central do Limite que diz que &#8220;toda soma de vari\u00e1veis aleat\u00f3rias independentes de m\u00e9dia finita e vari\u00e2ncia limitada \u00e9 aproximadamente Normal, desde que o n\u00famero de termos da soma seja suficientemente grande&#8221; (ver o teorema para um enunciado mais preciso).<\/p>\n<table class=\" aligncenter\" style=\"width: 500px;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7163\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7163\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/500px-Standard_deviation_diagram_svg.png\" data-orig-size=\"500,250\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Standard deviation diagram\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/500px-Standard_deviation_diagram_svg.png\" class=\"size-full wp-image-7163 aligncenter\" title=\"Standard deviation diagram\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/500px-Standard_deviation_diagram_svg.png\" alt=\"\" width=\"500\" height=\"250\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/500px-Standard_deviation_diagram_svg.png 500w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/500px-Standard_deviation_diagram_svg-300x150.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/500px-Standard_deviation_diagram_svg-150x75.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/500px-Standard_deviation_diagram_svg-400x200.png 400w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/td>\n<\/tr>\n<tr>\n<td>A \u00e1rea em azul escuro est\u00e1 a menos de um desvio padr\u00e3o (\u03c3) da m\u00e9dia. Em uma distribui\u00e7\u00e3o normal, isto representa cerca de 68% do conjunto, enquanto dois desvios padr\u00f5es desde a m\u00e9dia (azul m\u00e9dio e escuro) representam cerca de 95%, e tr\u00eas desvios padr\u00f5es (azul claro, m\u00e9dio e escuro) cobrem cerca de 99.7%.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">\n<ul>\n<li><a href=\"https:\/\/pt.wikipedia.org\/wiki\/Distribui%C3%A7%C3%A3o_normal\" target=\"_blank\" rel=\"noopener\">Distribui\u00e7\u00e3o normal<\/a> (Wikip\u00e9dia)<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><applet codebase=\"http:\/\/stat.wvu.edu\/srs\/modules\/Applets\/ContDistr\" code=\"continuousDist.class\"\r\nname=\"continuousDist\" width=500 height=375>\r\n\t<param name = \"distributionName\" value = \"Normal\" >\r\n\t<param name = \"mu\" value = 0.0  >\r\n\t<param name = \"sigma\" value = 1.0 >\r\n<\/applet><\/p>\n<ul>\n<li><a href=\"https:\/\/web.archive.org\/web\/20210222101702\/http:\/\/www.stat.wvu.edu\/SRS\/Modules\/Normal\/normal.html\" target=\"_parent\" rel=\"noopener\">West Virg\u00ednia University<\/a><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube-nocookie.com\/embed\/BXof869EC68?rel=0\" width=\"640\" height=\"480\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7162' onClick='GTTabs_show(1,7162)'>Folha de c\u00e1lculo &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7162'>\n<span class='GTTabs_titles'><b>Folha de c\u00e1lculo<\/b><\/span><!--more--><\/p>\n<p>[embeddoc url=&#8221;https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-B_N.xls&#8221; height=&#8221;500px&#8221; viewer=&#8221;microsoft&#8221;]<\/p>\n<ul>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-B_N.xls\">https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-B_N.xls<\/a><\/li>\n<\/ul>\n<p>\u00a0<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7162' onClick='GTTabs_show(0,7162)'>&lt;&lt; Distribui\u00e7\u00e3o normal<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7162' onClick='GTTabs_show(2,7162)'>GeoGebra &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_7162'>\n<span class='GTTabs_titles'><b>GeoGebra<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": 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