{"id":2732,"date":"2024-10-17T11:04:40","date_gmt":"2024-10-17T09:04:40","guid":{"rendered":"https:\/\/ellistat.com\/?post_type=ressource-pedagogiqu&#038;p=2732"},"modified":"2024-10-24T16:50:22","modified_gmt":"2024-10-24T14:50:22","slug":"leggi-della-probabilita","status":"publish","type":"ressource-pedagogiqu","link":"https:\/\/ellistat.com\/it\/ressource-pedagogiqu\/lois-de-probabilite\/","title":{"rendered":"Legge normale"},"content":{"rendered":"\n<p>En statistiques, la loi normale (ou distribution normale) est l&rsquo;une des distributions de probabilit\u00e9 les plus importantes et couramment utilis\u00e9es. Elle est \u00e9galement connue sous le nom loi naturelle, ou distribution gaussienne, en l&rsquo;honneur du math\u00e9maticien <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Carl_Friedrich_Gauss\">Carl Friedrich Gauss<\/a> qui a \u00e9tudi\u00e9 en d\u00e9tail ses propri\u00e9t\u00e9s.\u00a0<\/p>\n\n\n\n<p>La distribution normale est caract\u00e9ris\u00e9e par sa forme en cloche sym\u00e9trique, ce qui signifie que la plupart des valeurs se regroupent autour de la moyenne, et les valeurs s&rsquo;\u00e9loignent de la moyenne \u00e0 mesure qu&rsquo;elles deviennent plus grandes ou plus petites. La distribution normale est d\u00e9finie par deux param\u00e8tres :<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Moyenne (\u00b5) : C&rsquo;est le centre de la cloche, repr\u00e9sentant la valeur autour de laquelle les autres valeurs se regroupent. <\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u00c9cart-type (\u03c3) : C&rsquo;est une mesure de la dispersion des valeurs par rapport \u00e0 la moyenne. Plus l&rsquo;\u00e9cart-type est grand, plus la dispersion des valeurs est importante. <\/li>\n<\/ul>\n\n\n\n<p>La fonction de densit\u00e9 de probabilit\u00e9 de la distribution normale est donn\u00e9e par la formule math\u00e9matique suivante pour une variable al\u00e9atoire\u202f:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">f(x)=\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^{2}}{2\\sigma^{2}}}<\/span><\/p>\n\n\n\n<p>Cette distribution a plusieurs propri\u00e9t\u00e9s importantes :&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sym\u00e9trie : La distribution est sym\u00e9trique par rapport \u00e0 sa moyenne.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Forme en cloche : La plupart des valeurs se trouvent pr\u00e8s de la moyenne, et la probabilit\u00e9 de valeurs extr\u00eames diminue rapidement \u00e0 mesure que l&rsquo;on s&rsquo;\u00e9loigne de la moyenne.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>68-95-99.7 R\u00e8gle : Environ 68% des valeurs se situent \u00e0 un \u00e9cart-type de la moyenne, 95% \u00e0 deux \u00e9cart-types et 99.7% \u00e0 trois \u00e9cart-types.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>La distribution normale est utilis\u00e9e dans de nombreux domaines de la statistique, y compris l&rsquo;inf\u00e9rence statistique, la mod\u00e9lisation et les tests d&rsquo;hypoth\u00e8ses, en raison de ses propri\u00e9t\u00e9s math\u00e9matiques bien connues et de sa fr\u00e9quence d&rsquo;apparition dans de nombreux ph\u00e9nom\u00e8nes naturels et exp\u00e9rimentaux.<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"374\" height=\"385\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/image-10.png\" alt=\"\" class=\"wp-image-2733\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/image-10.png 374w, https:\/\/ellistat.com\/wp-content\/uploads\/image-10-291x300.png 291w, https:\/\/ellistat.com\/wp-content\/uploads\/image-10-12x12.png 12w\" sizes=\"auto, (max-width: 374px) 100vw, 374px\" \/><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\">Loi normale centr\u00e9e r\u00e9duite<\/h2>\n\n\n\n<p>La loi \u00ab\u00a0normale centr\u00e9e r\u00e9duite\u00a0\u00bb fait r\u00e9f\u00e9rence \u00e0 une distribution normale standardis\u00e9e, c&rsquo;est-\u00e0-dire une distribution normale avec une moyenne de 0 et un \u00e9cart-type de 1. C&rsquo;est l&rsquo;une des distributions les plus couramment utilis\u00e9es en statistiques.&nbsp;<\/p>\n\n\n\n<p>En effet toute variable normale peut \u00eatre transform\u00e9e en une normale centr\u00e9e r\u00e9duite en soustrayant la moyenne de la variable et en divisant par l&rsquo;\u00e9cart-type. Cette normalisation est utile car elle permet de comparer des variables qui initialement peuvent avoir des unit\u00e9s diff\u00e9rentes ou des \u00e9chelles diff\u00e9rentes. De plus, elle simplifie les calculs dans de nombreux contextes statistiques.&nbsp;<\/p>\n\n\n\n<p>Pour une variable al\u00e9atoire X suivant une distribution normale avec une moyenne \u03bc et un \u00e9cart-type \u03c3. La normalisation de X pour obtenir la normale centr\u00e9e r\u00e9duite (souvent not\u00e9e Z) se fait en utilisant la formule :&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Z=\\frac{X-\\mu}{\\sigma}<\/span><\/p>\n\n\n\n<p>La valeur de Z repr\u00e9sente le d\u00e9calage en nombre d\u2019\u00e9cart-type par rapport \u00e0 la moyenne. Elle peut \u00eatre positive ou n\u00e9gative.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"377\" height=\"344\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/image-11.png\" alt=\"\" class=\"wp-image-2735\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/image-11.png 377w, https:\/\/ellistat.com\/wp-content\/uploads\/image-11-300x274.png 300w, https:\/\/ellistat.com\/wp-content\/uploads\/image-11-13x12.png 13w\" sizes=\"auto, (max-width: 377px) 100vw, 377px\" \/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>Une valeur de Z=2, signifie que ce point est sup\u00e9rieur \u00e0 la moyenne \u00b5 et le&nbsp;d\u00e9calage par rapport \u00e0 cette derni\u00e8re est&nbsp;de 2&nbsp;\u00e9cart-types&nbsp;\u03c3.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Une valeur de Z=-3.5, signifie que ce point est inf\u00e9rieur \u00e0 la moyenne \u00b5 et le d\u00e9calage par rapport \u00e0 cette derni\u00e8re est de&nbsp;3.5&nbsp;\u00e9cart-types&nbsp;\u03c3.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p>Avec cette transformation, on va pouvoir utiliser la table de la loi normale centr\u00e9e r\u00e9duite. Cette table permet de d\u00e9terminer les valeurs de la fonction de r\u00e9partition de la loi normale F(x) en fonction de la valeur de Z.&nbsp;&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">F(Z)=\\int_{-\\infty }^{Z}\\frac{1}{\\sqrt{2\\Pi}}e^{-\\frac{u^{2}}{2}}<\/span><\/p>\n\n\n\n<p>Avec\u202f:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>F(Z)\u202f: La fonction de r\u00e9partition de la loi normale standard (ou normale centr\u00e9e r\u00e9duite). Elle est une fonction math\u00e9matique qui donne la probabilit\u00e9 qu&rsquo;une variable al\u00e9atoire suivant une distribution normale standard soit inf\u00e9rieure ou \u00e9gale \u00e0 une valeur donn\u00e9e.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc39(\ud835\udc4d)=\ud835\udc43(\ud835\udc67 \u2264 \ud835\udc4d)<\/p>\n\n\n\n<p>La valeur de F(Z) est toujours comprise entre 0 et 1, car il s\u2019agit d\u2019une probabilit\u00e9.&nbsp;<\/p>\n\n\n\n<p>Les valeurs de la fonction de r\u00e9partition F(Z) pour la distribution normale standard sont utilis\u00e9es dans de nombreux domaines de la statistique pour effectuer des calculs de probabilit\u00e9, notamment dans les tests d\u2019hypoth\u00e8ses, intervalles de confiance, l\u2019estimation du taux de non-conformit\u00e9, l\u2019estimation de la fiabilit\u00e9 des processus et d\u2019autres analyses statistiques.&nbsp;<\/p>\n\n\n\n<p>La fonction de r\u00e9partition F(Z) ne peut pas \u00eatre exprim\u00e9e en termes de fonctions \u00e9l\u00e9mentaires (telles que polyn\u00f4mes, exponentielles ou trigonom\u00e9triques) et n\u00e9cessite souvent l\u2019utilisation de tables statistiques ou de logiciels informatiques pour calculer les valeurs de probabilit\u00e9 associ\u00e9es \u00e0 des valeurs sp\u00e9cifiques de Z. Dans le cas de la loi normale, la table de loi normale centr\u00e9e r\u00e9duite, appel\u00e9e aussi la table de Z sera utilis\u00e9e pour le calcul de F(Z)\u202f:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"712\" height=\"880\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/image.jpeg\" alt=\"\" class=\"wp-image-2736\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/image.jpeg 712w, https:\/\/ellistat.com\/wp-content\/uploads\/image-243x300.jpeg 243w, https:\/\/ellistat.com\/wp-content\/uploads\/image-10x12.jpeg 10w\" sizes=\"auto, (max-width: 712px) 100vw, 712px\" \/><\/figure>\n\n\n\n<p>Exemple\u202f:&nbsp;<\/p>\n\n\n\n<p>Trouver les valeurs des probabilit\u00e9s suivantes en utilisant la loi normale\u202f:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc43(\ud835\udc67\u22640),&nbsp;\ud835\udc43(\ud835\udc67\u2264\u22122), \ud835\udc43(\ud835\udc67\u22651.55), \ud835\udc43(\u22122\u2264 \ud835\udc67 \u22641.55)&nbsp;<\/p>\n\n\n\n<p>Solution\u202f:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Probabilit\u00e9&nbsp;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ud835\udc43(\ud835\udc67\u22640) = 0.5Pz \u2264 0 = 0.5&nbsp;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ud835\udc43(\ud835\udc67\u2264\u22122)=\ud835\udc43(2\u2264\ud835\udc67)=1\u2212\ud835\udc43(\ud835\udc67\u22642) =&nbsp;1\u22120.9772=0.0228&nbsp;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ud835\udc43(\ud835\udc67\u22651.55) = 1\u2212\ud835\udc43(\ud835\udc67\u22641.55)= 1\u22120.9394 = 0,0606<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ud835\udc43(\u22122\u2264\ud835\udc67\u22641.55) = \ud835\udc43(\ud835\udc67\u22641.55)\u2212\ud835\udc43(\ud835\udc67\u2264\u22122) = 0.9394\u22120.0228 = 0.9166<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"450\" height=\"556\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/image-12.png\" alt=\"\" class=\"wp-image-2737\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/image-12.png 450w, https:\/\/ellistat.com\/wp-content\/uploads\/image-12-243x300.png 243w, https:\/\/ellistat.com\/wp-content\/uploads\/image-12-10x12.png 10w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\">Calcul du pourcentage hors tol\u00e9rance&nbsp;<\/h2>\n\n\n\n<p>Comme abord\u00e9 lors de l&rsquo;\u00e9tablissement des caract\u00e9ristiques de la distribution normale, celle-ci est pleinement caract\u00e9ris\u00e9e d\u00e8s que sa moyenne et son \u00e9cart-type sont connus. Plus sp\u00e9cifiquement :&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Environ 68.27% des observations se situent \u00e0 l&rsquo;int\u00e9rieur d&rsquo;un \u00e9cart-type de la moyenne.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Environ 95.45% des observations se situent \u00e0 l&rsquo;int\u00e9rieur de deux \u00e9carts-types de la moyenne.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Environ 99.73% des observations se situent \u00e0 l&rsquo;int\u00e9rieur de trois \u00e9carts-types de la moyenne.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p>Ces pourcentages d\u00e9crivent la fa\u00e7on dont les donn\u00e9es sont r\u00e9parties autour de la moyenne dans une distribution normale, fournissant des indications pr\u00e9cieuses sur la dispersion des valeurs par rapport \u00e0 la moyenne.&nbsp;<\/p>\n\n\n\n<p>Cependant, pour \u00e9valuer plus pr\u00e9cis\u00e9ment le pourcentage d&rsquo;\u00e9l\u00e9ments en dehors des limites tol\u00e9r\u00e9es dans une population, il est possible d&rsquo;utiliser le calcul du nombre z.&nbsp;<\/p>\n\n\n\n<p>Le nombre z se calcule ainsi :&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Z = \\frac{\\mu-\\text{tol\u00e9rance}}{\\sigma}<\/span><\/p>\n\n\n\n<p>Il repr\u00e9sente la mesure en termes d&rsquo;\u00e9cart-types entre la valeur moyenne de l&rsquo;\u00e9chantillon et la limite de tol\u00e9rance.&nbsp;<\/p>\n\n\n\n<p>Une fois le nombre z d\u00e9termin\u00e9, il est possible de calculer le pourcentage d&rsquo;\u00e9l\u00e9ments hors tol\u00e9rance en se r\u00e9f\u00e9rant \u00e0 la table de Gauss ou \u00e0 la table de la distribution normale centr\u00e9e r\u00e9duite. Cette table permet de trouver la proportion des valeurs au-del\u00e0 d&rsquo;une certaine distance (repr\u00e9sent\u00e9e par le nombre z) de la moyenne dans une distribution normale, ce qui aide \u00e0 \u00e9valuer le pourcentage d&rsquo;\u00e9l\u00e9ments en dehors des limites tol\u00e9r\u00e9es.&nbsp;<\/p>\n\n\n\n<p><br><strong>Exemple\u202f:&nbsp;<\/strong><\/p>\n\n\n\n<p>Trouver le pourcentage hors tol\u00e9rance total, sachant que le diam\u00e8tre moyen est \u00b5=10.1mm et l\u2019\u00e9cart type \u03c3=0.5mm et l\u2019intervalle de tol\u00e9rance\u202fIT=[9\u202f; 11].&nbsp;<\/p>\n\n\n\n<p>Calculons le z<sub>min<\/sub>\u202f:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Z_{min} = \\frac{\\mu-\\text{tol\u00e9rance}}{\\sigma} = \\frac{10.1-9}{0.5} = 2.2<\/span><\/p>\n\n\n\n<p>On en d\u00e9duit le pourcentage de pi\u00e8ces hors tol\u00e9rance min dans la table de Gauss :&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">% HT min = 100% &#8211; 98.61% = 1.39%<\/p>\n\n\n\n<p>Calculons le z<sub>max<\/sub>\u202f:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Z_{max} = \\frac{\\mu-\\text{tol\u00e9rance}}{\\sigma} = \\frac{10.1-11}{0.5} = 1.8<\/span><\/p>\n\n\n\n<p>On en d\u00e9duit le pourcentage de pi\u00e8ces hors tol\u00e9rance max dans la table de Gauss :&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">% HT max&nbsp;=100%\u221298,61% = 3.59%<\/p>\n\n\n\n<p>On d\u00e9duit donc le pourcentage hors tol\u00e9rance total\u202f:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">% HT=&nbsp;% HTmin +% HTmax<\/p>\n\n\n\n<p class=\"has-text-align-center\">% HT = 1.39%+3.59%\u22485%<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"475\" height=\"562\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/image-13.png\" alt=\"\" class=\"wp-image-2738\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/image-13.png 475w, https:\/\/ellistat.com\/wp-content\/uploads\/image-13-254x300.png 254w, https:\/\/ellistat.com\/wp-content\/uploads\/image-13-10x12.png 10w\" sizes=\"auto, (max-width: 475px) 100vw, 475px\" \/><\/figure>\n<\/div>","protected":false},"featured_media":0,"template":"","meta":{"_acf_changed":false},"menu-ressource-pedagogique":[27],"class_list":["post-2732","ressource-pedagogiqu","type-ressource-pedagogiqu","status-publish","hentry","menu-ressource-pedagogique-3-statistiques-descriptives"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - 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