{"id":2732,"date":"2024-10-17T11:04:40","date_gmt":"2024-10-17T09:04:40","guid":{"rendered":"https:\/\/clooma.ai\/?post_type=ressource-pedagogiqu&#038;p=2732"},"modified":"2024-10-24T16:50:22","modified_gmt":"2024-10-24T14:50:22","slug":"valoszinusegi-torvenyek","status":"publish","type":"ressource-pedagogiqu","link":"https:\/\/clooma.ai\/hu\/oktatasi-forras\/valoszinusegi-torvenyek\/","title":{"rendered":"Norm\u00e1l t\u00f6rv\u00e9ny"},"content":{"rendered":"<p>A statisztik\u00e1ban a norm\u00e1lis t\u00f6rv\u00e9ny (vagy norm\u00e1lis eloszl\u00e1s) az egyik legfontosabb \u00e9s leggyakrabban haszn\u00e1lt val\u00f3sz\u00edn\u0171s\u00e9gi eloszl\u00e1s. Term\u00e9szetes t\u00f6rv\u00e9nyk\u00e9nt vagy Gauss-eloszl\u00e1sk\u00e9nt is ismert, a matematikus tisztelet\u00e9re. <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Carl_Friedrich_Gauss\">Carl Friedrich Gauss<\/a> aki r\u00e9szletesen tanulm\u00e1nyozta a tulajdons\u00e1gait.\u00a0<\/p>\n\n\n\n<p>A norm\u00e1lis eloszl\u00e1st szimmetrikus harang alakja jellemzi, ami azt jelenti, hogy a legt\u00f6bb \u00e9rt\u00e9k az \u00e1tlag k\u00f6r\u00fcl csoportosul, \u00e9s az \u00e9rt\u00e9kek az \u00e1tlagt\u00f3l t\u00e1volodnak, ahogy nagyobbak vagy kisebbek lesznek. A norm\u00e1lis eloszl\u00e1st k\u00e9t param\u00e9ter hat\u00e1rozza meg:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u00c1tlag (\u00b5): Ez a harang k\u00f6z\u00e9ppontja, amely azt az \u00e9rt\u00e9ket k\u00e9pviseli, amely k\u00f6r\u00fcl a t\u00f6bbi \u00e9rt\u00e9k csoportosul. <\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Standard elt\u00e9r\u00e9s (\u03c3): Ez az \u00e9rt\u00e9kek sz\u00f3r\u00e1s\u00e1nak m\u00e9rt\u00e9ke az \u00e1tlaghoz k\u00e9pest. Min\u00e9l nagyobb a sz\u00f3r\u00e1s, ann\u00e1l nagyobb az \u00e9rt\u00e9kek sz\u00f3r\u00e1sa. <\/li>\n<\/ul>\n\n\n\n<p>A norm\u00e1lis eloszl\u00e1s val\u00f3sz\u00edn\u0171s\u00e9gi s\u0171r\u0171s\u00e9gf\u00fcggv\u00e9ny\u00e9t a k\u00f6vetkez\u0151 matematikai k\u00e9plet adja meg egy v\u00e9letlen v\u00e1ltoz\u00f3ra :&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>Ez az eloszl\u00e1s t\u00f6bb fontos tulajdons\u00e1ggal rendelkezik:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Szimmetria: Az eloszl\u00e1s szimmetrikus az \u00e1tlaga k\u00f6r\u00fcl.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Harang alak\u00fa: A legt\u00f6bb \u00e9rt\u00e9k az \u00e1tlaghoz k\u00f6zel van, \u00e9s a sz\u00e9ls\u0151\u00e9rt\u00e9kek val\u00f3sz\u00edn\u0171s\u00e9ge gyorsan cs\u00f6kken, ahogy t\u00e1volodunk az \u00e1tlagt\u00f3l.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>68-95-99,7 Szab\u00e1ly: Az \u00e9rt\u00e9kek megk\u00f6zel\u00edt\u0151leg 68% az \u00e1tlagt\u00f3l sz\u00e1m\u00edtott egy sz\u00f3r\u00e1son bel\u00fcl, 95% k\u00e9t sz\u00f3r\u00e1son bel\u00fcl \u00e9s 99,7% h\u00e1rom sz\u00f3r\u00e1son bel\u00fcl van.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A norm\u00e1lis eloszl\u00e1st a statisztika sz\u00e1mos ter\u00fclet\u00e9n haszn\u00e1lj\u00e1k, bele\u00e9rtve a statisztikai k\u00f6vetkeztet\u00e9st, a modellez\u00e9st \u00e9s a hipot\u00e9zisvizsg\u00e1latot, mivel j\u00f3l ismert matematikai tulajdons\u00e1gai \u00e9s sz\u00e1mos term\u00e9szeti \u00e9s k\u00eds\u00e9rleti jelens\u00e9gben val\u00f3 el\u0151fordul\u00e1s\u00e1nak gyakoris\u00e1ga miatt.<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png\" alt=\"\" class=\"wp-image-2733\"\/><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\">Cs\u00f6kkentett norm\u00e1lis eloszl\u00e1s<\/h2>\n\n\n\n<p>A \"centr\u00e1lis reduk\u00e1lt norm\u00e1lis\" eloszl\u00e1s a standard norm\u00e1lis eloszl\u00e1sra utal, azaz olyan norm\u00e1lis eloszl\u00e1sra, amelynek \u00e1tlaga 0 \u00e9s sz\u00f3r\u00e1sa 1. Ez az egyik leggyakrabban haszn\u00e1lt eloszl\u00e1s a statisztik\u00e1ban.&nbsp;<\/p>\n\n\n\n<p>B\u00e1rmely norm\u00e1lis v\u00e1ltoz\u00f3 \u00e1talak\u00edthat\u00f3 cs\u00f6kkentett centr\u00e1lis norm\u00e1liss\u00e1 a v\u00e1ltoz\u00f3 \u00e1tlag\u00e1nak kivon\u00e1s\u00e1val \u00e9s a sz\u00f3r\u00e1ssal val\u00f3 oszt\u00e1s\u00e1val. Ez a normaliz\u00e1l\u00e1s hasznos olyan v\u00e1ltoz\u00f3k \u00f6sszehasonl\u00edt\u00e1s\u00e1hoz, amelyek eredetileg k\u00fcl\u00f6nb\u00f6z\u0151 m\u00e9rt\u00e9kegys\u00e9gekkel vagy k\u00fcl\u00f6nb\u00f6z\u0151 sk\u00e1l\u00e1kkal rendelkeznek. Emellett sz\u00e1mos statisztikai \u00f6sszef\u00fcgg\u00e9sben egyszer\u0171s\u00edti a sz\u00e1m\u00edt\u00e1sokat.&nbsp;<\/p>\n\n\n\n<p>Egy X v\u00e9letlen v\u00e1ltoz\u00f3 eset\u00e9ben, amely norm\u00e1l eloszl\u00e1st k\u00f6vet \u03bc \u00e1tlaggal \u00e9s \u03c3 sz\u00f3r\u00e1ssal. Az X normaliz\u00e1l\u00e1sa a reduk\u00e1lt centr\u00e1lis norm\u00e1lis (gyakran Z-nek nevezik) kisz\u00e1m\u00edt\u00e1s\u00e1hoz a k\u00f6vetkez\u0151 k\u00e9plet seg\u00edts\u00e9g\u00e9vel t\u00f6rt\u00e9nik :&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>A Z \u00e9rt\u00e9ke az \u00e1tlagt\u00f3l val\u00f3 sz\u00f3r\u00e1sok sz\u00e1m\u00e1t jelenti. Lehet pozit\u00edv vagy negat\u00edv.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/clooma.ai\/wp-content\/uploads\/image-11.png\" alt=\"\" class=\"wp-image-2735\"\/><\/figure><\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>A Z=2 \u00e9rt\u00e9k azt jelenti, hogy ez a pont a \u00b5 \u00e1tlag felett van, \u00e9s az \u00e1tlagt\u00f3l val\u00f3 elt\u00e9r\u00e9s 2 \u03c3 standard elt\u00e9r\u00e9s.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A Z=-3,5 \u00e9rt\u00e9k azt jelenti, hogy ez a pont az \u00e1tlagos \u00b5 alatt van, \u00e9s az \u00e1tlagt\u00f3l val\u00f3 elt\u00e9r\u00e9s 3,5 \u03c3 standard elt\u00e9r\u00e9s.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p>Ezzel az \u00e1talak\u00edt\u00e1ssal haszn\u00e1lhatjuk a k\u00f6zpontos\u00edtott reduk\u00e1lt norm\u00e1lis eloszl\u00e1s t\u00e1bl\u00e1zat\u00e1t. Ezt a t\u00e1bl\u00e1zatot arra haszn\u00e1ljuk, hogy meghat\u00e1rozzuk az F(x) norm\u00e1lis eloszl\u00e1s eloszl\u00e1sf\u00fcggv\u00e9ny\u00e9nek \u00e9rt\u00e9keit Z \u00e9rt\u00e9k\u00e9nek f\u00fcggv\u00e9ny\u00e9ben.&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>A :&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>F(Z) : A standard norm\u00e1lis eloszl\u00e1s (vagy reduk\u00e1lt centr\u00e1lis norm\u00e1lis eloszl\u00e1s) eloszl\u00e1sf\u00fcggv\u00e9nye. Olyan matematikai f\u00fcggv\u00e9ny, amely megadja annak val\u00f3sz\u00edn\u0171s\u00e9g\u00e9t, hogy egy standard norm\u00e1lis eloszl\u00e1st k\u00f6vet\u0151 v\u00e9letlen v\u00e1ltoz\u00f3 kisebb vagy egyenl\u0151 egy adott \u00e9rt\u00e9kn\u00e9l.&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>Az F(Z) \u00e9rt\u00e9ke mindig 0 \u00e9s 1 k\u00f6z\u00f6tt van, mivel ez egy val\u00f3sz\u00edn\u0171s\u00e9g.&nbsp;<\/p>\n\n\n\n<p>A standard norm\u00e1lis eloszl\u00e1s F(Z) eloszl\u00e1sf\u00fcggv\u00e9ny\u00e9nek \u00e9rt\u00e9keit a statisztika sz\u00e1mos ter\u00fclet\u00e9n haszn\u00e1lj\u00e1k val\u00f3sz\u00edn\u0171s\u00e9gi sz\u00e1m\u00edt\u00e1sok elv\u00e9gz\u00e9s\u00e9re, bele\u00e9rtve a hipot\u00e9zisvizsg\u00e1latot, a konfidenciaintervallumokat, a nem-megfelel\u00e9si ar\u00e1ny becsl\u00e9s\u00e9t, a folyamat megb\u00edzhat\u00f3s\u00e1g\u00e1nak becsl\u00e9s\u00e9t \u00e9s m\u00e1s statisztikai elemz\u00e9seket.&nbsp;<\/p>\n\n\n\n<p>Az F(Z) eloszl\u00e1sf\u00fcggv\u00e9ny nem fejezhet\u0151 ki elemi f\u00fcggv\u00e9nyekkel (p\u00e9ld\u00e1ul polinom, exponenci\u00e1lis vagy trigonometrikus f\u00fcggv\u00e9nyekkel), \u00e9s gyakran statisztikai t\u00e1bl\u00e1zatok vagy sz\u00e1m\u00edt\u00f3g\u00e9pes szoftverek haszn\u00e1lata sz\u00fcks\u00e9ges a Z bizonyos \u00e9rt\u00e9keihez tartoz\u00f3 val\u00f3sz\u00edn\u0171s\u00e9gi \u00e9rt\u00e9kek kisz\u00e1m\u00edt\u00e1s\u00e1hoz. A norm\u00e1lis eloszl\u00e1s eset\u00e9ben az F(Z) kisz\u00e1m\u00edt\u00e1s\u00e1hoz a reduk\u00e1lt centr\u00e1lis norm\u00e1lis eloszl\u00e1si t\u00e1bl\u00e1zatot, m\u00e1s n\u00e9ven Z-t\u00e1bl\u00e1zatot kell haszn\u00e1lni:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/clooma.ai\/wp-content\/uploads\/image.jpeg\" alt=\"\" class=\"wp-image-2736\"\/><\/figure>\n\n\n\n<p>P\u00e9lda:&nbsp;<\/p>\n\n\n\n<p>Keresse meg a k\u00f6vetkez\u0151 val\u00f3sz\u00edn\u0171s\u00e9gek \u00e9rt\u00e9keit a norm\u00e1lis eloszl\u00e1s seg\u00edts\u00e9g\u00e9vel:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc43(\ud835\udc67\u22640), \ud835\udc43(\ud835\udc67\u2264-2), \ud835\udc43(\ud835\udc67\u22651,55), \ud835\udc43(-2\u2264 \ud835\udc67 \u22641,55)&nbsp;<\/p>\n\n\n\n<p>Megold\u00e1s:&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\">Val\u00f3sz\u00edn\u0171s\u00e9g&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\u2264-2)=\ud835\udc43(2\u2264\ud835\udc67)=1-\ud835\udc43(\ud835\udc67\u22642) = 1-0,9772=0,0228&nbsp;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ud835\udc43(\ud835\udc67\u22651.55) = 1-\ud835\udc43(\ud835\udc67\u22641.55)= 1-0.9394 = 0.0606<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ud835\udc43(-2\u2264\ud835\udc67\u22641,55) = \ud835\udc43(\ud835\udc67\u22641,55)-\ud835\udc43(\ud835\udc67\u2264-2) = 0,9394-0,0228 = 0,9166<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/clooma.ai\/wp-content\/uploads\/image-12.png\" alt=\"\" class=\"wp-image-2737\"\/><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\">A t\u0171r\u00e9shat\u00e1ron k\u00edv\u00fcli sz\u00e1zal\u00e9kos ar\u00e1ny kisz\u00e1m\u00edt\u00e1sa&nbsp;<\/h2>\n\n\n\n<p>Amint azt a norm\u00e1lis eloszl\u00e1s jellemz\u0151inek meg\u00e1llap\u00edt\u00e1sakor m\u00e1r eml\u00edtett\u00fck, a norm\u00e1lis eloszl\u00e1s teljes m\u00e9rt\u00e9kben jellemezhet\u0151, amint ismert az \u00e1tlaga \u00e9s a sz\u00f3r\u00e1sa. Pontosabban :&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A megfigyel\u00e9sek 68,27%-je az \u00e1tlagt\u00f3l sz\u00e1m\u00edtott egy sz\u00f3r\u00e1son bel\u00fcl van.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A megfigyel\u00e9sek mintegy 95,45%-je az \u00e1tlagt\u00f3l sz\u00e1m\u00edtott k\u00e9t sz\u00f3r\u00e1son bel\u00fcl van.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A megfigyel\u00e9sek 99,73%-je az \u00e1tlagt\u00f3l sz\u00e1m\u00edtott h\u00e1rom sz\u00f3r\u00e1son bel\u00fcl van.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p>Ezek a sz\u00e1zal\u00e9kok le\u00edrj\u00e1k, hogy az adatok hogyan oszlanak el az \u00e1tlag k\u00f6r\u00fcl egy norm\u00e1lis eloszl\u00e1sban, \u00e9rt\u00e9kes inform\u00e1ci\u00f3t szolg\u00e1ltatva az \u00e9rt\u00e9keknek az \u00e1tlaghoz viszony\u00edtott sz\u00f3r\u00e1s\u00e1r\u00f3l.&nbsp;<\/p>\n\n\n\n<p>Ahhoz azonban, hogy pontosabban meg lehessen \u00e1llap\u00edtani, hogy egy popul\u00e1ci\u00f3ban a megengedett hat\u00e1r\u00e9rt\u00e9keken k\u00edv\u00fcl es\u0151 elemek h\u00e1ny sz\u00e1zal\u00e9ka van, ki lehet sz\u00e1m\u00edtani a z-sz\u00e1mot.&nbsp;<\/p>\n\n\n\n<p>A z sz\u00e1mot a k\u00f6vetkez\u0151k\u00e9ppen kell kisz\u00e1m\u00edtani:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Z = \\frac{\\mu-\\text{tolerancia}}{{sigma}}<\/span><\/p>\n\n\n\n<p>A m\u00e9r\u00e9st a minta \u00e1tlag\u00e9rt\u00e9ke \u00e9s a t\u0171r\u00e9shat\u00e1r k\u00f6z\u00f6tti standard elt\u00e9r\u00e9sekben fejezi ki.&nbsp;<\/p>\n\n\n\n<p>A z sz\u00e1m meghat\u00e1roz\u00e1sa ut\u00e1n a Gauss-t\u00e1bl\u00e1zat vagy a k\u00f6zpontos\u00edtott reduk\u00e1lt norm\u00e1lis eloszl\u00e1si t\u00e1bl\u00e1zat seg\u00edts\u00e9g\u00e9vel kisz\u00e1m\u00edthat\u00f3 a t\u0171r\u00e9shat\u00e1ron k\u00edv\u00fcl es\u0151 elemek sz\u00e1zal\u00e9kos ar\u00e1nya. Ezt a t\u00e1bl\u00e1zatot arra haszn\u00e1lj\u00e1k, hogy megtal\u00e1lj\u00e1k a norm\u00e1lis eloszl\u00e1sban az \u00e1tlagt\u00f3l egy bizonyos t\u00e1vols\u00e1gon t\u00fali \u00e9rt\u00e9kek ar\u00e1ny\u00e1t (amelyet a z sz\u00e1m jelk\u00e9pez), ami seg\u00edt a t\u0171r\u00e9shat\u00e1ron k\u00edv\u00fcl es\u0151 elemek sz\u00e1zal\u00e9kos ar\u00e1ny\u00e1nak felm\u00e9r\u00e9s\u00e9ben.&nbsp;<\/p>\n\n\n\n<p><br><strong>P\u00e9lda:&nbsp;<\/strong><\/p>\n\n\n\n<p>Hat\u00e1rozza meg a teljes t\u0171r\u00e9shat\u00e1ron k\u00edv\u00fclis\u00e9g sz\u00e1zal\u00e9kos ar\u00e1ny\u00e1t, ha az \u00e1tlagos \u00e1tm\u00e9r\u0151 \u00b5=10,1 mm, a sz\u00f3r\u00e1s \u03c3=0,5 mm \u00e9s a t\u0171r\u00e9shat\u00e1r IT=[9; 11].&nbsp;<\/p>\n\n\n\n<p>Sz\u00e1m\u00edtsuk ki a z<sub>min<\/sub>:&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{tolerancia}}{\\sigma} = \\frac{10.1-9}{0.5} = 2.2<\/span><\/p>\n\n\n\n<p>Ez megadja a Gauss-t\u00e1bl\u00e1zatban a min. t\u0171r\u00e9shat\u00e1ron k\u00edv\u00fcl es\u0151 alkatr\u00e9szek sz\u00e1zal\u00e9kos ar\u00e1ny\u00e1t:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">% HT min = 100% - 98.61% = 1.39%<\/p>\n\n\n\n<p>Sz\u00e1m\u00edtsuk ki a z<sub>max<\/sub>:&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{tolerancia}}{sigma} = \\frac{10.1-11}{0.5} = 1.8<\/span><\/p>\n\n\n\n<p>A Gauss-t\u00e1bl\u00e1zatban szerepl\u0151 max. t\u0171r\u00e9shat\u00e1ron k\u00edv\u00fcl es\u0151 alkatr\u00e9szek sz\u00e1zal\u00e9kos ar\u00e1nya ebb\u0151l levezethet\u0151:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">% HT max =100%-98,61% = 3,59%<\/p>\n\n\n\n<p>A teljes t\u0171r\u00e9shat\u00e1ron t\u00fali sz\u00e1zal\u00e9kos ar\u00e1nyt ez\u00e9rt levonj\u00e1k :&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">% HT= % 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 decoding=\"async\" src=\"https:\/\/clooma.ai\/wp-content\/uploads\/image-13.png\" alt=\"\" class=\"wp-image-2738\"\/><\/figure><\/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 v25.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Loi normale - Clooma<\/title>\n<meta name=\"description\" content=\"En statistiques, la loi normale est l&#039;une des distributions de probabilit\u00e9 les plus importantes et couramment utilis\u00e9es.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/clooma.ai\/hu\/oktatasi-forras\/valoszinusegi-torvenyek\/\" \/>\n<meta property=\"og:locale\" content=\"hu_HU\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Loi normale - Clooma\" \/>\n<meta property=\"og:description\" content=\"En statistiques, la loi normale est l&#039;une des distributions de probabilit\u00e9 les plus importantes et couramment utilis\u00e9es.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/clooma.ai\/hu\/oktatasi-forras\/valoszinusegi-torvenyek\/\" \/>\n<meta property=\"og:site_name\" content=\"Clooma\" \/>\n<meta property=\"article:modified_time\" content=\"2024-10-24T14:50:22+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Becs\u00fclt olvas\u00e1si id\u0151\" \/>\n\t<meta name=\"twitter:data1\" content=\"7 perc\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/\",\"url\":\"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/\",\"name\":\"Loi normale - Clooma\",\"isPartOf\":{\"@id\":\"https:\/\/clooma.ai\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png\",\"datePublished\":\"2024-10-17T09:04:40+00:00\",\"dateModified\":\"2024-10-24T14:50:22+00:00\",\"description\":\"En statistiques, la loi normale est l'une des distributions de probabilit\u00e9 les plus importantes et couramment utilis\u00e9es.\",\"breadcrumb\":{\"@id\":\"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#breadcrumb\"},\"inLanguage\":\"hu\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"hu\",\"@id\":\"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#primaryimage\",\"url\":\"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png\",\"contentUrl\":\"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\/\/clooma.ai\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Loi normale\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/clooma.ai\/#website\",\"url\":\"https:\/\/clooma.ai\/\",\"name\":\"Clooma\",\"description\":\"Close the loop, master the future\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/clooma.ai\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"hu\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Loi normale - Clooma","description":"A statisztik\u00e1ban a norm\u00e1lis eloszl\u00e1s az egyik legfontosabb \u00e9s leggyakrabban haszn\u00e1lt val\u00f3sz\u00edn\u0171s\u00e9gi eloszl\u00e1s.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/clooma.ai\/hu\/oktatasi-forras\/valoszinusegi-torvenyek\/","og_locale":"hu_HU","og_type":"article","og_title":"Loi normale - Clooma","og_description":"En statistiques, la loi normale est l'une des distributions de probabilit\u00e9 les plus importantes et couramment utilis\u00e9es.","og_url":"https:\/\/clooma.ai\/hu\/oktatasi-forras\/valoszinusegi-torvenyek\/","og_site_name":"Clooma","article_modified_time":"2024-10-24T14:50:22+00:00","og_image":[{"url":"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png","type":"","width":"","height":""}],"twitter_card":"summary_large_image","twitter_misc":{"Becs\u00fclt olvas\u00e1si id\u0151":"7 perc"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/","url":"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/","name":"Loi normale - Clooma","isPartOf":{"@id":"https:\/\/clooma.ai\/#website"},"primaryImageOfPage":{"@id":"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#primaryimage"},"image":{"@id":"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#primaryimage"},"thumbnailUrl":"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png","datePublished":"2024-10-17T09:04:40+00:00","dateModified":"2024-10-24T14:50:22+00:00","description":"A statisztik\u00e1ban a norm\u00e1lis eloszl\u00e1s az egyik legfontosabb \u00e9s leggyakrabban haszn\u00e1lt val\u00f3sz\u00edn\u0171s\u00e9gi eloszl\u00e1s.","breadcrumb":{"@id":"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#breadcrumb"},"inLanguage":"hu","potentialAction":[{"@type":"ReadAction","target":["https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/"]}]},{"@type":"ImageObject","inLanguage":"hu","@id":"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#primaryimage","url":"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png","contentUrl":"https:\/\/clooma.ai\/wp-content\/uploads\/image-10.png"},{"@type":"BreadcrumbList","@id":"https:\/\/clooma.ai\/ressource-pedagogiqu\/lois-de-probabilite\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/clooma.ai\/"},{"@type":"ListItem","position":2,"name":"Loi normale"}]},{"@type":"WebSite","@id":"https:\/\/clooma.ai\/#website","url":"https:\/\/clooma.ai\/","name":"Clooma","description":"Z\u00e1rd be a hurkot, ir\u00e1ny\u00edtsd a j\u00f6v\u0151t","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/clooma.ai\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"hu"}]}},"_links":{"self":[{"href":"https:\/\/clooma.ai\/hu\/wp-json\/wp\/v2\/ressource-pedagogiqu\/2732","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clooma.ai\/hu\/wp-json\/wp\/v2\/ressource-pedagogiqu"}],"about":[{"href":"https:\/\/clooma.ai\/hu\/wp-json\/wp\/v2\/types\/ressource-pedagogiqu"}],"wp:attachment":[{"href":"https:\/\/clooma.ai\/hu\/wp-json\/wp\/v2\/media?parent=2732"}],"wp:term":[{"taxonomy":"menu-ressource-pedagogique","embeddable":true,"href":"https:\/\/clooma.ai\/hu\/wp-json\/wp\/v2\/menu-ressource-pedagogique?post=2732"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}