{"id":181866,"date":"2024-04-02T08:47:00","date_gmt":"2024-04-02T07:47:00","guid":{"rendered":"https:\/\/datascientest.com\/en\/?p=181866"},"modified":"2024-03-07T21:05:28","modified_gmt":"2024-03-07T20:05:28","slug":"demystifying-the-central-limit-theorem-a-guide-to-understanding-and-application","status":"publish","type":"post","link":"https:\/\/datascientest.com\/en\/demystifying-the-central-limit-theorem-a-guide-to-understanding-and-application","title":{"rendered":"Demystifying the Central Limit Theorem: A Guide to Understanding and Application"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"181866\" class=\"elementor elementor-181866\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-885d3a0 e-flex e-con-boxed e-con\" data-id=\"885d3a0\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0e84f3b elementor-widget elementor-widget-heading\" data-id=\"0e84f3b\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h2 class=\"elementor-heading-title elementor-size-default\">Reaffirming the importance of the normal distribution, the central limit theorem is one of the key concepts in statistics.<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-801f450 elementor-widget elementor-widget-text-editor\" data-id=\"801f450\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<p>So what exactly is it? How does it translate? And above all, how is it applied? That&#8217;s what we&#8217;ll be looking at in this article.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-828bb20 e-flex e-con-boxed e-con\" data-id=\"828bb20\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a2adc68 elementor-widget elementor-widget-heading\" data-id=\"a2adc68\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The central limit theorem and convergence to the normal distribution<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d33c7cf elementor-widget elementor-widget-text-editor\" data-id=\"d33c7cf\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>The <strong>central limit theorem<\/strong> establishes the convergence of a sequence of variables defined on the same space to a normal distribution.<\/p><p>In other words: the more independent and identically distributed random variables are added together, the closer the probability distribution of the new variable will be to a normal distribution (also known as a<strong> Gaussian curve or a bell curve).<\/strong><\/p><p>And this applies to all types of <a href=\"https:\/\/datascientest.com\/en\/bayesian-optimization-definition-and-operation\">probability distributions<\/a> for a random event, such as the continuous uniform distribution, the triangular distribution, the exponential distribution and so on.<\/p><p>As soon as this event is repeated often enough, its mean gradually converges to a normal distribution. Even if at first (after adding up the variables once or twice), the distribution was not normal, it gradually becomes so as the addition of variables increases.<\/p><p>It is precisely this principle that enables statisticians to assert that the normal law is the law of natural phenomena.<\/p><p><strong>A little history<\/strong>: The central limit theorem first came into being in 1718, when De Moivre demonstrated the importance of the normal law.<\/p><p>But it was not until the work of Pierre-Simon de Laplace in 1812 that this theorem was truly applied to special cases. It wasn&#8217;t until 1920 and George P\u00f3lya&#8217;s work (&#8220;Sur le th\u00e9or\u00e8me central du calcul probabiliste, parmi ceux ayant rapport \u00e0 la notion de limite, et le probl\u00e8me des moments&#8221;) that it was given the name Th\u00e9or\u00e8me central limite.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-56f1404 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"56f1404\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-scientist-course\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Learn more about normal distribution<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f1d1217 e-flex e-con-boxed e-con\" data-id=\"f1d1217\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-7f31e6c elementor-widget elementor-widget-heading\" data-id=\"7f31e6c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The mathematical application of the central limit theorem<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-38a749c elementor-widget elementor-widget-text-editor\" data-id=\"38a749c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>To calculate the central limit theorem, consider :<\/p><ul><li>X1, X2, &#8230; as a sequence of independent, identically distributed real random variables.<\/li><li>\u03c3 \u2260 0: the expectation \u03bc and standard deviation \u03c3 of the D distribution are not infinite.<\/li><li>Sn = X1+X2+X3 &#8230; as the sum of all random variables.<\/li><li>n\u03bc: corresponds to the expectation of Sn<\/li><li>\u03c3\u221an: corresponds to the standard deviation of Sn<\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3d04aaf elementor-widget elementor-widget-text-editor\" data-id=\"3d04aaf\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>The central limit theorem states:<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-db2ff1b elementor-widget elementor-widget-image\" data-id=\"db2ff1b\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-image{text-align:center}.elementor-widget-image a{display:inline-block}.elementor-widget-image a img[src$=\".svg\"]{width:48px}.elementor-widget-image img{vertical-align:middle;display:inline-block}<\/style>\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" src=\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/09\/theoreme-central-limite.jpg\" title=\"\" alt=\"\" loading=\"lazy\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9417bae elementor-widget elementor-widget-text-editor\" data-id=\"9417bae\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Here, Zn&#8217;s expectation is 0 and its standard deviation is 1. In other words, the variable is centered and reduced.<\/p><p>Zntends towards the normal distribution as ntends towards infinity.<\/p><p>So for any real z :<\/p><p><strong>lim n\u2192\u221e P (Zn \u2264 z) = \u03a6(z)<\/strong><\/p><p>where : \u03a6 is the distribution function of \u039d(0,1)<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-005fa6d e-flex e-con-boxed e-con\" data-id=\"005fa6d\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-90b6a39 elementor-widget elementor-widget-heading\" data-id=\"90b6a39\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Example of application of the central limit theorem<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d3a8877 elementor-widget elementor-widget-text-editor\" data-id=\"d3a8877\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>To help you better understand the central limit theorem, here&#8217;s a concrete example.<\/p><p>Marine and Julien are playing dice. Every time they hit even, they win \u20ac1, and every time they hit odd, they lose \u20ac1. In this case, the even and odd numbers are identically distributed random variables, since Marine and Julien have as much chance of winning as losing. In the first few games, the balance may tip more towards even numbers (and therefore wins) or odd numbers (and therefore loses). But as the game progresses, their probability of winning or losing is close to 0.<\/p><p>This <strong>central limit theorem<\/strong> can be found in many everyday situations. For example, the size and weight of individuals within a population, the distribution of salaries within a company, etc. The average of these results always ends up resembling a bell curve as the population sample is enlarged,<\/p><p>Because of the wide variety of these applications, the <strong>central limit theorem<\/strong> is an essential <a href=\"https:\/\/datascientest.com\/en\/understanding-kurtosis-calculating-outlier-frequency-in-statistical-distributions\">statistical tool for data scientists<\/a>, enabling them to model Machine Learning models. In addition to data science tools, it is therefore essential to master mathematical concepts. It&#8217;s precisely for this reason that training is essential. Find out more about our programs.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b9f213f elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"b9f213f\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-scientist-course\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Master the central limit theorem<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f15566b e-flex e-con-boxed e-con\" data-id=\"f15566b\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5f712a8 elementor-widget elementor-widget-heading\" data-id=\"5f712a8\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Things to remember<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9793d00 elementor-widget elementor-widget-text-editor\" data-id=\"9793d00\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\">Identically distributed random variables follow a normal<a href=\"https:\/\/datascientest.com\/en\/jensen-shannon-divergence-everything-you-need-to-know-about-this-ml-model\"> distribution<\/a> as they are added together.<\/li><li style=\"font-weight: 400;\" aria-level=\"1\">With this statement, the central limit theorem underlines the importance of the normal distribution.<\/li><li style=\"font-weight: 400;\" aria-level=\"1\">Moreover, this theorem applies to a wide variety of hypotheses, from a game of chance to determining the average wage or the size of a population.<\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3d5001e elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"3d5001e\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Discover our training courses<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-962c4fe e-flex e-con-boxed e-con\" data-id=\"962c4fe\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-54e316f elementor-widget elementor-widget-html\" data-id=\"54e316f\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"FAQPage\",\n  \"mainEntity\": [{\n    \"@type\": \"Question\",\n    \"name\": \"The central limit theorem and convergence to the normal distribution\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The central limit theorem establishes the convergence of a sequence of variables defined on the same space to a centered reduced normal distribution. In other words, the more independent and identically distributed random variables we add together, the closer the probability distribution of the new variable will be to a normal distribution (also known as a Gaussian curve or a bell curve). And this applies to any type of probability distribution for a random event, such as the continuous uniform distribution, the triangular distribution, the exponential distribution, etc.\"\n    }\n  },{\n    \"@type\": \"Question\",\n    \"name\": \"Example of application of the central limit theorem\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"To help you better understand the central limit theorem, here's a concrete example. Marine and Julien are playing dice. Every time they hit even, they win \u20ac1, and every time they hit odd, they lose \u20ac1. In this case, the even and odd numbers are identically distributed random variables, since Marine and Julien have as much chance of winning as losing. In the first few games, the balance may tip more towards even numbers (and therefore wins) or odd numbers (and therefore loses). But as they play, their probability of winning or losing is close to 0.\"\n    }\n  }]\n}\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Reaffirming the importance of the normal distribution, the central limit theorem is one of the key concepts in statistics. So what exactly is it? How does it translate? And above all, how is it applied? That&#8217;s what we&#8217;ll be looking at in this article. The central limit theorem and convergence to the normal distribution The [&hellip;]<\/p>\n","protected":false},"author":67,"featured_media":181870,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_theme","format":"standard","meta":{"content-type":"","inline_featured_image":false},"categories":[2425],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Demystifying the Central Limit Theorem: A Guide to Understanding and Application<\/title>\n<meta name=\"description\" content=\"Learn about the Central Limit Theorem, a fundamental concept in statistics. 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