{"id":31291,"date":"2021-09-22T06:05:14","date_gmt":"2021-09-22T06:05:14","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=31291"},"modified":"2021-11-30T17:10:16","modified_gmt":"2021-11-30T17:10:16","slug":"central-limit-theorem-2","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/","title":{"rendered":"Central Limit Theorem"},"content":{"rendered":"\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/QDmc4Pa92bs\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The central limit theorem asserts that when we have simple random samples, each of size <em>n<\/em> from a population with mean \u03bc and variance \u03c3<sup>2<\/sup>, the sample mean X approximately has a normal distribution with mean \u03bc and variance \u03c3<sup>2<\/sup>\/n as <em>n <\/em>(sample size) becomes large.<\/p>\n<p><!--more--><\/p>\n<h2><strong>Breaking Down the Central Limit Theorem<\/strong><\/h2>\n<p>Suppose we have a sequence of independent and regularly distributed variables X<sub>1<\/sub>, X<sub>2<\/sub>, X<sub>3<\/sub> \u2026, X<sub>n<\/sub> with a finite mean \u03bc and non-zero variance \u03c3<sup>2<\/sup>, then the distribution of \\( \\frac {(X &#8211; \\mu)}{\\left(\\frac {\\sigma}{\\sqrt n}\\right)}\\) approaches the standard normal distribution as <em>n<\/em> approaches infinity, i.e., \\(n \\rightarrow \\infty \\).<\/p>\n<p>Remember that \\(X=\\left( \\frac { 1 }{ n } \\right) \\ast \\sum { { X }_{ i } } \\).<\/p>\n<p>The central limit theorem provides very useful normal approximations to some common distributions including the binomial and poisson distributions.<\/p>\n<p>Note that while X is approximately normally distributed with mean \u03bc and variance \u03c3<sup>2<\/sup>\/n, \u03a3X<sub>i<\/sub> is approximately normally distributed with mean n\u03bc and variance n\u03c3<sup>2<\/sup>. In fact, we can show that both the mean and the variance are exact and it is only the shape of the curve that is an approximation.<\/p>\n<h3><strong>What is a Large Enough <em>n<\/em>?<\/strong><\/h3>\n<p>The answer to this question might not be very simple. Nevertheless, the widely accepted value is <em>n<\/em> \u2265 30. The truth is that the value of <em>n<\/em> depends on the shape of the population involved, i.e., the distribution of X<sub>i<\/sub> and its skewness.<\/p>\n<p>In a non-normal but fairly symmetric distribution, <em>n<\/em> = 10 can be considered large enough. With a very skewed distribution, the value of n can be 50 or even more.<\/p>\n<h2><strong>Application of the Central Limit Theorem<\/strong><\/h2>\n<h3><strong>Normal Approximation of the Binomial Distribution<\/strong><\/h3>\n<p>Imagine that we have a set of independent and regularly distributed variables X<sub>i<\/sub>, i = 1\u2026, <em>n<\/em> such that:<\/p>\n<p>\\(\\sum X_i \\sim \\text{binomial} (n,\\theta)\\). Applying the CLT, for large n:<\/p>\n<p>\\(X \\sim N \\left( \\mu, \\frac {\\sigma^2}{n} \\right)\\) and \\(\\sum X_i \\sim N(n \\mu, n \\sigma^2)\\)<\/p>\n<p>Additionally, note that the binomial distribution is a sequence of Bernoulli variables such that the mean is \u03b8 and the variance, \u03b8(1- \u03b8).<\/p>\n<p>Therefore, applying the CLT:<\/p>\n<p>\\(\\sum X_i \\sim N(n\\theta, n \\theta (1 \u2013 \\theta))\\) which is the normal approximation to the binomial and <em>n<\/em> = 10 is considered large enough for CLT application.<\/p>\n<h3><strong>Normal Approximation of the Poisson Distribution<\/strong><\/h3>\n<p>Under poisson distribution, \\(\\mu = \\sigma^2=\\lambda\\). Therefore, applying the CLT:<\/p>\n<p>$$ \\sum X_i \\sim N(n\\lambda, n \\lambda) $$<\/p>\n<p>The central limit theorem is a very useful tool, especially in the construction of confidence intervals or testing of hypotheses. As long as <em>n<\/em> is \u201csufficiently large,\u201d just about any non-normal distribution can be approximated as normal.<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[25],"tags":[],"class_list":["post-31291","post","type-post","status-publish","format-standard","hentry","category-corporate-issuers","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Central Limit Theorem - AnalystPrep | CFA\u00ae Exam Study Notes<\/title>\n<meta name=\"description\" content=\"The central limit theorem asserts that when we have simple random samples each of size n from a population with a mean \u03bc and variance ...\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Central Limit Theorem - AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"og:description\" content=\"The central limit theorem asserts that when we have simple random samples each of size n from a population with a mean \u03bc and variance ...\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2021-09-22T06:05:14+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-30T17:10:16+00:00\" \/>\n<meta name=\"author\" content=\"Kosikos Tuitoek\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kosikos Tuitoek\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/\"},\"author\":{\"name\":\"Kosikos Tuitoek\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\"},\"headline\":\"Central Limit Theorem\",\"datePublished\":\"2021-09-22T06:05:14+00:00\",\"dateModified\":\"2021-11-30T17:10:16+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/\"},\"wordCount\":434,\"articleSection\":[\"Corporate Issuers\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/\",\"name\":\"Central Limit Theorem - AnalystPrep | CFA\u00ae Exam Study Notes\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\"},\"datePublished\":\"2021-09-22T06:05:14+00:00\",\"dateModified\":\"2021-11-30T17:10:16+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\"},\"description\":\"The central limit theorem asserts that when we have simple random samples each of size n from a population with a mean \u03bc and variance ...\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Central Limit Theorem\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\",\"name\":\"AnalystPrep | CFA\u00ae Exam Study Notes\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\",\"name\":\"Kosikos Tuitoek\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"caption\":\"Kosikos Tuitoek\"},\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/kosikos-tuitoek-enockanalystprep-com\/\"}]}<\/script>\n<meta property=\"og:video\" content=\"https:\/\/www.youtube.com\/embed\/QDmc4Pa92bs\" \/>\n<meta property=\"og:video:type\" content=\"text\/html\" \/>\n<meta property=\"og:video:duration\" content=\"2261\" \/>\n<meta property=\"og:video:width\" content=\"480\" \/>\n<meta property=\"og:video:height\" content=\"270\" \/>\n<meta property=\"ya:ovs:adult\" content=\"false\" \/>\n<meta property=\"ya:ovs:upload_date\" content=\"2021-09-22T06:05:14+00:00\" \/>\n<meta property=\"ya:ovs:allow_embed\" content=\"true\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Central Limit Theorem - AnalystPrep | CFA\u00ae Exam Study Notes","description":"The central limit theorem asserts that when we have simple random samples each of size n from a population with a mean \u03bc and variance ...","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:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/","og_locale":"en_US","og_type":"article","og_title":"Central Limit Theorem - AnalystPrep | CFA\u00ae Exam Study Notes","og_description":"The central limit theorem asserts that when we have simple random samples each of size n from a population with a mean \u03bc and variance ...","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2021-09-22T06:05:14+00:00","article_modified_time":"2021-11-30T17:10:16+00:00","author":"Kosikos Tuitoek","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Kosikos Tuitoek","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/"},"author":{"name":"Kosikos Tuitoek","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"headline":"Central Limit Theorem","datePublished":"2021-09-22T06:05:14+00:00","dateModified":"2021-11-30T17:10:16+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/"},"wordCount":434,"articleSection":["Corporate Issuers"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/","name":"Central Limit Theorem - AnalystPrep | CFA\u00ae Exam Study Notes","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"datePublished":"2021-09-22T06:05:14+00:00","dateModified":"2021-11-30T17:10:16+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"description":"The central limit theorem asserts that when we have simple random samples each of size n from a population with a mean \u03bc and variance ...","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/corporate-issuers\/central-limit-theorem-2\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Central Limit Theorem"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/","name":"AnalystPrep | CFA\u00ae Exam Study Notes","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20","name":"Kosikos Tuitoek","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","caption":"Kosikos Tuitoek"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/kosikos-tuitoek-enockanalystprep-com\/"}]},"og_video":"https:\/\/www.youtube.com\/embed\/QDmc4Pa92bs","og_video_type":"text\/html","og_video_duration":"2261","og_video_width":"480","og_video_height":"270","ya_ovs_adult":"false","ya_ovs_upload_date":"2021-09-22T06:05:14+00:00","ya_ovs_allow_embed":"true"},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/31291","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/users\/15"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=31291"}],"version-history":[{"count":30,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/31291\/revisions"}],"predecessor-version":[{"id":38536,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/31291\/revisions\/38536"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=31291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=31291"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=31291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}