{"id":1573,"date":"2019-10-01T13:29:00","date_gmt":"2019-10-01T13:29:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1573"},"modified":"2025-05-15T14:09:18","modified_gmt":"2025-05-15T14:09:18","slug":"central-limit-theorem","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/central-limit-theorem\/","title":{"rendered":"The Central Limit Theorem"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Sampling and Estimation (2021 Level I CFA\u00ae Exam \u2013 Reading 10)\",\n  \"description\": \"This video lesson covers CFA Level 1 Quantitative Methods, focusing on sampling and estimation. It explains simple and stratified sampling, sampling errors, the central limit theorem, standard error, confidence intervals, and issues like data mining and survivorship bias. The lesson concludes by introducing hypothesis testing as the next topic.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/mgY_3CHHYBw\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=mgY_3CHHYBw\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/mgY_3CHHYBw\",\n  \"duration\": \"PT32M00S\"\n}\n<\/script>\n\n\n\n<p>[vsw id=&#8221;mgY_3CHHYBw&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;]<\/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 a 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 identically distributed variables X<sub>1<\/sub>, X<sub>2<\/sub>, X<sub>3<\/sub> \u2026, X<sub>n<\/sub> with mean 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 approximate.<\/p>\n<h3><strong>What is a Large Enough <em>n<\/em>?<\/strong><\/h3>\n<p>Although the widely accepted value is n \u2265 30, this answer might be too simple. The truth is that the value of n 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, n = 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>Suppose we have a set of independent and identically distributed variables X<sub>i<\/sub>, i = 1\u2026, n 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 n = 10 is considered large enough for CLT application.<\/p>\n<h3><strong>Normal Approximation of the Poisson Distribution<\/strong><\/h3>\n<p>Under Poisson, \\(\\mu = \\sigma^2=\\lambda\\)<\/p>\n<p>Therefore, applying 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 n is \u201csufficiently large,\u201d just about any non-normal distribution can be approximated as normal.<\/p>\n<p><em>Reading 10 LOS 10e:<\/em><\/p>\n<p><em>Explain the central limit theorem and its importance.<\/em><\/p>\n<div class=\"notes_inv\">\n<hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>[vsw id=&#8221;mgY_3CHHYBw&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;] 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 \u03c32, the sample mean X approximately has a normal&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-1573","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","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 with Examples | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Understand the central limit theorem and how it helps approximate non-normal distributions as normal for statistical analysis.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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