{"id":29798,"date":"2021-09-09T03:38:56","date_gmt":"2021-09-09T03:38:56","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29798"},"modified":"2025-03-22T21:23:27","modified_gmt":"2025-03-22T21:23:27","slug":"normal-distribution","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/normal-distribution\/","title":{"rendered":"Normal Distribution"},"content":{"rendered":"<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A random variable is said to have a normal distribution (Gaussian curve) if its values make a smooth curve that assumes a \u201cbell shape.\u201d A normal variable has a mean \\(\u03bc\\), pronounced as \u201cmu,\u201d and a standard deviation \\(\u03c3\\), pronounced as \u201csigma.\u201d All normal distributions have a distinguishable bell shape regardless of the mean, variance, and standard deviation.<\/p>\n<p><!--more--><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17012\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123.jpg\" alt=\"shape-of-the-normal-distribution\" width=\"1463\" height=\"1065\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123.jpg 1463w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123-300x218.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123-768x559.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123-1024x745.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123-400x291.jpg 400w\" sizes=\"auto, (max-width: 1463px) 100vw, 1463px\" \/><br \/>\nA normal distribution has certain properties that make it a useful tool in the world of finance.<\/p>\n<ol>\n<li>Its shorthand notation is \\(X \\sim N(\\mu, \\sigma^2)\\). This is read as \u201cthe random variable X has a normal distribution with mean \\(\u03bc\\) and variance \\(\u03c3^2\\).\u201d<\/li>\n<li>It has a symmetric shape, meaning it can be cut into two halves that are <strong>mirror images<\/strong> of each other; as such, skewness = 0.<\/li>\n<li>Kurtosis = 3. Remember that kurtosis is a measure of flatness and excess kurtosis is measured relative to 3, the \u201cnormal kurtosis.\u201d<\/li>\n<li>The mean, mode, and median are all equal and lie directly in the middle of the distribution.<\/li>\n<li>If \\(X \\sim N(\\mu_x, \\sigma^2_x)\\) and \\(Y\\sim N(\\mu_y, \\sigma^2_y)\\), then the two variables combined also have a normal distribution.<\/li>\n<li>The standard deviation measures the distance from the mean to the point of inflection, which is the point where the curve changes from an \u201cupside-down-bowl\u201d shape to a \u201cright-side-up-bowl\u201d shape.<\/li>\n<li>Probabilities follow the empirical rule. About 68% of the total values lie within one standard deviation of the mean, 95% of the values lie within two standard deviations of the mean, and 99.7% lie within three standard deviations of the mean.<\/li>\n<\/ol>\n<h2><strong>Area Under the Normal Distribution Curve <\/strong><\/h2>\n<p>To determine the probability that a random variable \\(X\\) lies between two points \\(a\\) and \\(b\\):<\/p>\n<p>$$ P\\left( a &lt; X &lt; b \\right) =\\int _{ a }^{ b }{ f\\left( x \\right)dx } $$<\/p>\n<p>The normal distribution is very important in statistical analysis, especially because of the central limit theorem. The theorem asserts that any distribution becomes normally distributed when the number of variables is <strong>sufficiently large<\/strong>. For instance, the binomial distribution tends to \u201cchange\u201d into the normal distribution with mean \\(n\u03b8\\) and variance \\(n\u03b8 (1 &#8211; \u03b8)\\).<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Which of the following is\u00a0<em>least likely\u00a0<\/em>a property of normal distribution?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>The excess kurtosis is equal to 3.<\/li>\n<li>The mean, mode, and median are all equal.<\/li>\n<li>About 68% of the total values lie within one standard deviation of the mean.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p><span data-preserver-spaces=\"true\">The normal distribution has an excess kurtosis of 0. Excess kurtosis is a value above kurtosis of 3. Since the kurtosis of the normal distribution is 3, the excess kurtosis of 0.<\/span><\/p>\n<p><strong><span data-preserver-spaces=\"true\">B and C are incorrect<\/span><\/strong>.\u00a0<span data-preserver-spaces=\"true\">They are the true properties of the normal distribution.<\/span><\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>A random variable is said to have a normal distribution (Gaussian curve) if its values make a smooth curve that assumes a \u201cbell shape.\u201d A normal variable has a mean \\(\u03bc\\), pronounced as \u201cmu,\u201d and a standard deviation \\(\u03c3\\), pronounced&#8230;<\/p>\n","protected":false},"author":15,"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-29798","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>Normal Distribution Explained | AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn the characteristics of a normal distribution curve, its bell shape, and how it applies to CFA exam probability and statistics concepts.\" \/>\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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