{"id":1236,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1236"},"modified":"2026-04-02T10:53:27","modified_gmt":"2026-04-02T10:53:27","slug":"standard-normal-distribution-calculations","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/standard-normal-distribution-calculations\/","title":{"rendered":"The Standard Normal Distribution: Calculation and Interpretation of Probability"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2021 Level I CFA\u00ae Exam \u2013 Reading 9)\",\n  \"description\": \"This video lesson covers common probability distributions in quantitative methods. It explains key concepts like discrete vs. continuous variables, probability and cumulative distribution functions, binomial and normal distributions, confidence intervals, Monte Carlo simulations, and more\u2014using practical financial examples to help analysts apply statistical reasoning in investment decisions.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/i.ytimg.com\/vi\/KbEfz3KiJDo\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=KbEfz3KiJDo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo\",\n  \"duration\": \"PT45M31S\"\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129.jpg\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129.jpg\",\n  \"width\": 1463,\n  \"height\": 970,\n  \"caption\": \"Standard normal distribution calculation example for CFA Level I quantitative methods\",\n  \"copyrightNotice\": \"\u00a9 AnalystPrep\",\n  \"creditText\": \"AnalystPrep\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-positive.png\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-positive.png\",\n  \"width\": 759,\n  \"height\": 793,\n  \"caption\": \"Standard normal distribution Z-table for positive z-values used in CFA Level I quantitative methods\",\n  \"copyrightNotice\": \"\u00a9 AnalystPrep\",\n  \"creditText\": \"AnalystPrep\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Calculate P(Z \u2264 \u22122.5)\",\n    \"text\": \"Calculate P(Z \u2264 \u22122.5).\\n\\nA. 0.9938\\nB. 0.0062\\nC. 0.06\",\n    \"answerCount\": 3,\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"A. 0.9938\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"B. 0.0062\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C. 0.06\"\n      }\n    ],\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"B. 0.0062\",\n      \"commentary\": \"Using the symmetry of the standard normal distribution, P(Z \u2264 \u22122.5) = 1 \u2212 P(Z \u2264 2.5). From standard normal tables, P(Z \u2264 2.5) = 0.9938. Therefore, P(Z \u2264 \u22122.5) = 1 \u2212 0.9938 = 0.0062.\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/standard-normal-distribution-calculations\/\"\n    },\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  referrerpolicy=\"strict-origin-when-cross-origin\"\n  allowfullscreen>\n<\/iframe>\n\n\n<p>The standard normal distribution refers to a normal distribution that has been standardized such that it has a mean of 0 and a standard deviation of 1. The shorthand notation used is:<\/p>\n<p><!--more--><\/p>\n<p>$$ N \\sim (0, 1) $$<\/p>\n<p>In the context of statistics and mathematics, standardization is the process of converting an observed value for a random variable into a z-value where:<\/p>\n<p>$$ \\begin{align*} Z &amp; = \\cfrac {(\\text{observed value} \u2013 \\text{population mean})} {\\text{standard deviation}} \\\\ &amp; =\\cfrac {(x \u2013 \\mu)}{\\sigma} \\\\ \\end{align*} $$<\/p>\n<p>The z-value, also referred to as the z-score in some books, represents the number of standard deviations a given observed value is from the population mean.<\/p>\n<div style=\"margin: 20px 0;\"><a style=\"display: block; width: 100%; text-align: center; padding: 10px; border: 2px solid #2f5bea; border-radius: 40px; font-size: 16px; color: #2f5bea; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Practice z-score questions with our free trial. <\/a><\/div>\n<h3><strong>Example: Standard Normal Distribution<\/strong><\/h3>\n<p>The returns on ABC stock are normally distributed where the mean is $0.60 with a standard deviation of $0.20. Calculate the z-scores for a return of $0.10.<\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>If the return is $0.10, then x = 0.1 (this is our observed value)<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*} z &amp; =\\cfrac {(x \u2013 \\mu)}{\\sigma} \\\\ &amp; =\\cfrac {(0.1 \u2013 0.6)}{0.2} \\\\ &amp; = -2.5 \\quad (\\text{The return of }$0.1 \\text{ is two and a half standard deviations below the mean}.) \\end{align*} $$<\/p>\n<p>We could get the z-score for any other observed value following a similar approach. For instance, the z-score for a return of $1 will be:<\/p>\n<p>$$ \\begin{align*} Z &amp; =\\cfrac {(1 \u2013 0.6)}{0.2} \\\\ &amp; = 2 \\quad (\\text{The return of } $1 \\text{ is two standard deviations above the mean}) \\\\ \\end{align*} $$<\/p>\n<h2><strong>Calculating Probabilities Using z-values under the Standard Normal Distribution<\/strong><\/h2>\n<p>Using the standard normal distribution table, we can confirm that a normally distributed <a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/introduction-probability-defining-basic-terms\/\">random variable<\/a> Z, with mean equal to 0 and variance equal to 1, is less than or equal to z, i.e., P(Z \u2264 z).<\/p>\n<p>However, the table does this only when we have positive values of z. Simply put, if an examiner asks you to find the probability behind a given positive z-value, you\u2019d have to look it up directly on the table.<\/p>\n<p>P(Z \u2264 z) = \u03b8(z) when z is positive<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17016\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129.jpg\" alt=\"calculating-probabilities\" width=\"1463\" height=\"970\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129.jpg 1463w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129-300x199.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129-768x509.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129-1024x679.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129-400x265.jpg 400w\" sizes=\"auto, (max-width: 1463px) 100vw, 1463px\" \/><\/p>\n<h3><strong>Example: Using the z-score Table<\/strong><\/h3>\n<p>Using the data from our first example, calculate the probability that the return is less than $1.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First, you would be required to calculate the z-value (2 in this case).<\/p>\n<p>P(Z \u2264 2) can be read off directly from the table.<\/p>\n<p>You just move down and locate the z-value that lies to the right of \u201c2,\u201d i.e., 0.9772.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17236\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-positive.png\" alt=\"z-table-positive\" width=\"759\" height=\"793\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-positive.png 759w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-positive-287x300.png 287w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-positive-400x418.png 400w\" sizes=\"auto, (max-width: 759px) 100vw, 759px\" \/><\/p>\n<h3><strong>Negative z-values<\/strong><\/h3>\n<p>If we have a negative z-value and do not have access to the negative values from the table (as shown below), we can still calculate the corresponding probability by noting that:<\/p>\n<p>$$ P(Z\u00a0 \\le -z) = 1 \u2013 P(Z \\le z) \\text{ or} $$<\/p>\n<p>$$ \\theta(\u2013z) = 1 \u2013 \\theta(z) $$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17018\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table.png\" alt=\"z-table\" width=\"759\" height=\"781\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table.png 759w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-292x300.png 292w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-400x412.png 400w\" sizes=\"auto, (max-width: 759px) 100vw, 759px\" \/><\/p>\n<p>This relationship is true when we consider the following facts:<\/p>\n<ol>\n<li>The total area (probability) under the standard normal distribution is 1.<\/li>\n<li>The standardized normal distribution is symmetrical about the mean.<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17017\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130.jpg\" alt=\"calculating-probabilities-2\" width=\"3546\" height=\"909\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130.jpg 3546w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130-300x77.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130-768x197.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130-1024x262.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130-400x103.jpg 400w\" sizes=\"auto, (max-width: 3546px) 100vw, 3546px\" \/><\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Calculate P(Z\u00a0 \u2264 -2.5)<\/p>\n<p>A. 0.9938<\/p>\n<p>B. 0.0062<\/p>\n<p>C. 0.06<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>$$ \\begin{align*} P(Z\u00a0 \\le -2.5) &amp; = 1 &#8211; P(Z\u00a0 \\le 2.5) \\\\ &amp; = 1 \u2013 0.9938 \\\\ &amp; = 0.0062 \\\\ \\end{align*} $$<\/p>\n<\/blockquote>\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-block; padding: 10px 26px; background: #3f78d7; color: #fff; border-radius: 40px; font-size: 16px; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a>\n<p style=\"margin-top: 10px; max-width: 600px; margin-left: auto; margin-right: auto; font-size: 14px;\">Solve CFA-style questions on z-scores, standard normal distribution, and probability calculations.<\/p>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>The standard normal distribution refers to a normal distribution that has been standardized such that it has a mean of 0 and a standard deviation of 1. The shorthand notation used is:<\/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-1236","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>Standard Normal Distribution | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn how to standardize a normal distribution with a mean of 0 and a standard deviation of 1 for probability calculations.\" \/>\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\/quantitative-methods\/standard-normal-distribution-calculations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Standard Normal Distribution | CFA Level 1 - 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