{"id":29817,"date":"2021-09-09T04:51:16","date_gmt":"2021-09-09T04:51:16","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29817"},"modified":"2025-12-07T18:22:54","modified_gmt":"2025-12-07T18:22:54","slug":"the-standard-normal-distribution","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/","title":{"rendered":"The Standard Normal Distribution"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"The returns on ABC stock are distributed normally. The mean is $0.60 with a standard deviation of $0.20. The z-score for a return of $0.10 is closest to:\",\n    \"text\": \"The returns on ABC stock are distributed normally. The mean is $0.60 with a standard deviation of $0.20. The z-scores for a return of $0.10 is closest to:\\n\\nA. -0.5\\nB. -2.5\\nC. 2.5\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The z-score is computed as (0.10 \u2212 0.60) \/ 0.20 = \u22122.5, meaning the return of $0.10 is two and a half standard deviations below the mean.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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>$$ N \\sim (0, 1) $$<\/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<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>The returns on ABC stock are distributed normally. The mean is $0.60 with a standard deviation of $0.20. The z-scores for a return of $0.10 is&nbsp;<em>closest to:<\/em><\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>-0.5.<\/li>\n<li>-2.5.<\/li>\n<li>2.5.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is&nbsp;<strong>B.<\/strong><\/p>\n<p>If the return is $0.10, then x = 0.1 (this is our observed value). 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><\/blockquote>","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: $$ N \\sim (0, 1) $$ In the&#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-29817","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>The Standard Normal Distribution - AnalystPrep | CFA\u00ae Exam Study Notes<\/title>\n<meta name=\"description\" content=\"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.\" \/>\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\/the-standard-normal-distribution\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Standard Normal Distribution - AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"og:description\" content=\"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.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2021-09-09T04:51:16+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-07T18:22:54+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=\"1 minute\" \/>\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\/quantitative-methods\/the-standard-normal-distribution\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/\"},\"author\":{\"name\":\"Kosikos Tuitoek\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\"},\"headline\":\"The Standard Normal Distribution\",\"datePublished\":\"2021-09-09T04:51:16+00:00\",\"dateModified\":\"2025-12-07T18:22:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/\"},\"wordCount\":185,\"articleSection\":[\"Quantitative Methods\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/\",\"name\":\"The Standard Normal Distribution - AnalystPrep | CFA\u00ae Exam Study Notes\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\"},\"datePublished\":\"2021-09-09T04:51:16+00:00\",\"dateModified\":\"2025-12-07T18:22:54+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\"},\"description\":\"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.\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"The Standard Normal Distribution\"}]},{\"@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\/TXO9ODcLWiU\" \/>\n<meta property=\"og:video:type\" content=\"text\/html\" \/>\n<meta property=\"og:video:duration\" content=\"3648\" \/>\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=\"2021-09-09T04:51:16+00:00\" \/>\n<meta property=\"ya:ovs:allow_embed\" content=\"true\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"The Standard Normal Distribution - AnalystPrep | CFA\u00ae Exam Study Notes","description":"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.","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\/quantitative-methods\/the-standard-normal-distribution\/","og_locale":"en_US","og_type":"article","og_title":"The Standard Normal Distribution - AnalystPrep | CFA\u00ae Exam Study Notes","og_description":"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.","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2021-09-09T04:51:16+00:00","article_modified_time":"2025-12-07T18:22:54+00:00","author":"Kosikos Tuitoek","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Kosikos Tuitoek","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/"},"author":{"name":"Kosikos Tuitoek","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"headline":"The Standard Normal Distribution","datePublished":"2021-09-09T04:51:16+00:00","dateModified":"2025-12-07T18:22:54+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/"},"wordCount":185,"articleSection":["Quantitative Methods"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/","name":"The Standard Normal Distribution - AnalystPrep | CFA\u00ae Exam Study Notes","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"datePublished":"2021-09-09T04:51:16+00:00","dateModified":"2025-12-07T18:22:54+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"description":"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.","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/the-standard-normal-distribution\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"The Standard Normal Distribution"}]},{"@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\/TXO9ODcLWiU","og_video_type":"text\/html","og_video_duration":"3648","og_video_width":"480","og_video_height":"270","ya_ovs_adult":"false","ya_ovs_upload_date":"2021-09-09T04:51:16+00:00","ya_ovs_allow_embed":"true"},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/29817","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=29817"}],"version-history":[{"count":18,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/29817\/revisions"}],"predecessor-version":[{"id":56613,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/29817\/revisions\/56613"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=29817"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=29817"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=29817"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}