{"id":29822,"date":"2021-09-09T05:14:58","date_gmt":"2021-09-09T05:14:58","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29822"},"modified":"2025-12-01T19:07:00","modified_gmt":"2025-12-01T19:07:00","slug":"using-standard-normal-distribution-to-calculate-probabilities","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/using-standard-normal-distribution-to-calculate-probabilities\/","title":{"rendered":"Using the Standard Normal Distribution to Calculate Probabilities"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 4)\",\n  \"description\": \"Common Probability Distributions | Level I CFA\u00ae Exam Prep. 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Ideal for CFA candidates mastering quantitative methods.\",\n  \"uploadDate\": \"2021-11-09T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/TXO9ODcLWiU\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=TXO9ODcLWiU\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/TXO9ODcLWiU\",\n  \"duration\": \"PT1H47S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@graph\": [\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130-1024x262.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130-1024x262.jpg\",\n      \"caption\": \"Calculating probabilities (2)\",\n      \"width\": 1024,\n      \"height\": 262,\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\",\n        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table.png\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table.png\",\n      \"caption\": \"Z-table\",\n      \"width\": 759,\n      \"height\": 781,\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\",\n        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-positive.png\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/z-table-positive.png\",\n      \"caption\": \"Z-table positive\",\n      \"width\": 759,\n      \"height\": 793,\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\",\n        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129.jpg\",\n      \"caption\": \"Calculating probabilities\",\n      \"width\": 1463,\n      \"height\": 970,\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\",\n        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    }\n  ]\n}\n<\/script>\n\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 -2.5).\",\n    \"text\": \"Calculate P(Z \u2264 -2.5).\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B: 0.0062. Since P(Z \u2264 -2.5) = 1 \u2013 P(Z \u2264 2.5) = 1 \u2013 0.9938 = 0.0062. ([analystprep.com](https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/using-standard-normal-distribution-to-calculate-probabilities\/))\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"A: 0.9938 \u2014 incorrect, this corresponds to P(Z \u2264 2.5), not P(Z \u2264 -2.5).\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C: 0.06 \u2014 incorrect; not consistent with the standard normal table result for z = \u20132.5.\"\n      }\n    ],\n    \"dateCreated\": \"2025-12-01T00:00:00Z\"\n  },\n  \"dateCreated\": \"2025-12-01T00:00:00Z\",\n  \"author\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Using the standard normal distribution table, we can confirm that a normally distributed random variable \\(Z\\), with a 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 will have to look it up directly on the table, knowing that \\(P(Z \u2264 z) = \u03b8(z)\\) when \\(z\\) is positive.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-17016 size-full\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-129.jpg\" alt=\"Image showing the normal distribution table for 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>Positive z-values<\/strong><\/h3>\n<h4><strong>Example: Using the z-score Table<\/strong><\/h4>\n<p>The returns on ABC stock are normally distributed, where the mean is $0.60 with a standard deviation of $0.20. The probability that the return is less than $1 is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>If the return is $0.10, then \\(x = 0.1\\) (this is our observed value). Therefore,<\/p>\n<p>$$\\begin{align} z &amp;=\\frac{x-\\mu}{\\sigma}\\\\&amp;=\\frac{1-0.6}{0.2}\\\\ &amp;=2\\ \\text{(The return of \\$1 is two and a half standard deviations below the mean)}\\end{align}$$<\/p>\n<p>First, you\u2019d be required to calculate the z-value (2 in this case). \\(P(Z \u2264 1)\\) can be read 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 wp-image-17236 size-full\" 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>$$ \\begin{align} P(Z\u00a0 \\le -z) &amp; = 1 \u2013 P(Z \\le z) \\\\\u00a0 \\theta(\u2013z) &amp; = 1 \u2013 \\theta(z) \\\\ 0.0228 &amp; = 1 &#8211; 0.9772 \\end{align}$$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-17018 size-full\" 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 wp-image-17017 size-full\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-130.jpg\" alt=\"Normal Distribution tables 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 <strong>B<\/strong>.<\/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>","protected":false},"excerpt":{"rendered":"<p>Using the standard normal distribution table, we can confirm that a normally distributed random variable \\(Z\\), with a mean equal to 0 and variance equal to 1, is less than or equal to \\(z\\), i.e., \\(P(Z \u2264 z)\\). However, 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-29822","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Standard Normal Distribution &amp; Probability | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn what is a standard normal distribution and how to calculate probability from experts at AnalystPrep (Using the z-score Table)\" \/>\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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