{"id":348,"date":"2019-08-17T13:28:00","date_gmt":"2019-08-17T13:28:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=348"},"modified":"2025-12-17T12:52:40","modified_gmt":"2025-12-17T12:52:40","slug":"chebyshevs-inequality","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/chebyshevs-inequality\/","title":{"rendered":"Chebyshev\u2019s Inequality"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Statistical Concepts and Market Returns (2021 Level I CFA\u00ae Exam \u2013 Reading 7)\",\n  \"description\": \"This video lesson covers statistical concepts and market returns, explaining descriptive and inferential statistics, population vs. sample, and measurement scales. It details frequency distributions, measures of central tendency, variance, standard deviation, skewness, and kurtosis. The lesson emphasizes using statistics for data interpretation and informed decision-making in finance.\",\n  \"uploadDate\": \"2020-07-20T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/alD9eAT2lQU\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/alD9eAT2lQU\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/alD9eAT2lQU\",\n  \"duration\": \"PT31M21S\"\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\": \"Using Chebyshev\u2019s inequality, calculate the percentage of observations that would fall outside 3 standard deviations of the mean.\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B.\\n\\nWorking:\\nUsing Chebyshev\u2019s inequality:\\nP = 1 \u2013 1\/3\u00b2 = 89%.\\nBut we are interested in the percentage outside the range, i.e., 1 \u2013 0.89 = 0.11 or 11%.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"11%\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"89%\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"90%\"\n      }\n    ],\n    \"answerCount\": 3\n  }\n}\n<\/script>\n\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/alD9eAT2lQU?si=IvI8Bgk73NpzOrhp\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n\n<p>Chebyshev&#8217;s inequality is a probability theorem used to characterize the dispersion or spread of data away from the mean. It was developed by a Russian mathematician called Pafnuty Chebyshev. The theorem states that:<\/p>\n<p><!--more--><\/p>\n<p><em>For any set of observations, whether sample or population data and regardless of the shape of the distribution, the percentage of the observations that lie within k<\/em> <em>standard deviations of the mean is at least \\(1 \u2013 \\cfrac {1}{k^2}\\) for all \\(k &gt; 1\\).<\/em><\/p>\n<p>Simply put, this means that you can use the formula to determine the percentage of observations lying inside or outside a given number of standard deviations. Remember that the standard deviation tells us how far values are from the arithmetic mean. For example, two-thirds of the observations fall within one standard deviation on either side of the mean in a normal distribution. However, Chebyshev&#8217;s inequality goes slightly against the 68-95-99.7 rule commonly applied to the normal distribution.<\/p>\n<h2><strong>Chebyshev&#8217;s Inequality Formula<\/strong><\/h2>\n<p>$$ P = 1 \u2013 \\cfrac {1}{k^2} $$<\/p>\n<p><em>Where <\/em><\/p>\n<p><em>P is the percentage of observations<\/em><\/p>\n<p><em>K is the number of standard deviations<\/em><\/p>\n<h3>Example: Chebyshev&#8217;s Inequality<\/h3>\n<p>Suppose we wish to find the percentage of observations lying within two standard deviations of the mean:<\/p>\n<p>k = 2<\/p>\n<p>Hence,<\/p>\n<p>$$ \\begin{align*} P &amp; = 1 \u2013 \\cfrac {1}{2^2} \\\\ &amp; = 0.75 = 75\\% \\\\ \\end{align*} $$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Using Chebyshev\u2019s inequality, calculate the percentage of observations that would fall outside 3 standard deviations of the mean.<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>11%<\/li>\n<li>89%<\/li>\n<li>90%<\/li>\n<\/ol>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>Working:<\/p>\n<p>note that the question asks for the percentage that would fall <strong>outside <\/strong>3 standard deviations. Therefore:<\/p>\n<p>$$ P = 1 \u2013 \\cfrac {1}{3^2} = 89\\% $$<\/p>\n<p>But we are interested in 1 \u2013 p\u00a0, i.e., the outside observations.<\/p>\n<p>Therefore,<\/p>\n<p>$$ 1 \u2013 0.89 = 0.11 \\text{ or } 11\\% $$<\/p>\n<\/blockquote>\n<div class=\"notes_inv\"><hr \/>\n<p>\u00a0<\/p>\n<\/div>\n<p>\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Chebyshev&#8217;s inequality is a probability theorem used to characterize the dispersion or spread of data away from the mean. It was developed by a Russian mathematician called Pafnuty Chebyshev. The theorem states that:<\/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-348","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>Chebyshev&#039;s Inequality: Formula &amp; Applications | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Chebyshev&#039;s inequality estimates the minimum proportion of values within a certain number of standard deviations from the mean in any data distribution.\" \/>\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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