{"id":394,"date":"2019-08-17T13:28:00","date_gmt":"2019-08-17T13:28:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=394"},"modified":"2026-03-26T18:17:24","modified_gmt":"2026-03-26T18:17:24","slug":"kurtosis-and-skewness-types-of-distributions","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/kurtosis-and-skewness-types-of-distributions\/","title":{"rendered":"Kurtosis and Skewness"},"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 CFA Level 1 video lesson covers key statistical concepts, including descriptive and inferential statistics, measures of central tendency and dispersion, skewness, kurtosis, frequency distributions, and coefficient of variation. Learn how these concepts are applied to data analysis and investment decision-making.\",\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  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/wp-content\/uploads\/logo.png\",\n      \"width\": 250,\n      \"height\": 37\n    }\n  },\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/05085139\/page-64.png\",\n  \"caption\": \"Image showing Kurtosis graph\",\n  \"width\": 1632,\n  \"height\": 972,\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  }\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\": \"Using the data from the example above (12, 13, 54, 56, 25), determine the type of kurtosis present.\",\n    \"text\": \"Using the data from the example above (12, 13, 54, 56, 25), determine the type of kurtosis present.\\n\\nA. Mesokurtic distribution\\nB. Platykurtic distribution\\nC. Leptokurtic distribution\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B (Platykurtic distribution). Using the sample, the computed sample kurtosis is approximately 0.7861, so the excess kurtosis is 0.7861 \u2212 3 = \u22122.2139. Because the excess kurtosis is negative, the distribution is platykurtic.\",\n      \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/kurtosis-and-skewness-types-of-distributions\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\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\/alD9eAT2lQU\" \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\n<p>Kurtosis refers to measuring the degree to which a given distribution is more or less \u2018peaked\u2019 relative to the normal distribution. The concept of kurtosis is very useful in decision-making.&nbsp; In this regard, we have 3 categories of distributions:<\/p>\n\n\n\n<!--more-->\n\n\n\n<ol class=\"wp-block-list\">\n<li>Leptokurtic<\/li>\n\n\n\n<li>Mesokurtic<\/li>\n\n\n\n<li>Platykurtic<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"1632\" height=\"972\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-64.png\" alt=\"Image showing Kurtosis graph\" class=\"wp-image-16952\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-64.png 1632w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-64-300x179.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-64-768x457.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-64-1024x610.png 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-64-400x238.png 400w\" sizes=\"auto, (max-width: 1632px) 100vw, 1632px\" \/><\/figure>\n<\/div>\n\n\n<div style=\"text-align:center; background:#f3f5f9; padding:20px 16px; margin:24px 0;\">\n  <div style=\"max-width:760px; margin:0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display:inline-flex; align-items:center; justify-content:center; width:100%; padding:12px 32px; border:2px solid #2f6fdd; border-radius:999px; color:#2f6fdd; text-decoration:none; font-size:15.5px; font-weight:500; line-height:1;\">\n      Practice skewness and kurtosis concepts with a Free Trial.\n    <\/a>\n  <\/div>\n<\/div>\n\n\n<\/p>\n<h2><strong>Leptokurtic<\/strong><\/h2>\n<p>A leptokurtic distribution is more peaked than the normal distribution. The higher peak results from the clustering of data points along the X-axis. The tails are also fatter than those of a normal distribution. The coefficient of kurtosis is usually found to be <strong>more than 3<\/strong>.<\/p>\n<p>The term \u201claptop\u201d means thin or skinny. When analyzing historical returns, a leptokurtic distribution means that small changes are less frequent since historical values are clustered around the mean. However, there are also large fluctuations represented by the fat tails.<\/p>\n<h2><strong>Platykurtic<\/strong><\/h2>\n<p>A platykurtic distribution has extremely dispersed points along the X-axis, resulting in a lower peak when compared to the normal distribution. \u201cPlaty\u201d means broad. Hence, the prefix fits the distribution\u2019s shape, which is wide and flat. The points are less clustered around the mean compared to the leptokurtic distribution. The coefficient of kurtosis is usually <strong>less than 3. <\/strong><\/p>\n<p>Returns that follow this type of distribution have fewer major fluctuations compared to leptokurtic returns. However, you should note that fluctuations represent the riskiness of an asset. More fluctuations represent more risk and vice versa. Therefore, platykurtic returns are less risky than leptokurtic returns.<\/p>\n<h2><strong>Mesokurtic<\/strong><\/h2>\n<p>Lastly, mesokurtic distributions have a curve that\u2019s <strong>similar<\/strong> to that of the normal distribution. In other words, the distribution is largely normal.<\/p>\n<h2><strong>Measures of Sample Skewness and Kurtosis<\/strong><\/h2>\n<p>Exam tip: The learning outcome statement reads, <em>&#8220;<\/em>e<em>xplain measures of sample skewness and kurtosis.&#8221;\u00a0<\/em>However, the calculations will have you better understand those concepts.<\/p>\n<h3><strong>Sample skewness<\/strong><\/h3>\n<p><em><span class=\"dropcap\">\\({ S }_{ k }=\\frac { 1 }{ n } \\frac { \\sum _{ i=1 }^{ n }{ { \\left( { X }_{ i }-\\bar { X } \\right) }^{ 3 } } }{ { S }^{ 3 } } \\)<\/span><\/em><\/p>\n<p>Where: \\(\\bar { X }\\) is the sample mean;<\/p>\n<p>S is the sample standard deviation; and<\/p>\n<p>n is the number of observations.<\/p>\n<p><em>Note: the numerator is raised to the third power, and as such, it can either be positive or negative. However, the denominator (s<sup>3<\/sup>) is always positive.<\/em><\/p>\n<p>Interpretation: A positive value indicates positive skewness. A \u2018zero\u2019 value indicates the data is not skewed. Lastly, a negative value indicates negative skewness or rather a negatively skewed distribution.<\/p>\n<h3><strong>Sample Kurtosis<\/strong><\/h3>\n<p>Sample kurtosis is always measured relative to the kurtosis of a normal distribution, which is 3. Therefore, we are always interested in the \u201cexcess\u201c kurtosis, i.e.,<\/p>\n<p><em>Excess kurtosis = sample kurtosis \u2013 3, where:<\/em><\/p>\n<p><em>\\({ S }_{ kr }=\\frac { 1 }{ n } \\frac { \\sum _{ i=1 }^{ n }{ { \\left( { X }_{ i }-\\bar { X } \\right) }^{ 4 } } }{ { S }^{ 4 } } \\)<\/em><\/p>\n<p>Interpretation: A positive excess kurtosis indicates a leptokurtic distribution. A zero value indicates a mesokurtic distribution. Lastly, a negative excess kurtosis represents a platykurtic distribution.<\/p>\n<h3><strong>Example<\/strong><\/h3>\n<p>Suppose we have the following observations:<\/p>\n<p>{12\u00a0\u00a0 13\u00a0\u00a0 54\u00a0\u00a0 56\u00a0\u00a0 25}<\/p>\n<p>Determine the skewness of the data.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First, we must determine the sample mean and the sample standard deviation:<\/p>\n<p>$$ X =\\cfrac {(12 + 13 +\u2026+25)}{5} =\\cfrac {160}{5} = 32 $$<\/p>\n<p>$$ S^2= \\cfrac{{(12 \u2013 32)^2 + \u2026+ (25 \u2013 32)^2}}{4} = 467.5 $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ S = 467.5^{\\frac{1}{2}} = 21.62 $$<\/p>\n<p>Now we can work out the skewness:<\/p>\n<p><em><span class=\"dropcap\">\\({ S }_{ k }=\\frac { 1 }{ n } \\frac { \\sum _{ i=1 }^{ n }{ { \\left( { X }_{ i }-\\bar { X } \\right) }^{ 3 } } }{ { S }^{ 3 } } =\\frac { 1 }{ 5 } \\frac { { -20 }^{ 3 }+{ (-19 }^{ 3 })+{ 22 }^{ 3 }+{ 24 }^{ 3 }+{ (-7 }^{ 3 }) }{ { 21.62 }^{ 3 } } =0.1835 \\)<\/span><\/em><\/p>\n<p>Skewness is positive. Hence, the data has a positively skewed distribution.<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Using the data from the example above (12\u00a0 13\u00a0 54\u00a0 56\u00a0 25), determine the type of kurtosis present.<\/p>\n<p>A. Mesokurtic distribution<\/p>\n<p>B. Platykurtic distribution<\/p>\n<p>C. Leptokurtic distribution<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>$$ X =\\cfrac {(12 + 13 +\u2026+25)}{5} =\\cfrac {160}{5} = 32 $$<\/p>\n<p>$$ S^2=\\cfrac {{(12 \u2013 32)^2 + \u2026+ (25 \u2013 32)^2}}{4} = 467.5 $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ S = 467.5^{\\frac{1}{2}} = 21.62 $$<\/p>\n<p>\\({ S }_{ kr }=\\frac { 1 }{ n } \\frac { \\sum _{ i=1 }^{ n }{ { \\left( { X }_{ i }-\\bar { X } \\right) }^{ 4 } } }{ { S }^{ 4 } } =\\frac { 1 }{ 5 } \\frac { { -20 }^{ 4 }+{ (-19 }^{ 4 })+{ 22 }^{ 4 }+{ 24 }^{ 4 }+{ (-7 }^{ 4 }) }{ { 21.62 }^{ 4 } } =0.7861\\)<\/p>\n<p>Next, we subtract 3 from the sample kurtosis and get the excess kurtosis.<\/p>\n<p>Thus,\\(\\text {excess kurtosis} = 0.7861 \u2013 3 = -2.2139\\)<\/p>\n<p>Since the excess kurtosis is negative, we have a platykurtic distribution.<\/p>\n<\/blockquote>\n<div class=\"notes_inv\">\n<hr \/><\/div>\n\n\n<div style=\"text-align:center; background:#f3f5f9; padding:44px 20px 24px; margin:40px 0;\">\n  \n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-flex; align-items:center; justify-content:center; background:#4274d8; color:#ffffff; text-decoration:none; padding:14px 36px; border-radius:999px; font-size:16px; font-weight:700; line-height:1;\">\n    Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:640px; margin:14px auto 0; font-size:15.5px; line-height:1.5; color:#1f2937;\">\n    Strengthen skewness and kurtosis analysis with\n    exam-focused CFA Level I practice.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Kurtosis refers to measuring the degree to which a given distribution is more or less \u2018peaked\u2019 relative to the normal distribution. The concept of kurtosis is very useful in decision-making.&nbsp; In this regard, we have 3 categories of distributions:<\/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-394","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>Kurtosis and Skewness in Distributions | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Skewness measures distribution symmetry, indicating left or right lean. 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