{"id":25233,"date":"2021-08-12T13:47:52","date_gmt":"2021-08-12T13:47:52","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=25233"},"modified":"2026-04-03T15:26:53","modified_gmt":"2026-04-03T15:26:53","slug":"kurtosis","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/kurtosis\/","title":{"rendered":"Kurtosis"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Organizing, Visualizing, and Describing Data \u2013 Part II: Measures of Central Tendency (Level I CFA\u00ae Exam)\",\n  \"description\": \"This lesson covers Measures of Central Tendency as part of the Level I CFA\u00ae Quantitative Methods curriculum. It explains how to calculate and interpret measures of central tendency, evaluate alternative definitions of the mean for investment problems, and calculate quantiles and interpret related visualizations. The video also covers measures of dispersion, target downside deviation, skewness, kurtosis, and correlation between two variables, using clear, exam-focused explanations throughout.\",\n  \"uploadDate\": \"2021-10-08T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/M0gKgPztSoM\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/M0gKgPztSoM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/M0gKgPztSoM\",\n  \"duration\": \"PT41M38S\"\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\": \"The skewness of the normal distribution is most likely:\",\n    \"text\": \"The skewness of the normal distribution is most likely:\\n\\nA. zero.\\nB. positive.\\nC. negative.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A.\\n\\nSince the normal curve is symmetric about its mean, its skewness is zero.\"\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\": \"A frequency distribution in which there are too few scores at the extremes of the distribution is most likely called:\",\n    \"text\": \"A frequency distribution in which there are too few scores at the extremes of the distribution is most likely called:\\n\\nA. platykurtic.\\nB. leptokurtic.\\nC. mesokurtic.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A.\\n\\nA platykurtic distribution has \u201cthin\u201d tails and is flatter compared to a normal distribution. It implies that there are fewer scores at the extremes of the distribution, which aligns with the question\u2019s description.\"\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\": \"When most of the data are concentrated on the left of the distribution, it is most likely called:\",\n    \"text\": \"When most of the data are concentrated on the left of the distribution, it is most likely called:\\n\\nA. symmetric distribution.\\nB. positively skewed distribution.\\nC. negatively skewed distribution.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B.\\n\\nA distribution is said to be skewed to the right, or positively skewed, when most of the data are concentrated on the left of the distribution. A distribution is said to be skewed to the left, or negatively skewed, if most of the data are concentrated on the right of the distribution. The left tail clearly extends farther from the distribution\u2019s center than the right tail.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/04\/Img_1.jpg\",\n  \"caption\": \"Kurtosis\",\n  \"width\": 1590,\n  \"height\": 1108,\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\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-61a.jpg\",\n  \"caption\": \"Positively Skewed Distribution\",\n  \"width\": 1373,\n  \"height\": 860,\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\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/M0gKgPztSoM\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Kurtosis refers to the measurement of 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.\u00a0 In this regard, we have 3 categories of distributions:<\/p>\n<ul>\n<li>Leptokurtic;<\/li>\n<li>Mesokurtic; or<\/li>\n<li>Platykurtic.<\/li>\n<\/ul>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-16952\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/04\/Img_1.jpg\" alt=\"kurtosis, platykurtic, leptokurtic graph\" \/><\/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 <em>x<\/em>-axis. The tails are also fatter than those of a normal distribution. The coefficient of kurtosis is usually more than 3.<\/p>\n<p>The term \u201clepto\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 <em>x<\/em>-axis, resulting in a lower peak when compared to a 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 a leptokurtic distribution. The coefficient of kurtosis is usually less than 3.<\/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 is similar to that of a normal distribution. In other words, the distribution is largely normal.<\/p>\n<h2><strong>Measures of Sample Skewness and Kurtosis<\/strong><\/h2>\n<p><em><strong>Exam tip: <\/strong><\/em>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><span class=\"dropcap\">$${ S }_{ k }=\\frac { 1 }{ n } \\frac { \\sum _{ i=1 }^{ n }{ { \\left( { X }_{ i }-\\bar { X } \\right) }^{ 3 } } }{ { S }^{ 3 } } $$<\/span><\/p>\n<p>Where:<\/p>\n<p>\\(\\bar { X }\\) = Sample mean;<\/p>\n<p>\\(S\\) = Sample standard deviation; and<\/p>\n<p>\\(n\\) = Number of observations.<\/p>\n<p><em><strong>Note:<\/strong> <\/em>The formula is very similar to the formula for standard deviation, but raised to the 3rd power.<\/p>\n<p><em><strong>Interpretation:<\/strong> <\/em>A positive value indicates positive skewness. A \u2018zero\u2019 value indicates that 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>\\(\\text{ Excess kurtosis = Sample kurtosis \u2013 3 }\\), where:<\/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><em><strong>Interpretation: <\/strong><\/em>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<h4><strong>Example: Calculating Skewness<\/strong><\/h4>\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>What is 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><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><\/p>\n<p>Skewness is positive. Hence, the data has a positively skewed distribution.<\/p>\n<h4><strong>Example: Calculating Kurtosis<\/strong><\/h4>\n<p>Using the data from the example above { 12\u00a0\u00a0 13\u00a0\u00a0 54\u00a0\u00a0 56\u00a0\u00a0 25 }, determine the type of kurtosis present.<\/p>\n<p><strong style=\"font-size: revert; color: initial;\">Solution<\/strong><\/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>\\(\\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<h2><strong>Question 1<\/strong><\/h2>\n<p>The skewness of the normal distribution is <em>most likely:<\/em><\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>zero.<\/li>\n<li>positive.<\/li>\n<li>negative.<\/li>\n<\/ol>\n<p><strong>\u00a0 Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>Since the normal curve is symmetric about its mean, its skewness is zero.<\/p>\n<p><strong>B is incorrect<\/strong> because a positively skewed distribution has positive skewness.<\/p>\n<p><strong>C is incorrect<\/strong> because a negatively skewed distribution has negative skewness.<\/p>\n<h2><strong>Question 2<\/strong><\/h2>\n<p>A frequency distribution in which there are too few scores at the extremes of the distribution is <em>most likely<\/em> called:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>platykurtic.<\/li>\n<li>leptokurtic.<\/li>\n<li>mesokurtic.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>A platykurtic distribution has &#8220;thin&#8221; tails and is flatter compared to a normal distribution. It implies that there are fewer scores at the extremes of the distribution, which aligns with the question&#8217;s description.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/04\/Img_1.jpg\" alt=\"kurtosis, platykurtic, leptokurtic graph\" \/><\/p>\n<h3><strong>Question 3<\/strong><\/h3>\n<p>When most of the data are concentrated on the left of the distribution, it is <em>most likely<\/em> called:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>symmetric distribution.<\/li>\n<li>positively skewed distribution.<\/li>\n<li>negatively skewed distribution.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>A distribution is said to be skewed to the right, or\u00a0positively skewed<em>,<\/em>\u00a0when most of the data are concentrated on the left of the distribution. A distribution is said to be skewed to the left, or\u00a0negatively skewed<em>,<\/em>\u00a0if most of the data are concentrated on the right of the distribution. The left tail clearly extends farther from the distribution&#8217;s center than the right tail.<\/p>\n<p><img decoding=\"async\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-61a.jpg\" \/><\/p>\n<p><strong>A is incorrect.<\/strong> A symmetric distribution is one in which the left and right sides mirror each other.<\/p>\n<p><strong>C is incorrect.<\/strong> A distribution is said to be skewed to the left, or negatively skewed<em>,<\/em> if most of the data are concentrated on the right of the distribution. The left tail extends farther away from the mean than the right tail.<\/p>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Kurtosis refers to the measurement of 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.\u00a0 In this regard, we have 3 categories of&#8230;<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-25233","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 in Statistics | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn about kurtosis, its types, and how it measures the peakedness of a distribution relative to a normal distribution.\" \/>\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\/kurtosis\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Kurtosis in Statistics | CFA Level 1 - 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