{"id":5459,"date":"2021-01-28T02:02:10","date_gmt":"2021-01-28T02:02:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=5459"},"modified":"2026-02-09T02:37:57","modified_gmt":"2026-02-09T02:37:57","slug":"4-17-the-normal-distribution-and-the-empirical-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/4-17-the-normal-distribution-and-the-empirical-rule\/","title":{"raw":"4.17 The Normal Distribution and the Empirical Rule","rendered":"4.17 The Normal Distribution and the Empirical Rule"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"text-align: center;\">Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>What is a normal distribution?<\/li>\r\n \t<li>The Empirical Rule (the 68 - 95 - 99.7 Rule)<\/li>\r\n \t<li>Drawing a normal distribution for a particular scenario<\/li>\r\n \t<li>Finding the percentage as it pertains to a particular scenario<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>What is a normal distribution?<\/h3>\r\nEarly statisticians noticed the same shape coming up over and over again in different distributions\u2014so they named it the normal distribution.\r\n<p class=\"indent\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28021657\/blank-normal-curve.gif\"><img class=\"size-medium wp-image-5467 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28021657\/blank-normal-curve-300x154.gif\" alt=\"\" width=\"300\" height=\"154\" \/><\/a><\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n\r\nFeatures of the Normal Distribution\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">It is bell-shaped<\/li>\r\n \t<li style=\"font-weight: 400;\">It is symmetric about the mean.\u00a0 In other words, if you divide the graph in half by drawing a vertical line at the mean, there will be two identical halves<\/li>\r\n \t<li style=\"font-weight: 400;\">The mean, median and mode are equal and located at the center of the distribution<\/li>\r\n \t<li style=\"font-weight: 400;\">It has no gaps or holes so we would say the distribution is continuous and extends out along the horizontal axis from\u00a0 \u2212\u221e to +\u221e<\/li>\r\n \t<li style=\"font-weight: 400;\">\u00a0An area under the distribution that falls between two data values is the same as the percentage of the data (in decimal form) between the same data values<\/li>\r\n \t<li style=\"font-weight: 400;\">\u00a0The total area under a normal distribution is exactly 1 or 100%<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n<h3>THE EMPIRICAL RULE<\/h3>\r\nFor data that is bell-shaped and symmetrical (Normal):\r\n<ul>\r\n \t<li>Approximately 68% of the data is within +\/- 1 standard deviations of the mean.<\/li>\r\n \t<li>Approximately 95% of the data is within +\/- 2 standard deviations of the mean.<\/li>\r\n \t<li>Approximately 99.7% of the data is within +\/- 3 standard deviations of the mean.<\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/12003706\/Empirical-rule-2.jpg\"><img class=\" wp-image-6403 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/12003706\/Empirical-rule-2.jpg\" alt=\"\" width=\"760\" height=\"337\" \/><\/a>\r\n<div class=\"textbox examples\">\r\n<h3>Example - Drawing a Normal Distribution<\/h3>\r\nThe trunk diameter of a certain variety of pine tree is normally distributed with a mean of \u03bc=150cm and a standard deviation of \u03c3=30cm.\u00a0 Sketch a normal curve that describes this distribution.\r\n<ul>\r\n \t<li>Step 1:\u00a0 Sketch a blank normal distribution<\/li>\r\n \t<li>Step 2:\u00a0\u00a0The mean of 150 cm goes in the middle<\/li>\r\n \t<li>Step 3:\u00a0 Label each standard deviation.\u00a0\u00a0Each standard deviation is a distance of 30 cm<\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28024235\/Drawing-the-Normal-Curve.png\"><img class=\"size-medium wp-image-5476 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28024235\/Drawing-the-Normal-Curve-300x217.png\" alt=\"\" width=\"300\" height=\"217\" \/><\/a>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example - Finding Percentages<\/h3>\r\nA certain variety of pine tree has a mean trunk diameter of \u03bc=150cm and a standard deviation of \u03c3=30cm\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">a)\u00a0 Approximately what percent of these trees have a diameter greater than 210?<\/span>\r\n\r\nb)\u00a0 Approximately what percent of these trees have a diameter between 90 and 210 centimeters?\r\n\r\na)\r\n<ul>\r\n \t<li>Step 1:\u00a0 Sketch a normal curve with a mean of 150 cm and a standard deviation of 30 cm<\/li>\r\n \t<li>Step 2:\u00a0 Shade the region corresponding to a diameter greater than 210 cm.<\/li>\r\n \t<li>Step 3:\u00a0 Add the percentages in the shaded area<\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28025529\/Image-for-area-greater-than-210.png\"><img class=\"size-medium wp-image-5477 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28025529\/Image-for-area-greater-than-210.png\" alt=\"\" width=\"300\" height=\"167\" \/><\/a>\r\n\r\n2.35%+0.15%=2.5%\r\n\r\nAbout 2.5% of these trees have a diameter greater than 210 cm.\r\n\r\nb)\r\n<ul>\r\n \t<li>Step 1:\u00a0 Use the same distribution from part a<\/li>\r\n \t<li>Step 2:\u00a0 Shade the region between 90 and 210 cm<\/li>\r\n \t<li>Step 3:\u00a0 Add the percentages in the shaded area.<\/li>\r\n<\/ul>\r\n13.5% + 34% + 34% + 13.5% = 95%\r\n\r\nAbout 95% of these trees have a diameter between 90 and 210 cm.\r\n\r\n<\/div>\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>\r\n\r\n<\/div>\r\n<div><\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3 style=\"text-align: center;\">Learning Objectives<\/h3>\n<ul>\n<li>What is a normal distribution?<\/li>\n<li>The Empirical Rule (the 68 &#8211; 95 &#8211; 99.7 Rule)<\/li>\n<li>Drawing a normal distribution for a particular scenario<\/li>\n<li>Finding the percentage as it pertains to a particular scenario<\/li>\n<\/ul>\n<\/div>\n<h3>What is a normal distribution?<\/h3>\n<p>Early statisticians noticed the same shape coming up over and over again in different distributions\u2014so they named it the normal distribution.<\/p>\n<p class=\"indent\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28021657\/blank-normal-curve.gif\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5467 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28021657\/blank-normal-curve-300x154.gif\" alt=\"\" width=\"300\" height=\"154\" \/><\/a><\/p>\n<div class=\"textbox shaded\">\n<p>Features of the Normal Distribution<\/p>\n<ul>\n<li style=\"font-weight: 400;\">It is bell-shaped<\/li>\n<li style=\"font-weight: 400;\">It is symmetric about the mean.\u00a0 In other words, if you divide the graph in half by drawing a vertical line at the mean, there will be two identical halves<\/li>\n<li style=\"font-weight: 400;\">The mean, median and mode are equal and located at the center of the distribution<\/li>\n<li style=\"font-weight: 400;\">It has no gaps or holes so we would say the distribution is continuous and extends out along the horizontal axis from\u00a0 \u2212\u221e to +\u221e<\/li>\n<li style=\"font-weight: 400;\">\u00a0An area under the distribution that falls between two data values is the same as the percentage of the data (in decimal form) between the same data values<\/li>\n<li style=\"font-weight: 400;\">\u00a0The total area under a normal distribution is exactly 1 or 100%<\/li>\n<\/ul>\n<\/div>\n<div>\n<h3>THE EMPIRICAL RULE<\/h3>\n<p>For data that is bell-shaped and symmetrical (Normal):<\/p>\n<ul>\n<li>Approximately 68% of the data is within +\/- 1 standard deviations of the mean.<\/li>\n<li>Approximately 95% of the data is within +\/- 2 standard deviations of the mean.<\/li>\n<li>Approximately 99.7% of the data is within +\/- 3 standard deviations of the mean.<\/li>\n<\/ul>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/12003706\/Empirical-rule-2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6403 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/12003706\/Empirical-rule-2.jpg\" alt=\"\" width=\"760\" height=\"337\" \/><\/a><\/p>\n<div class=\"textbox examples\">\n<h3>Example &#8211; Drawing a Normal Distribution<\/h3>\n<p>The trunk diameter of a certain variety of pine tree is normally distributed with a mean of \u03bc=150cm and a standard deviation of \u03c3=30cm.\u00a0 Sketch a normal curve that describes this distribution.<\/p>\n<ul>\n<li>Step 1:\u00a0 Sketch a blank normal distribution<\/li>\n<li>Step 2:\u00a0\u00a0The mean of 150 cm goes in the middle<\/li>\n<li>Step 3:\u00a0 Label each standard deviation.\u00a0\u00a0Each standard deviation is a distance of 30 cm<\/li>\n<\/ul>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28024235\/Drawing-the-Normal-Curve.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5476 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28024235\/Drawing-the-Normal-Curve-300x217.png\" alt=\"\" width=\"300\" height=\"217\" \/><\/a><\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example &#8211; Finding Percentages<\/h3>\n<p>A certain variety of pine tree has a mean trunk diameter of \u03bc=150cm and a standard deviation of \u03c3=30cm<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">a)\u00a0 Approximately what percent of these trees have a diameter greater than 210?<\/span><\/p>\n<p>b)\u00a0 Approximately what percent of these trees have a diameter between 90 and 210 centimeters?<\/p>\n<p>a)<\/p>\n<ul>\n<li>Step 1:\u00a0 Sketch a normal curve with a mean of 150 cm and a standard deviation of 30 cm<\/li>\n<li>Step 2:\u00a0 Shade the region corresponding to a diameter greater than 210 cm.<\/li>\n<li>Step 3:\u00a0 Add the percentages in the shaded area<\/li>\n<\/ul>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28025529\/Image-for-area-greater-than-210.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5477 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/28025529\/Image-for-area-greater-than-210.png\" alt=\"\" width=\"300\" height=\"167\" \/><\/a><\/p>\n<p>2.35%+0.15%=2.5%<\/p>\n<p>About 2.5% of these trees have a diameter greater than 210 cm.<\/p>\n<p>b)<\/p>\n<ul>\n<li>Step 1:\u00a0 Use the same distribution from part a<\/li>\n<li>Step 2:\u00a0 Shade the region between 90 and 210 cm<\/li>\n<li>Step 3:\u00a0 Add the percentages in the shaded area.<\/li>\n<\/ul>\n<p>13.5% + 34% + 34% + 13.5% = 95%<\/p>\n<p>About 95% of these trees have a diameter between 90 and 210 cm.<\/p>\n<\/div>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>\n<\/div>\n<div><\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-5459\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>The Normal Distribution Article. <strong>Provided by<\/strong>: Khan Academy. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.khanacademy.org\/math\/statistics-probability\/modeling-distributions-of-data\/normal-distributions-library\/a\/normal-distributions-review\">https:\/\/www.khanacademy.org\/math\/statistics-probability\/modeling-distributions-of-data\/normal-distributions-library\/a\/normal-distributions-review<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em><\/li><li>Introduction to Statistics - Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":383311,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"The Normal Distribution Article\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/www.khanacademy.org\/math\/statistics-probability\/modeling-distributions-of-data\/normal-distributions-library\/a\/normal-distributions-review\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Introduction to Statistics - Lumen Learning\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5459","chapter","type-chapter","status-publish","hentry"],"part":329,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/5459","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/users\/383311"}],"version-history":[{"count":19,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/5459\/revisions"}],"predecessor-version":[{"id":7094,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/5459\/revisions\/7094"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/parts\/329"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/5459\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/media?parent=5459"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapter-type?post=5459"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/contributor?post=5459"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/license?post=5459"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}