{"id":893,"date":"2023-03-20T19:18:06","date_gmt":"2023-03-20T19:18:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/z-score-and-the-empirical-rule-learn-it-5\/"},"modified":"2025-05-08T03:20:28","modified_gmt":"2025-05-08T03:20:28","slug":"z-score-and-the-empirical-rule-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/z-score-and-the-empirical-rule-learn-it-5\/","title":{"raw":"Z-Score and the Empirical Rule: Learn It 5","rendered":"Z-Score and the Empirical Rule: Learn It 5"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate z-scores to explain the location of data points.<\/li>\r\n\t<li>Compare observations using z-scores and the Empirical Rule.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Empirical Rule<\/h2>\r\n<p>If a distribution of a variable [latex]X[\/latex] is bell-shaped, unimodal, and symmetric, then we can estimate how many observations are within a certain number of standard deviations.<\/p>\r\n<p>The <strong>Empirical Rule <\/strong>(also known as the [latex]68-95-99.7[\/latex] rule) is a guideline that predicts the percentage of observations within a certain number of standard deviations.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>empirical rule<\/h3>\r\n<p>The Empirical Rule states that:<\/p>\r\n<ul>\r\n\t<li>About [latex]68\\%[\/latex] of observations in a data set will be within <strong>one<\/strong> standard deviation of the mean.<\/li>\r\n\t<li>About [latex]95\\%[\/latex] of the observations in a data set will be within <strong>two<\/strong> standard deviations of the mean.<\/li>\r\n\t<li>About [latex]99.7\\%[\/latex] of the observations in a data set will be within <strong>three<\/strong> standard deviations of the mean.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>Graphically, the Empirical Rule can be expressed like this:<\/p>\r\n\r\n[caption id=\"attachment_5694\" align=\"aligncenter\" width=\"1200\"]<img class=\"wp-image-5694 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/30233929\/4.5.L.Histogram1-1.png\" alt=\"An image displaying the distribution set by the Empirical Rule.\" width=\"1200\" height=\"1426\" \/> Figure 1. A normal distribution curve showing the empirical rule: about 68% of data fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 standard deviations.[\/caption]\r\n\r\n<p>Now, try applying the Empirical Rule to the Movie Runtime data set.<\/p>\r\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\">[ohm2_question hide_question_numbers=1]2754[\/ohm2_question]<\/span><\/section>\r\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\">[ohm2_question hide_question_numbers=1]2155[\/ohm2_question]<\/span><\/section>\r\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\">[ohm2_question hide_question_numbers=1]2753[\/ohm2_question]<\/span><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate z-scores to explain the location of data points.<\/li>\n<li>Compare observations using z-scores and the Empirical Rule.<\/li>\n<\/ul>\n<\/section>\n<h2>The Empirical Rule<\/h2>\n<p>If a distribution of a variable [latex]X[\/latex] is bell-shaped, unimodal, and symmetric, then we can estimate how many observations are within a certain number of standard deviations.<\/p>\n<p>The <strong>Empirical Rule <\/strong>(also known as the [latex]68-95-99.7[\/latex] rule) is a guideline that predicts the percentage of observations within a certain number of standard deviations.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>empirical rule<\/h3>\n<p>The Empirical Rule states that:<\/p>\n<ul>\n<li>About [latex]68\\%[\/latex] of observations in a data set will be within <strong>one<\/strong> standard deviation of the mean.<\/li>\n<li>About [latex]95\\%[\/latex] of the observations in a data set will be within <strong>two<\/strong> standard deviations of the mean.<\/li>\n<li>About [latex]99.7\\%[\/latex] of the observations in a data set will be within <strong>three<\/strong> standard deviations of the mean.<\/li>\n<\/ul>\n<\/section>\n<p>Graphically, the Empirical Rule can be expressed like this:<\/p>\n<figure id=\"attachment_5694\" aria-describedby=\"caption-attachment-5694\" style=\"width: 1200px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5694 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/30233929\/4.5.L.Histogram1-1.png\" alt=\"An image displaying the distribution set by the Empirical Rule.\" width=\"1200\" height=\"1426\" \/><figcaption id=\"caption-attachment-5694\" class=\"wp-caption-text\">Figure 1. A normal distribution curve showing the empirical rule: about 68% of data fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 standard deviations.<\/figcaption><\/figure>\n<p>Now, try applying the Empirical Rule to the Movie Runtime data set.<\/p>\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\"><iframe loading=\"lazy\" id=\"ohm2754\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2754&theme=lumen&iframe_resize_id=ohm2754&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/span><\/section>\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\"><iframe loading=\"lazy\" id=\"ohm2155\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2155&theme=lumen&iframe_resize_id=ohm2155&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/span><\/section>\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\"><iframe loading=\"lazy\" id=\"ohm2753\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2753&theme=lumen&iframe_resize_id=ohm2753&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/span><\/section>\n","protected":false},"author":13,"menu_order":43,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":834,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/893"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/893\/revisions"}],"predecessor-version":[{"id":6480,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/893\/revisions\/6480"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/834"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/893\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=893"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=893"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=893"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=893"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}