{"id":897,"date":"2023-03-20T19:18:10","date_gmt":"2023-03-20T19:18:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/z-score-and-the-empirical-rule-apply-it-2\/"},"modified":"2025-05-11T22:50:27","modified_gmt":"2025-05-11T22:50:27","slug":"z-score-and-the-empirical-rule-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/z-score-and-the-empirical-rule-apply-it-2\/","title":{"raw":"Z-Score and the Empirical Rule: Apply It 2","rendered":"Z-Score and the Empirical Rule: Apply It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate z-scores to explain the location of data points&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate [latex]z[\/latex]-scores to explain the location of data points.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Compare observations using z-scores and the Empirical Rule&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Compare observations using [latex]z[\/latex]-scores and the Empirical Rule.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 id=\"useEmp\">The Empirical Rule<\/h2>\r\n<section class=\"textbox recall\">Before beginning the activity below, recall the definition of the <strong>Empirical Rule<\/strong>. What does it state? [reveal-answer q=\"725555\"]Explain the Empirical Rule[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"725555\"]The Empirical Rule states that,\u00a0in a bell-shaped, unimodal distribution, almost all the observed data values, [latex]x[\/latex], lie within three standard deviations, [latex]\\sigma[\/latex], to either side of the mean, [latex]\\mu[\/latex].<br \/>\r\n[caption id=\"attachment_1501\" align=\"aligncenter\" width=\"501\"]<img class=\"wp-image-1501\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/01171355\/4.5.L.Histogram1.png\" alt=\"A bar graph with the highest bars in the middle and lower bars to either side. In the center, the x-axis is labeled &quot;mu.&quot; Three bars to the left, it is labeled &quot;mu - sigma,&quot; three more bars to the left it is labeled &quot;mu - 2 sigma,&quot; and three more to the left, it's labeled &quot;mu - 3 sigma.&quot; Three to the right of the center, it is labeled &quot;mu + sigma.&quot; Three more to the right and it is labeled &quot;mu + 2 sigma&quot; and three more to the right, it is labeled &quot;mu + 3 sigma.&quot; The center six bars are all green and labeled as 68&amp; collectively. The three leftmost center bars are labeled 34.1%, and the three rightmost center bars are also labeled 34.1%. The next three bars out on either side of the center six are each labeled 13.6% and the center 12 are all labeled 95% collectively. Lastly, the next three out on either side of the center twelve are each labeled 2.1% and all 18 are collectively labeled 99.7% \u2248 100%\" width=\"501\" height=\"596\" \/> 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<br \/>\r\nSpecifically,\r\n\r\n<ul>\r\n\t<li>[latex]68[\/latex]% of the observations lie within one standard deviation of the mean. [latex]\\left(\\mu\\pm\\sigma\\right)[\/latex]<\/li>\r\n\t<li>[latex]95[\/latex]% of the observations lie within two standard deviations of the mean. [latex]\\left(\\mu\\pm2\\sigma\\right)[\/latex]<\/li>\r\n\t<li>[latex]99.7[\/latex]% of the observations lie within three standard deviations of the mean. [latex]\\left(\\mu\\pm3\\sigma\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<p>For this reason, the Empirical Rule is sometimes called the\u00a0[latex]68-95-99.7[\/latex] rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Consider the weights of the livers and spleens in [latex]26[\/latex]-week-old female C57BL\/6J laboratory mice. The mean liver weight is [latex]0.999[\/latex] grams (g) with a standard deviation of [latex]0.087[\/latex] g, and the mean spleen weight is [latex]0.086[\/latex] g with a standard deviation of [latex]0.007[\/latex] g.<\/p>\r\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\">[ohm2_question hide_question_numbers=1]2158[\/ohm2_question]<\/span><\/section>\r\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\">[ohm2_question hide_question_numbers=1]2665[\/ohm2_question]<\/span><\/section>\r\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\">[ohm2_question hide_question_numbers=1]2159[\/ohm2_question]<\/span><\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2160[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate z-scores to explain the location of data points&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate [latex]z[\/latex]-scores to explain the location of data points.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Compare observations using z-scores and the Empirical Rule&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Compare observations using [latex]z[\/latex]-scores and the Empirical Rule.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 id=\"useEmp\">The Empirical Rule<\/h2>\n<section class=\"textbox recall\">Before beginning the activity below, recall the definition of the <strong>Empirical Rule<\/strong>. What does it state? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q725555\">Explain the Empirical Rule<\/button><\/p>\n<div id=\"q725555\" class=\"hidden-answer\" style=\"display: none\">The Empirical Rule states that,\u00a0in a bell-shaped, unimodal distribution, almost all the observed data values, [latex]x[\/latex], lie within three standard deviations, [latex]\\sigma[\/latex], to either side of the mean, [latex]\\mu[\/latex].<\/p>\n<figure id=\"attachment_1501\" aria-describedby=\"caption-attachment-1501\" style=\"width: 501px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1501\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/01171355\/4.5.L.Histogram1.png\" alt=\"A bar graph with the highest bars in the middle and lower bars to either side. In the center, the x-axis is labeled &quot;mu.&quot; Three bars to the left, it is labeled &quot;mu - sigma,&quot; three more bars to the left it is labeled &quot;mu - 2 sigma,&quot; and three more to the left, it's labeled &quot;mu - 3 sigma.&quot; Three to the right of the center, it is labeled &quot;mu + sigma.&quot; Three more to the right and it is labeled &quot;mu + 2 sigma&quot; and three more to the right, it is labeled &quot;mu + 3 sigma.&quot; The center six bars are all green and labeled as 68&amp; collectively. The three leftmost center bars are labeled 34.1%, and the three rightmost center bars are also labeled 34.1%. The next three bars out on either side of the center six are each labeled 13.6% and the center 12 are all labeled 95% collectively. Lastly, the next three out on either side of the center twelve are each labeled 2.1% and all 18 are collectively labeled 99.7% \u2248 100%\" width=\"501\" height=\"596\" \/><figcaption id=\"caption-attachment-1501\" 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>\nSpecifically,<\/p>\n<ul>\n<li>[latex]68[\/latex]% of the observations lie within one standard deviation of the mean. [latex]\\left(\\mu\\pm\\sigma\\right)[\/latex]<\/li>\n<li>[latex]95[\/latex]% of the observations lie within two standard deviations of the mean. [latex]\\left(\\mu\\pm2\\sigma\\right)[\/latex]<\/li>\n<li>[latex]99.7[\/latex]% of the observations lie within three standard deviations of the mean. [latex]\\left(\\mu\\pm3\\sigma\\right)[\/latex]<\/li>\n<\/ul>\n<p>For this reason, the Empirical Rule is sometimes called the\u00a0[latex]68-95-99.7[\/latex] rule.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Consider the weights of the livers and spleens in [latex]26[\/latex]-week-old female C57BL\/6J laboratory mice. The mean liver weight is [latex]0.999[\/latex] grams (g) with a standard deviation of [latex]0.087[\/latex] g, and the mean spleen weight is [latex]0.086[\/latex] g with a standard deviation of [latex]0.007[\/latex] g.<\/p>\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; text-align: initial;\"><iframe loading=\"lazy\" id=\"ohm2158\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2158&theme=lumen&iframe_resize_id=ohm2158&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=\"ohm2665\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2665&theme=lumen&iframe_resize_id=ohm2665&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=\"ohm2159\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2159&theme=lumen&iframe_resize_id=ohm2159&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/span><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2160\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2160&theme=lumen&iframe_resize_id=ohm2160&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":45,"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":"apply_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\/897"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/897\/revisions"}],"predecessor-version":[{"id":6640,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/897\/revisions\/6640"}],"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\/897\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=897"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=897"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=897"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=897"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}