{"id":478,"date":"2021-12-20T14:38:11","date_gmt":"2021-12-20T14:38:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=478"},"modified":"2022-02-11T21:23:18","modified_gmt":"2022-02-11T21:23:18","slug":"summary-of-4e","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/summary-of-4e\/","title":{"raw":"Summary of Z-Score and the Empirical Rule: 4E","rendered":"Summary of Z-Score and the Empirical Rule: 4E"},"content":{"raw":"This page would contain resource information like a glossary of terms from the section, key equations, and a reminder of concepts that were covered.\r\n\r\nMake this more relevant to what students want -- help them to build their processes, study guides, mnemonics, and memory dump material.\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>Standardizing the value includes finding the difference between the given value and the mean, and dividing that distance by the standard deviation. The resulting value is a number of standard deviations, and has no units associated with it.<\/li>\r\n \t<li>Standardized scores can result in positive and negative values. A negative can be thought of as indicating a value that lies to the left of the mean, and a positive indicates a value that lies to the right of the mean.<\/li>\r\n \t<li>An estimate of how many observations are within a certain number of standard deviations can be made if a distribution is bell shaped, unimodal, and symmetric.<\/li>\r\n \t<li>The Empirical Rule states that:\r\n<ul>\r\n \t<li>about 68% of observations in a dataset will be within one standard deviation of the mean<\/li>\r\n \t<li>about 95% of observations in a dataset will be within two standard deviations of the mean<\/li>\r\n \t<li>about 99.7% of the observations in a dataset will be within three standard deviations of the mean<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Converting values into standardized scores<\/strong><\/li>\r\n<\/ul>\r\n[latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572229168\" class=\"definition\">\r\n \t<dt>standardized value<\/dt>\r\n \t<dd id=\"fs-id1170572229174\">the number of standard deviations an observation is away from the mean. Also referred to as a <strong>z-score<\/strong>.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572229190\" class=\"definition\">\r\n \t<dt>Empirical Rule<\/dt>\r\n \t<dd id=\"fs-id1170572229195\">a guideline that predicts the percentage of observations within a certain number of standard deviations. Also known as the\u00a0<strong>68-95-99.7 Rule<\/strong> which states that\u00a0in a bell-shaped, unimodal distribution, almost all of the observed data values, [latex]x[\/latex], lie within three standard deviations, [latex]\\sigma[\/latex], to either side of the mean, [latex]\\mu[\/latex]. More specifically, about 68% of observations in a dataset will be within one standard deviation of the mean [latex]\\left(\\mu\\pm\\sigma\\right)[\/latex],\u00a0about 95% of the observations in a dataset will be within two standard deviations of the mean [latex]\\left(\\mu\\pm2\\sigma\\right)[\/latex], and\u00a0about 99.7% of the observations in a dataset will be within three standard deviations of the mean [latex]\\left(\\mu\\pm3\\sigma\\right)[\/latex].<\/dd>\r\n<\/dl>\r\nPut formal DCMP I Can statements to prepare for the self-check.\r\n\r\n<span style=\"background-color: #ffff00;\">These I Can Statements are new (the first one is the \"you will understand\" rephrased as an I Can):<\/span>\r\n<ul>\r\n \t<li>I can utilize standardized scores and the Empirical Rule to determine if an observation is usual.<\/li>\r\n \t<li>I can utilize standardized scores and the Empirical Rule to determine if an observation is unusual.<\/li>\r\n \t<li>I can compare two observations by calculating and comparing the standardized score.<\/li>\r\n<\/ul>","rendered":"<p>This page would contain resource information like a glossary of terms from the section, key equations, and a reminder of concepts that were covered.<\/p>\n<p>Make this more relevant to what students want &#8212; help them to build their processes, study guides, mnemonics, and memory dump material.<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>Standardizing the value includes finding the difference between the given value and the mean, and dividing that distance by the standard deviation. The resulting value is a number of standard deviations, and has no units associated with it.<\/li>\n<li>Standardized scores can result in positive and negative values. A negative can be thought of as indicating a value that lies to the left of the mean, and a positive indicates a value that lies to the right of the mean.<\/li>\n<li>An estimate of how many observations are within a certain number of standard deviations can be made if a distribution is bell shaped, unimodal, and symmetric.<\/li>\n<li>The Empirical Rule states that:\n<ul>\n<li>about 68% of observations in a dataset will be within one standard deviation of the mean<\/li>\n<li>about 95% of observations in a dataset will be within two standard deviations of the mean<\/li>\n<li>about 99.7% of the observations in a dataset will be within three standard deviations of the mean<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Converting values into standardized scores<\/strong><\/li>\n<\/ul>\n<p>[latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.<\/p>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572229168\" class=\"definition\">\n<dt>standardized value<\/dt>\n<dd id=\"fs-id1170572229174\">the number of standard deviations an observation is away from the mean. Also referred to as a <strong>z-score<\/strong>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572229190\" class=\"definition\">\n<dt>Empirical Rule<\/dt>\n<dd id=\"fs-id1170572229195\">a guideline that predicts the percentage of observations within a certain number of standard deviations. Also known as the\u00a0<strong>68-95-99.7 Rule<\/strong> which states that\u00a0in a bell-shaped, unimodal distribution, almost all of the observed data values, [latex]x[\/latex], lie within three standard deviations, [latex]\\sigma[\/latex], to either side of the mean, [latex]\\mu[\/latex]. More specifically, about 68% of observations in a dataset will be within one standard deviation of the mean [latex]\\left(\\mu\\pm\\sigma\\right)[\/latex],\u00a0about 95% of the observations in a dataset will be within two standard deviations of the mean [latex]\\left(\\mu\\pm2\\sigma\\right)[\/latex], and\u00a0about 99.7% of the observations in a dataset will be within three standard deviations of the mean [latex]\\left(\\mu\\pm3\\sigma\\right)[\/latex].<\/dd>\n<\/dl>\n<p>Put formal DCMP I Can statements to prepare for the self-check.<\/p>\n<p><span style=\"background-color: #ffff00;\">These I Can Statements are new (the first one is the &#8220;you will understand&#8221; rephrased as an I Can):<\/span><\/p>\n<ul>\n<li>I can utilize standardized scores and the Empirical Rule to determine if an observation is usual.<\/li>\n<li>I can utilize standardized scores and the Empirical Rule to determine if an observation is unusual.<\/li>\n<li>I can compare two observations by calculating and comparing the standardized score.<\/li>\n<\/ul>\n","protected":false},"author":25777,"menu_order":33,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-478","chapter","type-chapter","status-publish","hentry"],"part":621,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/478","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/478\/revisions"}],"predecessor-version":[{"id":3092,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/478\/revisions\/3092"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/621"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/478\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=478"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=478"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=478"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=478"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}