{"id":1755,"date":"2021-09-07T19:08:43","date_gmt":"2021-09-07T19:08:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1755"},"modified":"2023-12-05T09:19:36","modified_gmt":"2023-12-05T09:19:36","slug":"summary-the-standard-normal-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-the-standard-normal-distribution\/","title":{"raw":"Summary: The Standard Normal Distribution","rendered":"Summary: The Standard Normal Distribution"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">A standard normal distribution has a mean of 0 and a standard deviation of 1.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">A normal distribution is bell-shaped and the total area under the normal distribution curve is 1.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">A <em>z<\/em>-score tells you how many standard deviations a value is above or below the mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Transforming a value to its <em>z<\/em>-score allows you to apply the empirical rule.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The empirical rule states that for a normal distribution: 68% of the distribution lies within one standard deviation of the mean, 95% of the distribution lies within two standard deviations of the mean and 99.7% of the distribution lies within three standard deviations of the mean.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>standard normal distribution:<\/strong> a continuous random variable (RV) X ~ N(0, 1); when <em>X<\/em> follows the standard normal distribution, it is often noted as Z ~ N(0, 1).\r\n\r\n<strong><em>z<\/em>-score:<\/strong> the linear transformation of the form [latex]z=\\frac{x-M}{\\sigma}[\/latex];\u00a0if this transformation is applied to any normal distribution X ~ N(\u03bc, \u03c3) the result is the standard normal distribution Z ~ N(0,1). If this transformation is applied to any specific value <em>x<\/em> of the RV with mean <em>\u03bc<\/em> and standard deviation \u03c3, the result is called the <em>z<\/em>-score of <em>x<\/em>. The <em>z<\/em>-score allows us to compare data that are normally distributed but scaled differently.","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">A standard normal distribution has a mean of 0 and a standard deviation of 1.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">A normal distribution is bell-shaped and the total area under the normal distribution curve is 1.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">A <em>z<\/em>-score tells you how many standard deviations a value is above or below the mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Transforming a value to its <em>z<\/em>-score allows you to apply the empirical rule.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The empirical rule states that for a normal distribution: 68% of the distribution lies within one standard deviation of the mean, 95% of the distribution lies within two standard deviations of the mean and 99.7% of the distribution lies within three standard deviations of the mean.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>standard normal distribution:<\/strong> a continuous random variable (RV) X ~ N(0, 1); when <em>X<\/em> follows the standard normal distribution, it is often noted as Z ~ N(0, 1).<\/p>\n<p><strong><em>z<\/em>-score:<\/strong> the linear transformation of the form [latex]z=\\frac{x-M}{\\sigma}[\/latex];\u00a0if this transformation is applied to any normal distribution X ~ N(\u03bc, \u03c3) the result is the standard normal distribution Z ~ N(0,1). If this transformation is applied to any specific value <em>x<\/em> of the RV with mean <em>\u03bc<\/em> and standard deviation \u03c3, the result is called the <em>z<\/em>-score of <em>x<\/em>. The <em>z<\/em>-score allows us to compare data that are normally distributed but scaled differently.<\/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-1755\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/6-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/6-key-terms<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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":169134,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/6-key-terms\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1755","chapter","type-chapter","status-publish","hentry"],"part":256,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1755","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1755\/revisions"}],"predecessor-version":[{"id":3668,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1755\/revisions\/3668"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/256"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1755\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1755"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1755"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1755"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}