{"id":2705,"date":"2021-11-12T13:34:57","date_gmt":"2021-11-12T13:34:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2705"},"modified":"2023-12-05T09:20:37","modified_gmt":"2023-12-05T09:20:37","slug":"putting-it-together-the-normal-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/putting-it-together-the-normal-distribution\/","title":{"raw":"Putting It Together: The Normal Distribution","rendered":"Putting It Together: The Normal Distribution"},"content":{"raw":"<h2>Let\u2019s Summarize<\/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 measures how far <em>X<\/em> is from the mean in standard deviations. In other words, the <em>z<\/em>-score is the number of standard deviations <em>X<\/em> is from the mean of the distribution. For example, <em>Z<\/em> = 1 means the <em>x<\/em>-value is one standard deviation above the mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">We use a <em>normal density curve<\/em> to model the probability distribution for many variables, such as weight, shoe sizes, foot lengths, and other physical characteristics. For a normal curve, the empirical rule for normal curves tells us that 68% of the observations fall within 1 standard deviation of the mean, 95% within 2 standard deviations of the mean, and 99.7% within 3 standard deviations of the mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Normal distribution tables and technology can be used to calculate probabilities of find percentiles associated with the normal distribution.<\/li>\r\n<\/ul>","rendered":"<h2>Let\u2019s Summarize<\/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 measures how far <em>X<\/em> is from the mean in standard deviations. In other words, the <em>z<\/em>-score is the number of standard deviations <em>X<\/em> is from the mean of the distribution. For example, <em>Z<\/em> = 1 means the <em>x<\/em>-value is one standard deviation above the mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">We use a <em>normal density curve<\/em> to model the probability distribution for many variables, such as weight, shoe sizes, foot lengths, and other physical characteristics. For a normal curve, the empirical rule for normal curves tells us that 68% of the observations fall within 1 standard deviation of the mean, 95% within 2 standard deviations of the mean, and 99.7% within 3 standard deviations of the mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Normal distribution tables and technology can be used to calculate probabilities of find percentiles associated with the normal distribution.<\/li>\n<\/ul>\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-2705\">\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>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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":14,"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\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"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-2705","chapter","type-chapter","status-publish","hentry"],"part":256,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2705","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":2,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2705\/revisions"}],"predecessor-version":[{"id":3676,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2705\/revisions\/3676"}],"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\/2705\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2705"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2705"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2705"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2705"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}