{"id":2344,"date":"2021-10-11T19:33:16","date_gmt":"2021-10-11T19:33:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2344"},"modified":"2023-12-05T09:47:10","modified_gmt":"2023-12-05T09:47:10","slug":"the-regression-equation-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation-3\/","title":{"raw":"The Coefficient of Determination","rendered":"The Coefficient of Determination"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul>\r\n \t<li>Interpret the coefficient of determination in context<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>The Coefficient of Determination<\/h2>\r\n<strong>The variable [latex]\\mathbf{r^2}[\/latex] is called the coefficient of determination<\/strong> and is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. It has an interpretation in the context of the data:\r\n<ul>\r\n \t<li><em>r<\/em><sup>2<\/sup>, when expressed as a percent, represents the percent of variation in the dependent (predicted) variable <em>y<\/em> that can be explained by variation in the independent (explanatory) variable <em>x<\/em> using the regression (best-fit) line.<\/li>\r\n \t<li>1 \u2013 <em>r<\/em><sup>2<\/sup>, when expressed as a percentage, represents the percent of variation in <em>y<\/em> that is NOT explained by variation in <em>x<\/em> using the regression line. This can be seen as the scattering of the observed data points about the regression line.<\/li>\r\n<\/ul>\r\n<p id=\"fs-idp19642608\" class=\" \">Consider the example (example 2 aka the Third Exam vs Final Exam Example) introduced in the previous section:<\/p>\r\n\r\n<ul>\r\n \t<li class=\" \">The line of best fit is\u00a0[latex]\\displaystyle\\hat{{y}}=-{173.51}+{4.83}{x}[\/latex]<\/li>\r\n \t<li class=\" \">The correlation coefficient is\u00a0<em style=\"font-size: 1rem; orphans: 1; text-align: initial;\">r<\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\"> = 0.6631<\/span><\/li>\r\n \t<li class=\" \"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">The coefficient of determination is <\/span><em style=\"font-size: 1rem; orphans: 1; text-align: initial;\">r<\/em><sup style=\"orphans: 1; text-align: initial;\">2<\/sup><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\"> = 0.66312 = 0.4397<\/span><\/li>\r\n \t<li class=\" \"><strong style=\"font-size: 1rem; orphans: 1; text-align: initial;\">Interpretation of <em data-redactor-tag=\"em\">r<\/em><sup>2<\/sup> in the context of this example:<\/strong><\/li>\r\n \t<li class=\" \"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">Approximately 44% of the variation (0.4397 is approximately 0.44) in the final-exam grades can be explained by the variation in the grades on the third exam using the best-fit regression line.<\/span><\/li>\r\n \t<li class=\" \"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">Therefore, approximately 56% of the variation (1 \u2013 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam using the best-fit regression line. (This is seen as the scattering of the points about the line).<\/span><\/li>\r\n<\/ul>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul>\n<li>Interpret the coefficient of determination in context<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>The Coefficient of Determination<\/h2>\n<p><strong>The variable [latex]\\mathbf{r^2}[\/latex] is called the coefficient of determination<\/strong> and is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. It has an interpretation in the context of the data:<\/p>\n<ul>\n<li><em>r<\/em><sup>2<\/sup>, when expressed as a percent, represents the percent of variation in the dependent (predicted) variable <em>y<\/em> that can be explained by variation in the independent (explanatory) variable <em>x<\/em> using the regression (best-fit) line.<\/li>\n<li>1 \u2013 <em>r<\/em><sup>2<\/sup>, when expressed as a percentage, represents the percent of variation in <em>y<\/em> that is NOT explained by variation in <em>x<\/em> using the regression line. This can be seen as the scattering of the observed data points about the regression line.<\/li>\n<\/ul>\n<p id=\"fs-idp19642608\" class=\"\">Consider the example (example 2 aka the Third Exam vs Final Exam Example) introduced in the previous section:<\/p>\n<ul>\n<li class=\"\">The line of best fit is\u00a0[latex]\\displaystyle\\hat{{y}}=-{173.51}+{4.83}{x}[\/latex]<\/li>\n<li class=\"\">The correlation coefficient is\u00a0<em style=\"font-size: 1rem; orphans: 1; text-align: initial;\">r<\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\"> = 0.6631<\/span><\/li>\n<li class=\"\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">The coefficient of determination is <\/span><em style=\"font-size: 1rem; orphans: 1; text-align: initial;\">r<\/em><sup style=\"orphans: 1; text-align: initial;\">2<\/sup><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\"> = 0.66312 = 0.4397<\/span><\/li>\n<li class=\"\"><strong style=\"font-size: 1rem; orphans: 1; text-align: initial;\">Interpretation of <em data-redactor-tag=\"em\">r<\/em><sup>2<\/sup> in the context of this example:<\/strong><\/li>\n<li class=\"\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">Approximately 44% of the variation (0.4397 is approximately 0.44) in the final-exam grades can be explained by the variation in the grades on the third exam using the best-fit regression line.<\/span><\/li>\n<li class=\"\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">Therefore, approximately 56% of the variation (1 \u2013 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam using the best-fit regression line. (This is seen as the scattering of the points about the line).<\/span><\/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-2344\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>The Regression Equation. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-3-the-regression-equation\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-3-the-regression-equation<\/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><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\/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":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"The Regression Equation\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-3-the-regression-equation\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"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-2344","chapter","type-chapter","status-publish","hentry"],"part":303,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2344","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":5,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2344\/revisions"}],"predecessor-version":[{"id":4015,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2344\/revisions\/4015"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/303"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2344\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2344"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2344"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2344"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2344"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}