{"id":267,"date":"2023-02-20T17:14:15","date_gmt":"2023-02-20T17:14:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/coefficient-of-determination-learn-it-2\/"},"modified":"2025-05-11T23:18:53","modified_gmt":"2025-05-11T23:18:53","slug":"coefficient-of-determination-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/coefficient-of-determination-learn-it-2\/","title":{"raw":"Coefficient of Determination: Learn It 2","rendered":"Coefficient of Determination: Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Describe how the slope, shape of the data, and the coefficient of determination are connected.<\/li>\r\n\t<li>Find [latex]R^2[\/latex] and describe how [latex]R^2[\/latex] describes the relationship in a data set.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>[latex]R^{2}[\/latex] and Scatterplot Shape<\/h2>\r\n<p>The <strong>coefficient of determination<\/strong>, [latex]R^2[\/latex], is a measure of the proportion of the variation of a response variable in linearly related bivariate data that can be explained by its relationship with the explanatory variable. You should understand that:<\/p>\r\n<ul>\r\n\t<li>[latex]R^2[\/latex] is equivalent to the square of the correlation coefficient\u00a0[latex]r[\/latex] and will always be a positive number between [latex]0\\%[\/latex] and\u00a0[latex]100\\%[\/latex].<\/li>\r\n\t<li>[latex]R^2[\/latex] should be interpreted and written as a percentage.<\/li>\r\n<\/ul>\r\n<p>Consider what you already understand about the shape and spread of a scatterplot.<\/p>\r\n<ul>\r\n\t<li>The strongest linear relationships appear in plots as data that is roughly linear in shape with data points that lie very close to some line.<\/li>\r\n\t<li>Weaker relationships may be very roughly linear in shape and more spread out, with data points that lie further from some line.<\/li>\r\n\t<li>Non-linear relationships have data points that either form other shapes or are randomly scattered across the plot.<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1303[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Describe how the slope, shape of the data, and the coefficient of determination are connected.<\/li>\n<li>Find [latex]R^2[\/latex] and describe how [latex]R^2[\/latex] describes the relationship in a data set.<\/li>\n<\/ul>\n<\/section>\n<h2>[latex]R^{2}[\/latex] and Scatterplot Shape<\/h2>\n<p>The <strong>coefficient of determination<\/strong>, [latex]R^2[\/latex], is a measure of the proportion of the variation of a response variable in linearly related bivariate data that can be explained by its relationship with the explanatory variable. You should understand that:<\/p>\n<ul>\n<li>[latex]R^2[\/latex] is equivalent to the square of the correlation coefficient\u00a0[latex]r[\/latex] and will always be a positive number between [latex]0\\%[\/latex] and\u00a0[latex]100\\%[\/latex].<\/li>\n<li>[latex]R^2[\/latex] should be interpreted and written as a percentage.<\/li>\n<\/ul>\n<p>Consider what you already understand about the shape and spread of a scatterplot.<\/p>\n<ul>\n<li>The strongest linear relationships appear in plots as data that is roughly linear in shape with data points that lie very close to some line.<\/li>\n<li>Weaker relationships may be very roughly linear in shape and more spread out, with data points that lie further from some line.<\/li>\n<li>Non-linear relationships have data points that either form other shapes or are randomly scattered across the plot.<\/li>\n<\/ul>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1303\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1303&theme=lumen&iframe_resize_id=ohm1303&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":225,"module-header":"learn_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\/267"}],"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\/12"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/267\/revisions"}],"predecessor-version":[{"id":6657,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/267\/revisions\/6657"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/225"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/267\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=267"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=267"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=267"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}