{"id":268,"date":"2023-02-20T17:14:16","date_gmt":"2023-02-20T17:14:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/coefficient-of-determination-learn-it-3\/"},"modified":"2025-05-11T23:19:52","modified_gmt":"2025-05-11T23:19:52","slug":"coefficient-of-determination-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/coefficient-of-determination-learn-it-3\/","title":{"raw":"Coefficient of Determination: Learn It 3","rendered":"Coefficient of Determination: Learn It 3"},"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<p>Now that you have developed some intuition about\u00a0[latex]R^2[\/latex] and the shape of a plot, let's explore how the spread of a plot affects the value of\u00a0[latex]R^2[\/latex].<\/p>\r\n<section class=\"textbox interact\">Steps to explore the [latex]R^2[\/latex] value in the statistical tool:\r\n\r\n<p class=\"para\" style=\"text-indent: -0.25in; margin: 6pt 0in 6pt 0.25in; padding-left: 40px;\"><strong>Step 1: <\/strong>From the drop-down menu, select <strong>Linear Relationship<\/strong>.<\/p>\r\n<p class=\"para\" style=\"text-indent: -0.25in; margin: 6pt 0in 6pt 0.25in; padding-left: 40px;\"><strong>Step 2: <\/strong>Select the boxes that will display [latex]r[\/latex] and [latex]r^2[\/latex].<\/p>\r\n<p class=\"para\" style=\"text-indent: -0.25in; margin: 6pt 0in 6pt 0.25in; padding-left: 40px;\"><strong>Step 3: <\/strong>Toggle back and forth between the different options for spread (small, medium, or large) and click the <strong>Refresh<\/strong>\u00a0button, if needed. Make a note for yourself about how [latex]r^2[\/latex] changes as you change the spread from large to medium to small. As you do this, note that squaring [latex]r[\/latex] does, in fact, yield [latex]r^2[\/latex].<\/p>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/explorelinreg\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/explorelinreg\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1304[\/ohm2_question]<\/section>\r\n<section>\r\n<h2>Possible values of [latex]R^2[\/latex]<\/h2>\r\n<p>Depending on the tools you use, [latex]R^{2}[\/latex]\u00a0may be expressed as a decimal or as a percentage. Even though the tool expresses [latex]R^{2}[\/latex]\u00a0using a percentage, it is important to be able to read values of [latex]R^{2}[\/latex]\u00a0in both decimal and percentage form.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1309[\/ohm2_question]<\/section>\r\n<p>Recall that [latex]R^{2}[\/latex] is the proportion of the variation in the response variable that can be explained by its linear relationship with the explanatory variable. It doesn't make sense for this proportion to be negative. Can more than [latex]100\\%[\/latex] of the variation in the response variable be explained by its linear relationship with the explanatory variable? No, it can't. For these reasons, the value of [latex]R^{2}[\/latex] will always be between [latex]0[\/latex] and [latex]1[\/latex] or, in percentage form, [latex]0\\%[\/latex] and [latex]100\\%[\/latex].<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1310[\/ohm2_question]<\/section>\r\n<\/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<p>Now that you have developed some intuition about\u00a0[latex]R^2[\/latex] and the shape of a plot, let&#8217;s explore how the spread of a plot affects the value of\u00a0[latex]R^2[\/latex].<\/p>\n<section class=\"textbox interact\">Steps to explore the [latex]R^2[\/latex] value in the statistical tool:<\/p>\n<p class=\"para\" style=\"text-indent: -0.25in; margin: 6pt 0in 6pt 0.25in; padding-left: 40px;\"><strong>Step 1: <\/strong>From the drop-down menu, select <strong>Linear Relationship<\/strong>.<\/p>\n<p class=\"para\" style=\"text-indent: -0.25in; margin: 6pt 0in 6pt 0.25in; padding-left: 40px;\"><strong>Step 2: <\/strong>Select the boxes that will display [latex]r[\/latex] and [latex]r^2[\/latex].<\/p>\n<p class=\"para\" style=\"text-indent: -0.25in; margin: 6pt 0in 6pt 0.25in; padding-left: 40px;\"><strong>Step 3: <\/strong>Toggle back and forth between the different options for spread (small, medium, or large) and click the <strong>Refresh<\/strong>\u00a0button, if needed. Make a note for yourself about how [latex]r^2[\/latex] changes as you change the spread from large to medium to small. As you do this, note that squaring [latex]r[\/latex] does, in fact, yield [latex]r^2[\/latex].<\/p>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/explorelinreg\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/explorelinreg\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1304\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1304&theme=lumen&iframe_resize_id=ohm1304&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Possible values of [latex]R^2[\/latex]<\/h2>\n<p>Depending on the tools you use, [latex]R^{2}[\/latex]\u00a0may be expressed as a decimal or as a percentage. Even though the tool expresses [latex]R^{2}[\/latex]\u00a0using a percentage, it is important to be able to read values of [latex]R^{2}[\/latex]\u00a0in both decimal and percentage form.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1309\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1309&theme=lumen&iframe_resize_id=ohm1309&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Recall that [latex]R^{2}[\/latex] is the proportion of the variation in the response variable that can be explained by its linear relationship with the explanatory variable. It doesn&#8217;t make sense for this proportion to be negative. Can more than [latex]100\\%[\/latex] of the variation in the response variable be explained by its linear relationship with the explanatory variable? No, it can&#8217;t. For these reasons, the value of [latex]R^{2}[\/latex] will always be between [latex]0[\/latex] and [latex]1[\/latex] or, in percentage form, [latex]0\\%[\/latex] and [latex]100\\%[\/latex].<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1310\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1310&theme=lumen&iframe_resize_id=ohm1310&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":23,"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\/268"}],"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":18,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/268\/revisions"}],"predecessor-version":[{"id":6659,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/268\/revisions\/6659"}],"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\/268\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=268"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=268"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=268"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=268"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}