{"id":273,"date":"2023-02-20T17:14:19","date_gmt":"2023-02-20T17:14:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/coefficient-of-determination-dig-deeper\/"},"modified":"2025-05-11T23:21:12","modified_gmt":"2025-05-11T23:21:12","slug":"coefficient-of-determination-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/coefficient-of-determination-fresh-take\/","title":{"raw":"Coefficient of Determination: Fresh Take","rendered":"Coefficient of Determination: Fresh Take"},"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>Decimals and Percentages<\/h2>\r\n<p>The variable [latex]R^2[\/latex] is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form.\u00a0Depending 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]\u00a0as a percentage, it is important to be able to convert between the two forms.<\/p>\r\n<section class=\"textbox example\"><strong>(a)<\/strong> Convert [latex]0.489[\/latex] to a percentage.[reveal-answer q=\"962164\"]Show solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"962164\"]If you are given a number as a decimal and want to convert it to a percentage, multiply the number by 100 and use the % symbol afterward. For example, the decimal [latex]0.489[\/latex] is converted to a percentage as follows:\r\n\r\n<p style=\"text-align: center;\">[latex]0.489 \\rightarrow 0.489 \\times 100 \\% \\rightarrow 48.9 \\%[\/latex][\/hidden-answer]<\/p>\r\n<p><strong>(b)\u00a0<\/strong>Convert\u00a0[latex]67\\%[\/latex] to a decimal.<\/p>\r\n<p>[reveal-answer q=\"340779\"]Show solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"340779\"]If you are given a number as a percentage and want to convert it to a decimal, divide the percentage by 100 and remove the % symbol. For example, the percentage [latex]67\\%[\/latex] is converted to a decimal as follows:<\/p>\r\n<p style=\"text-align: center;\">[latex]67 \\% \\rightarrow 67\\div 100\\rightarrow 0.67[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1288[\/ohm2_question]<\/section>\r\n<h2>Squaring Numbers<\/h2>\r\n<p>Since the coefficient of determination is equal to the square of the correlation coefficient, we will examine the operation of squaring. Squaring a number is the same as multiplying that number by itself.<\/p>\r\n<section class=\"textbox example\">Square the following numbers: [latex]5^2[\/latex], [latex](-2)^2[\/latex], [latex]1^2[\/latex][reveal-answer q=\"592187\"]Show solution[\/reveal-answer][hidden-answer a=\"592187\"][latex]5^2=5\\cdot 5=25[\/latex][latex](-2)^2=(-2)\\cdot (-2)=4[\/latex][latex]1^2=1\\cdot 1=1[\/latex][\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1289[\/ohm2_question]<\/section>\r\n<p>Do you suppose it may be true that squaring a number always yields a larger number as in the examples above? Let's explore that idea by trying to find an example in which squaring a number yields either a smaller or the same number in return.<\/p>\r\n<p>A <strong>counterexample<\/strong> is an example that contradicts or disproves a general statement. For instance, suppose someone proposes the following general statement: \u201cAll people like ice cream\u201d. A counterexample to this statement would be someone who doesn\u2019t like ice cream. It only takes one counterexample to show that a statement is false.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1290[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1291[\/ohm2_question]<\/section>\r\n<h2>Values of\u00a0the Coefficient of Determination, [latex]R^{2}[\/latex]<\/h2>\r\n<section class=\"textbox interact\">Let's find and interpret the coefficient of determination ([latex]R^2[\/latex]) using our statistical tool. Create a scatterplot with at least five data points that lie on (or very close to) a line with non-zero slope.\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 <strong>Initial Relationship<\/strong>, select <strong>Draw Your Own (Click in Graph)<\/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>Check the boxes for <strong>Linear Regression Line<\/strong>, <strong>Show Correlation Coefficient [latex]r[\/latex]<\/strong>, and <strong>Squared Correlation Coefficient [latex]r^{2}[\/latex]<\/strong>.<\/p>\r\n\r\n[caption id=\"attachment_1187\" align=\"aligncenter\" width=\"486\"]<img class=\"wp-image-1187\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/27205204\/Picture116.png\" alt=\"\" width=\"486\" height=\"334\" \/> Figure 1. The blue line is the best fit\u2014but the points are so spread out, it barely tells us anything useful about the relationship between x and y.[\/caption]\r\n\r\n<p>[reveal-answer q=\"27878\"]Sample scatterplots[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"27878\"]<\/p>\r\n\r\n[caption id=\"attachment_1188\" align=\"aligncenter\" width=\"484\"]<img class=\"wp-image-1188\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/27205434\/Picture117.png\" alt=\"\" width=\"484\" height=\"336\" \/> Figure 2. A curved pattern is visible in the data, but the line of best fit doesn\u2019t follow it, which is why the coefficient of determination is nearly zero.[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/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]1292[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1293[\/ohm2_question]<\/section>\r\n<section>\r\n<h2>Correlation Coefficient ([latex]r[\/latex]) vs. Coefficient of Determination ([latex]R^{2}[\/latex])<\/h2>\r\n<p><strong>The variable [latex]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.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1284[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1285[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1286[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1287[\/ohm2_question]<\/section>\r\n<h2><strong>Let\u2019s Summarize<\/strong><\/h2>\r\n<p>When the\u00a0form of a relationship is linear, we use the correlation coefficient, [latex]r[\/latex], to measure the strength and direction of the linear relationship. The correlation coefficient, [latex]r[\/latex], ranges between [latex]\u22121[\/latex] and [latex]1[\/latex].\u00a0The square of the correlation,\u00a0[latex]R^2[\/latex], is the coefficient of determination. [latex]R^2[\/latex] is the proportion of the variation in the response variable that is explained by the least-squares regression line. The\u00a0coefficient of determination ranges between [latex]0\\%[\/latex] and [latex]100\\%[\/latex].<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1296[\/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>Decimals and Percentages<\/h2>\n<p>The variable [latex]R^2[\/latex] is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form.\u00a0Depending 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]\u00a0as a percentage, it is important to be able to convert between the two forms.<\/p>\n<section class=\"textbox example\"><strong>(a)<\/strong> Convert [latex]0.489[\/latex] to a percentage.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q962164\">Show solution<\/button><\/p>\n<div id=\"q962164\" class=\"hidden-answer\" style=\"display: none\">If you are given a number as a decimal and want to convert it to a percentage, multiply the number by 100 and use the % symbol afterward. For example, the decimal [latex]0.489[\/latex] is converted to a percentage as follows:<\/p>\n<p style=\"text-align: center;\">[latex]0.489 \\rightarrow 0.489 \\times 100 \\% \\rightarrow 48.9 \\%[\/latex]<\/div>\n<\/div>\n<p><strong>(b)\u00a0<\/strong>Convert\u00a0[latex]67\\%[\/latex] to a decimal.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q340779\">Show solution<\/button><\/p>\n<div id=\"q340779\" class=\"hidden-answer\" style=\"display: none\">If you are given a number as a percentage and want to convert it to a decimal, divide the percentage by 100 and remove the % symbol. For example, the percentage [latex]67\\%[\/latex] is converted to a decimal as follows:<\/p>\n<p style=\"text-align: center;\">[latex]67 \\% \\rightarrow 67\\div 100\\rightarrow 0.67[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1288\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1288&theme=lumen&iframe_resize_id=ohm1288&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Squaring Numbers<\/h2>\n<p>Since the coefficient of determination is equal to the square of the correlation coefficient, we will examine the operation of squaring. Squaring a number is the same as multiplying that number by itself.<\/p>\n<section class=\"textbox example\">Square the following numbers: [latex]5^2[\/latex], [latex](-2)^2[\/latex], [latex]1^2[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q592187\">Show solution<\/button><\/p>\n<div id=\"q592187\" class=\"hidden-answer\" style=\"display: none\">[latex]5^2=5\\cdot 5=25[\/latex][latex](-2)^2=(-2)\\cdot (-2)=4[\/latex][latex]1^2=1\\cdot 1=1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1289\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1289&theme=lumen&iframe_resize_id=ohm1289&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Do you suppose it may be true that squaring a number always yields a larger number as in the examples above? Let&#8217;s explore that idea by trying to find an example in which squaring a number yields either a smaller or the same number in return.<\/p>\n<p>A <strong>counterexample<\/strong> is an example that contradicts or disproves a general statement. For instance, suppose someone proposes the following general statement: \u201cAll people like ice cream\u201d. A counterexample to this statement would be someone who doesn\u2019t like ice cream. It only takes one counterexample to show that a statement is false.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1290\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1290&theme=lumen&iframe_resize_id=ohm1290&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1291\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1291&theme=lumen&iframe_resize_id=ohm1291&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Values of\u00a0the Coefficient of Determination, [latex]R^{2}[\/latex]<\/h2>\n<section class=\"textbox interact\">Let&#8217;s find and interpret the coefficient of determination ([latex]R^2[\/latex]) using our statistical tool. Create a scatterplot with at least five data points that lie on (or very close to) a line with non-zero slope.<\/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 <strong>Initial Relationship<\/strong>, select <strong>Draw Your Own (Click in Graph)<\/strong>.<\/p>\n<p class=\"para\" style=\"text-indent: -0.25in; margin: 6pt 0in 6pt 0.25in; padding-left: 40px;\"><strong>Step 2: <\/strong>Check the boxes for <strong>Linear Regression Line<\/strong>, <strong>Show Correlation Coefficient [latex]r[\/latex]<\/strong>, and <strong>Squared Correlation Coefficient [latex]r^{2}[\/latex]<\/strong>.<\/p>\n<figure id=\"attachment_1187\" aria-describedby=\"caption-attachment-1187\" style=\"width: 486px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1187\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/27205204\/Picture116.png\" alt=\"\" width=\"486\" height=\"334\" \/><figcaption id=\"caption-attachment-1187\" class=\"wp-caption-text\">Figure 1. The blue line is the best fit\u2014but the points are so spread out, it barely tells us anything useful about the relationship between x and y.<\/figcaption><\/figure>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q27878\">Sample scatterplots<\/button><\/p>\n<div id=\"q27878\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_1188\" aria-describedby=\"caption-attachment-1188\" style=\"width: 484px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1188\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/27205434\/Picture117.png\" alt=\"\" width=\"484\" height=\"336\" \/><figcaption id=\"caption-attachment-1188\" class=\"wp-caption-text\">Figure 2. A curved pattern is visible in the data, but the line of best fit doesn\u2019t follow it, which is why the coefficient of determination is nearly zero.<\/figcaption><\/figure>\n<\/div>\n<\/div>\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=\"ohm1292\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1292&theme=lumen&iframe_resize_id=ohm1292&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1293\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1293&theme=lumen&iframe_resize_id=ohm1293&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Correlation Coefficient ([latex]r[\/latex]) vs. Coefficient of Determination ([latex]R^{2}[\/latex])<\/h2>\n<p><strong>The variable [latex]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.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1284\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1284&theme=lumen&iframe_resize_id=ohm1284&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1285\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1285&theme=lumen&iframe_resize_id=ohm1285&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1286\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1286&theme=lumen&iframe_resize_id=ohm1286&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1287\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1287&theme=lumen&iframe_resize_id=ohm1287&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2><strong>Let\u2019s Summarize<\/strong><\/h2>\n<p>When the\u00a0form of a relationship is linear, we use the correlation coefficient, [latex]r[\/latex], to measure the strength and direction of the linear relationship. The correlation coefficient, [latex]r[\/latex], ranges between [latex]\u22121[\/latex] and [latex]1[\/latex].\u00a0The square of the correlation,\u00a0[latex]R^2[\/latex], is the coefficient of determination. [latex]R^2[\/latex] is the proportion of the variation in the response variable that is explained by the least-squares regression line. The\u00a0coefficient of determination ranges between [latex]0\\%[\/latex] and [latex]100\\%[\/latex].<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1296\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1296&theme=lumen&iframe_resize_id=ohm1296&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":27,"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":"fresh_take","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\/273"}],"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":10,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/273\/revisions"}],"predecessor-version":[{"id":6661,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/273\/revisions\/6661"}],"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\/273\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=273"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=273"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=273"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=273"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}