{"id":283,"date":"2023-02-20T17:14:27","date_gmt":"2023-02-20T17:14:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/assessing-the-fit-of-a-line-dig-deeper\/"},"modified":"2025-05-11T23:25:40","modified_gmt":"2025-05-11T23:25:40","slug":"assessing-the-fit-of-a-line-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/assessing-the-fit-of-a-line-fresh-take\/","title":{"raw":"Assessing the Fit of a Line: Fresh Take","rendered":"Assessing the Fit of a Line: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Describe the connection between the residual and the position of a data point relative to the line of best fit.<\/li>\r\n\t<li>Create and use a residual plot to identify influential points and determine the most appropriate regression model.<\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine the reliability of predictions from the line of best fit using the residuals and standard error of the residuals&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Determine the reliability of predictions from the line of best fit using the residuals and standard error of the residuals.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>We have used scatterplots of data and constructed lines of best fit to describe the relationship in bivariate data. We have also learned about the correlation coefficient [latex]r[\/latex] and the coefficient of determination [latex]R^2[\/latex]. Let's review some of these ideas before looking into residuals.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1311[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1312[\/ohm2_question]<\/section>\r\n<h2>Residuals<\/h2>\r\n<p>The following are different ways of expressing the same idea:<\/p>\r\n<ul>\r\n\t<li>The residual is the difference between the observed value and the predicted value.<\/li>\r\n\t<li>The residual is the vertical distance between the observed value and the predicted value.<\/li>\r\n<\/ul>\r\n<p>In all cases, in order to calculate the residual, you must subtract the predicted value from the observed value.<\/p>\r\n<section class=\"textbox example\">\r\n<h3>Residual Plots<\/h3>\r\n<p>The graph below shows a scatterplot and the regression line for a set of ten points. The blue points represent our original data set (our observed values). The red points, lying directly on the regression line, are the predicted values.<\/p>\r\n\r\n[caption id=\"attachment_1221\" align=\"aligncenter\" width=\"327\"]<img class=\"wp-image-1221 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/30184146\/m3_examining_relationships_topic_3_3_asses_fit_line_u1_m2_assessfitline_image3.gif\" alt=\"The appropriate alternative text can be seen in the description of the image.\" width=\"327\" height=\"219\" \/> Figure 1. The vertical arrows represent residuals\u2014upward arrows indicate positive residuals, and downward arrows indicate negative residuals.[\/caption]\r\n\r\n<p>The vertical arrows from the predicted to observed values represent the residuals. The up arrows correspond to positive residuals, and the down arrows correspond to negative residuals.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1316[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1317[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1318[\/ohm2_question]<\/section>\r\n<p><span style=\"font-size: 1rem; font-weight: normal; text-align: initial; color: #373d3f;\">A residual can be positive or negative. A data point has a positive residual when the data point is located above the line of best fit. A data point has a negative residual when the data point is located below the line of best fit.<\/span><\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1321[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1322[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1323[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1319[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Describe the connection between the residual and the position of a data point relative to the line of best fit.<\/li>\n<li>Create and use a residual plot to identify influential points and determine the most appropriate regression model.<\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine the reliability of predictions from the line of best fit using the residuals and standard error of the residuals&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Determine the reliability of predictions from the line of best fit using the residuals and standard error of the residuals.<\/span><\/li>\n<\/ul>\n<\/section>\n<p>We have used scatterplots of data and constructed lines of best fit to describe the relationship in bivariate data. We have also learned about the correlation coefficient [latex]r[\/latex] and the coefficient of determination [latex]R^2[\/latex]. Let&#8217;s review some of these ideas before looking into residuals.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1311\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1311&theme=lumen&iframe_resize_id=ohm1311&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1312\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1312&theme=lumen&iframe_resize_id=ohm1312&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Residuals<\/h2>\n<p>The following are different ways of expressing the same idea:<\/p>\n<ul>\n<li>The residual is the difference between the observed value and the predicted value.<\/li>\n<li>The residual is the vertical distance between the observed value and the predicted value.<\/li>\n<\/ul>\n<p>In all cases, in order to calculate the residual, you must subtract the predicted value from the observed value.<\/p>\n<section class=\"textbox example\">\n<h3>Residual Plots<\/h3>\n<p>The graph below shows a scatterplot and the regression line for a set of ten points. The blue points represent our original data set (our observed values). The red points, lying directly on the regression line, are the predicted values.<\/p>\n<figure id=\"attachment_1221\" aria-describedby=\"caption-attachment-1221\" style=\"width: 327px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1221 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/30184146\/m3_examining_relationships_topic_3_3_asses_fit_line_u1_m2_assessfitline_image3.gif\" alt=\"The appropriate alternative text can be seen in the description of the image.\" width=\"327\" height=\"219\" \/><figcaption id=\"caption-attachment-1221\" class=\"wp-caption-text\">Figure 1. The vertical arrows represent residuals\u2014upward arrows indicate positive residuals, and downward arrows indicate negative residuals.<\/figcaption><\/figure>\n<p>The vertical arrows from the predicted to observed values represent the residuals. The up arrows correspond to positive residuals, and the down arrows correspond to negative residuals.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1316\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1316&theme=lumen&iframe_resize_id=ohm1316&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1317\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1317&theme=lumen&iframe_resize_id=ohm1317&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1318\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1318&theme=lumen&iframe_resize_id=ohm1318&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p><span style=\"font-size: 1rem; font-weight: normal; text-align: initial; color: #373d3f;\">A residual can be positive or negative. A data point has a positive residual when the data point is located above the line of best fit. A data point has a negative residual when the data point is located below the line of best fit.<\/span><\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1321\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1321&theme=lumen&iframe_resize_id=ohm1321&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1322\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1322&theme=lumen&iframe_resize_id=ohm1322&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1323\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1323&theme=lumen&iframe_resize_id=ohm1323&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1319\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1319&theme=lumen&iframe_resize_id=ohm1319&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":36,"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\/283"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/283\/revisions"}],"predecessor-version":[{"id":6667,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/283\/revisions\/6667"}],"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\/283\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=283"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=283"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=283"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}