{"id":1162,"date":"2015-11-12T18:35:31","date_gmt":"2015-11-12T18:35:31","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1162"},"modified":"2015-11-12T18:35:31","modified_gmt":"2015-11-12T18:35:31","slug":"distinguish-between-linear-and-nonlinear-relations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/distinguish-between-linear-and-nonlinear-relations\/","title":{"raw":"Distinguish between linear and nonlinear relations","rendered":"Distinguish between linear and nonlinear relations"},"content":{"raw":"<section id=\"fs-id1165137594434\" data-depth=\"1\"><p id=\"fs-id1165135160844\">As we saw in \"<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/find-the-line-of-best-fit\/\" target=\"_blank\">Find the line of best fit<\/a>\"\u00a0with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the <strong>correlation coefficient<\/strong>, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a \"diagnostic on\" selection to find the correlation coefficient, which mathematicians label as <em>r<\/em>. The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.<\/p>\nWe should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of <em>r<\/em>\u00a0and the graph of the data, the image below\u00a0shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.\n\n[caption id=\"\" align=\"aligncenter\" width=\"901\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201206\/CNX_Precalc_Figure_02_04_0072.jpg\" alt=\"A series of scatterplot graphs. Some are linear and some are not.\" width=\"901\" height=\"401\" data-media-type=\"image\/jpg\"\/><b>Figure 7.<\/b> Plotted data and related correlation coefficients. (credit: \"DenisBoigelot,\" Wikimedia Commons)[\/caption]\n\n<div id=\"fs-id1165137443573\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Correlation Coefficient<\/h3>\n<p id=\"fs-id1165137416387\">The <strong>correlation coefficient<\/strong> is a value, <em>r<\/em>, between \u20131 and 1.<\/p>\n\n<ul id=\"eip-id1165133093343\"><li><em data-effect=\"italics\">r<\/em> &gt; 0 suggests a positive (increasing) relationship<\/li>\n\t<li><em data-effect=\"italics\">r<\/em> &lt; 0 suggests a negative (decreasing) relationship<\/li>\n\t<li>The closer the value is to 0, the more scattered the data.<\/li>\n\t<li>The closer the value is to 1 or \u20131, the less scattered the data is.<\/li>\n<\/ul><\/div>\n<div id=\"Example_02_04_05\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137387185\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137680583\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 5: Finding a Correlation Coefficient<\/h3>\n<p id=\"fs-id1165137734908\">Calculate the correlation coefficient for cricket-chirp data in the table below.<\/p>\n\n<table summary=\"Two rows and ten columns. The first row is labeled, 'chirps'. The second row is labeled is labeled, 'Temp'. Reading the remaining rows as ordered pairs (i.e., (chirps, Temp), we have the following values: (44, 80.5), (35, 70.5), (20.4, 57), (33, 66), (31, 68), (35, 72), (18.5, 52), (37, 73.5) and (26, 53).\"><colgroup><col data-width=\"10%\"\/><col data-width=\"10%\"\/><col data-width=\"10%\"\/><col data-width=\"10%\"\/><col data-width=\"10%\"\/><col data-width=\"10%\"\/><col data-width=\"10%\"\/><col data-width=\"10%\"\/><col data-width=\"10%\"\/><col data-width=\"10%\"\/><\/colgroup><tbody><tr><td><strong>Chirps<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr><tr><td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr><\/tbody><\/table><\/div>\n<div id=\"fs-id1165137639432\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137471985\">Because the data appear to follow a linear pattern, we can use technology to calculate <em>r<\/em>. Enter the inputs and corresponding outputs and select the Linear Regression. The calculator will also provide you with the correlation coefficient, <em>r\u00a0<\/em>= 0.9509. This value is very close to 1, which suggests a strong increasing linear relationship.<\/p>\n<p id=\"fs-id1165137473276\">Note: For some calculators, the Diagnostics must be turned \"on\" in order to get the correlation coefficient when linear regression is performed: [2nd]&gt;[0]&gt;[alpha][x\u20131], then scroll to DIAGNOSTICSON.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/section><section id=\"fs-id1165137731906\" data-depth=\"1\"><div id=\"fs-id1165137724542\" class=\"note precalculus try\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"\/>\n<\/section>","rendered":"<section id=\"fs-id1165137594434\" data-depth=\"1\">\n<p id=\"fs-id1165135160844\">As we saw in &#8220;<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/find-the-line-of-best-fit\/\" target=\"_blank\">Find the line of best fit<\/a>&#8221;\u00a0with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the <strong>correlation coefficient<\/strong>, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a &#8220;diagnostic on&#8221; selection to find the correlation coefficient, which mathematicians label as <em>r<\/em>. The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.<\/p>\n<p>We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of <em>r<\/em>\u00a0and the graph of the data, the image below\u00a0shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.<\/p>\n<div style=\"width: 911px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201206\/CNX_Precalc_Figure_02_04_0072.jpg\" alt=\"A series of scatterplot graphs. Some are linear and some are not.\" width=\"901\" height=\"401\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7.<\/b> Plotted data and related correlation coefficients. (credit: &#8220;DenisBoigelot,&#8221; Wikimedia Commons)<\/p>\n<\/div>\n<div id=\"fs-id1165137443573\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Correlation Coefficient<\/h3>\n<p id=\"fs-id1165137416387\">The <strong>correlation coefficient<\/strong> is a value, <em>r<\/em>, between \u20131 and 1.<\/p>\n<ul id=\"eip-id1165133093343\">\n<li><em data-effect=\"italics\">r<\/em> &gt; 0 suggests a positive (increasing) relationship<\/li>\n<li><em data-effect=\"italics\">r<\/em> &lt; 0 suggests a negative (decreasing) relationship<\/li>\n<li>The closer the value is to 0, the more scattered the data.<\/li>\n<li>The closer the value is to 1 or \u20131, the less scattered the data is.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_02_04_05\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137387185\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137680583\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 5: Finding a Correlation Coefficient<\/h3>\n<p id=\"fs-id1165137734908\">Calculate the correlation coefficient for cricket-chirp data in the table below.<\/p>\n<table summary=\"Two rows and ten columns. The first row is labeled, 'chirps'. The second row is labeled is labeled, 'Temp'. Reading the remaining rows as ordered pairs (i.e., (chirps, Temp), we have the following values: (44, 80.5), (35, 70.5), (20.4, 57), (33, 66), (31, 68), (35, 72), (18.5, 52), (37, 73.5) and (26, 53).\">\n<colgroup>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/>\n<col data-width=\"10%\" \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137639432\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137471985\">Because the data appear to follow a linear pattern, we can use technology to calculate <em>r<\/em>. Enter the inputs and corresponding outputs and select the Linear Regression. The calculator will also provide you with the correlation coefficient, <em>r\u00a0<\/em>= 0.9509. This value is very close to 1, which suggests a strong increasing linear relationship.<\/p>\n<p id=\"fs-id1165137473276\">Note: For some calculators, the Diagnostics must be turned &#8220;on&#8221; in order to get the correlation coefficient when linear regression is performed: [2nd]&gt;[0]&gt;[alpha][x\u20131], then scroll to DIAGNOSTICSON.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137731906\" data-depth=\"1\">\n<div id=\"fs-id1165137724542\" class=\"note precalculus try\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<\/div>\n<\/section>\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-1162\">\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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":276,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1162","chapter","type-chapter","status-publish","hentry"],"part":1151,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1162","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1162\/revisions"}],"predecessor-version":[{"id":2441,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1162\/revisions\/2441"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1151"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1162\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=1162"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1162"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1162"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=1162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}