{"id":169,"date":"2017-04-15T03:18:26","date_gmt":"2017-04-15T03:18:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/linear-relationships-2-of-4\/"},"modified":"2017-05-10T19:35:28","modified_gmt":"2017-05-10T19:35:28","slug":"linear-relationships-2-of-4","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-hccc-wm-concepts-statistics\/chapter\/linear-relationships-2-of-4\/","title":{"raw":"Linear Relationships (2 of 4)","rendered":"Linear Relationships (2 of 4)"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use a correlation coefficient to describe the direction and strength of a linear relationship. Recognize its limitations as a measure of the relationship between two quantitative variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>The Correlation Coefficient (<em>r<\/em>)<\/h3>\r\nThe numerical measure that assesses the strength of a linear relationship is called the <strong>correlation coefficient<\/strong> and is denoted by <em><strong>r<\/strong><\/em>. In this section, we\r\n<ul>\r\n \t<li>define <em>r<\/em>.<\/li>\r\n \t<li>discuss the calculation of <em>r<\/em>.<\/li>\r\n \t<li>explain how to interpret the value of <em>r<\/em>.<\/li>\r\n \t<li>talk about some of the properties of <em>r<\/em>.<\/li>\r\n<\/ul>\r\n<div class=\"oli_definition\">\r\n<dl>\r\n \t<dt>Correlation coefficient (r)<\/dt>\r\n \t<dd><span class=\"def-lbl\">(Definition)<\/span>\r\n<div class=\"meaning\">\r\n\r\nThe correlation coefficient (<em>r<\/em>) is a numeric measure that measures the <em>strength<\/em> and <em>direction<\/em> of a <em>linear<\/em> relationship between two quantitative variables.\r\n\r\n<strong>Calculation: <\/strong><em>r<\/em> is calculated using the following formula: [latex]r=\\frac{\u2211\\left(\\frac{x-\\stackrel{\u00af}{x}}{{s}_{x}}\\right)\\left(\\frac{y-\\stackrel{\u00af}{y}}{{s}_{y}}\\right)}{n-1}[\/latex]\r\n\r\n<\/div><\/dd>\r\n<\/dl>\r\n<\/div>\r\nwhere n is the sample size; x is a data value for the explanatory variable; [latex]\\stackrel{\u00af}{x}[\/latex] is the mean of the x-values; [latex]{s}_{x}[\/latex] is the standard deviation of the x-values; similarly, for the terms involving y. To calculate r, the term [latex]\\left(\\frac{x-\\stackrel{\u00af}{x}}{{s}_{x}}\\right)\\left(\\frac{y-\\stackrel{\u00af}{y}}{{s}_{y}}\\right)[\/latex] is calculated for each individual. These terms are added together, then the sum is divided by (n\u20131).\r\n\r\nHowever, the calculation of <em>r<\/em> is not the focus of this course. We use a statistics package to calculate the correlation coefficient for us, and the emphasis of this course is on the <em>interpretation<\/em> of <em>r<\/em>\u2019s value.\r\n<h3>Interpretation<\/h3>\r\nOnce we obtain the value of <em>r<\/em>, its interpretation with respect to the strength of linear relationships is quite simple, as this walkthrough illustrates:\r\n\r\nhttps:\/\/youtu.be\/Bt-Ey2ebfvs\r\n\r\nUse the simulation below to investigate how the value of [latex]r[\/latex] relates to the direction and strength of the relationship between the two variables in the scatterplot.\r\n\r\nIn the simulation, use the slider bar at the top of the simulation to change the value of the correlation coefficient (<em>r<\/em>) between \u22121 and 1. Observe the effect on the scatterplot. Click on the \"Switch Sign\" button to jump between positive and negative relationships of the same strength.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/linear_relationships_2_of_4\/linear_relationships_scatterplot.html\" target=\"new\">Click here to open this simulation in its own window.<\/a>\r\n\r\n<center><iframe id=\"_i_2f\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/linear_relationships_2_of_4\/linear_relationships_scatterplot.html\" width=\"800\" height=\"750\"><\/iframe><\/center>&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3477\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3478\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3479\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3480\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3481\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3482\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use a correlation coefficient to describe the direction and strength of a linear relationship. Recognize its limitations as a measure of the relationship between two quantitative variables.<\/li>\n<\/ul>\n<\/div>\n<h3>The Correlation Coefficient (<em>r<\/em>)<\/h3>\n<p>The numerical measure that assesses the strength of a linear relationship is called the <strong>correlation coefficient<\/strong> and is denoted by <em><strong>r<\/strong><\/em>. In this section, we<\/p>\n<ul>\n<li>define <em>r<\/em>.<\/li>\n<li>discuss the calculation of <em>r<\/em>.<\/li>\n<li>explain how to interpret the value of <em>r<\/em>.<\/li>\n<li>talk about some of the properties of <em>r<\/em>.<\/li>\n<\/ul>\n<div class=\"oli_definition\">\n<dl>\n<dt>Correlation coefficient (r)<\/dt>\n<dd><span class=\"def-lbl\">(Definition)<\/span><\/p>\n<div class=\"meaning\">\n<p>The correlation coefficient (<em>r<\/em>) is a numeric measure that measures the <em>strength<\/em> and <em>direction<\/em> of a <em>linear<\/em> relationship between two quantitative variables.<\/p>\n<p><strong>Calculation: <\/strong><em>r<\/em> is calculated using the following formula: [latex]r=\\frac{\u2211\\left(\\frac{x-\\stackrel{\u00af}{x}}{{s}_{x}}\\right)\\left(\\frac{y-\\stackrel{\u00af}{y}}{{s}_{y}}\\right)}{n-1}[\/latex]<\/p>\n<\/div>\n<\/dd>\n<\/dl>\n<\/div>\n<p>where n is the sample size; x is a data value for the explanatory variable; [latex]\\stackrel{\u00af}{x}[\/latex] is the mean of the x-values; [latex]{s}_{x}[\/latex] is the standard deviation of the x-values; similarly, for the terms involving y. To calculate r, the term [latex]\\left(\\frac{x-\\stackrel{\u00af}{x}}{{s}_{x}}\\right)\\left(\\frac{y-\\stackrel{\u00af}{y}}{{s}_{y}}\\right)[\/latex] is calculated for each individual. These terms are added together, then the sum is divided by (n\u20131).<\/p>\n<p>However, the calculation of <em>r<\/em> is not the focus of this course. We use a statistics package to calculate the correlation coefficient for us, and the emphasis of this course is on the <em>interpretation<\/em> of <em>r<\/em>\u2019s value.<\/p>\n<h3>Interpretation<\/h3>\n<p>Once we obtain the value of <em>r<\/em>, its interpretation with respect to the strength of linear relationships is quite simple, as this walkthrough illustrates:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Interpreting the value of r\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Bt-Ey2ebfvs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Use the simulation below to investigate how the value of [latex]r[\/latex] relates to the direction and strength of the relationship between the two variables in the scatterplot.<\/p>\n<p>In the simulation, use the slider bar at the top of the simulation to change the value of the correlation coefficient (<em>r<\/em>) between \u22121 and 1. Observe the effect on the scatterplot. Click on the &#8220;Switch Sign&#8221; button to jump between positive and negative relationships of the same strength.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/linear_relationships_2_of_4\/linear_relationships_scatterplot.html\" target=\"new\">Click here to open this simulation in its own window.<\/a><\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" id=\"_i_2f\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/linear_relationships_2_of_4\/linear_relationships_scatterplot.html\" width=\"800\" height=\"750\"><\/iframe><\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>\t<iframe id=\"lumen_assessment_3477\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3477&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3477\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3478\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3478&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3478\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3479\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3479&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3479\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3480\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3480&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3480\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3481\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3481&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3481\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3482\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3482&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3482\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\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-169\">\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>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":163,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in 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