{"id":306,"date":"2021-07-14T15:59:11","date_gmt":"2021-07-14T15:59:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/scatter-plots\/"},"modified":"2023-12-05T09:46:06","modified_gmt":"2023-12-05T09:46:06","slug":"scatter-plots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/scatter-plots\/","title":{"raw":"Displaying Data: Scatterplots","rendered":"Displaying Data: Scatterplots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning outcomes<\/h3>\r\n<ul>\r\n \t<li>Create a scatterplot for bivariate data<\/li>\r\n \t<li>Given a scatterplot, describe the strength, direction, and form of the relationship between the variables<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Ordered Pairs as Data Points<\/h3>\r\nWhen expressing pairs of inputs and outputs on a graph, they take the form of (<em>input<\/em>, <em>output<\/em>). In scatter plots, the two variables relate to create each data point,\u00a0(<em>variable 1<\/em>, <em>variable 2<\/em>), but it is often not necessary to declare that one is dependent on the other. In the example below, each\u00a0<em>Age<\/em>\u00a0coordinate corresponds to a\u00a0<em>Final Exam Score <\/em>in the form (<em>age<\/em>,\u00a0<em>score<\/em>). Each corresponding pair is plotted on the graph.\r\n\r\n<\/div>\r\nBefore we take up the discussion of linear regression and correlation, we need to examine a way to display the relationship between two variables\u00a0<em>x<\/em> and <em>y<\/em>. The most common and easiest way is a <strong>scatter plot<\/strong>. The following example illustrates a scatter plot.\r\n<div class=\"textbox exercises\">\r\n<h3>Example 1<\/h3>\r\nIn Europe and Asia, m-commerce is popular. M-commerce users have special mobile phones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or soda from a machine to banking to checking sports scores on the Internet. For the years 2000 through 2004, was there a relationship between the year and the number of m-commerce users? Construct a scatter plot. Let\u00a0<em>x<\/em> = the year and let <em>y<\/em> = the number of m-commerce users, in millions.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>(year)<\/th>\r\n<th>(# of users)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>2000<\/td>\r\n<td>0.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2002<\/td>\r\n<td>20.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2003<\/td>\r\n<td>33.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2004<\/td>\r\n<td>47.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTable showing the number of m-commerce users (in millions) by year.\r\n\r\n<img class=\"aligncenter wp-image-2293 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145614\/d663df2918f84749b9d6e2700a905415c10a8a20.jpeg\" alt=\"This is a scatter plot for the data provided. The x-axis represents the year and the y-axis represents the number of m-commerce users in millions. There are four points plotted, at (2000, 0.5), (2002, 20.0), (2003, 33.0), (2004, 47.0).\" width=\"487\" height=\"323\" \/>\r\n\r\nScatter plot showing the number of m-commerce users (in millions) by year.\r\n\r\n<\/div>\r\n<h2>Creating a Scatter Plot USING THE TI-83, 83+, 84, 84+ CALCULATOR<\/h2>\r\n<ol>\r\n \t<li>Enter your X data into list L1 and your Y data into list L2.<\/li>\r\n \t<li>Press 2nd STATPLOT ENTER to use Plot 1. On the input screen for PLOT 1, highlight On and press ENTER. (Make sure the other plots are OFF).<\/li>\r\n \t<li>For TYPE: highlight the very first icon, which is the scatter plot, and press ENTER.<\/li>\r\n \t<li>For Xlist:, enter L1 ENTER and for Ylist: L2 ENTER.<\/li>\r\n \t<li>For Mark: it does not matter which symbol you highlight, but the square is the easiest to see. Press ENTER.<\/li>\r\n \t<li>Make sure there are no other equations that could be plotted. Press Y = and clear any equations out.<\/li>\r\n \t<li>Press the ZOOM key and then the number 9 (for menu item \"ZoomStat\"); the calculator will fit the window to the data. You can press WINDOW to see the scaling of the axes.<\/li>\r\n<\/ol>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it 1<\/h3>\r\nAmelia plays basketball for her high school. She wants to improve to play at the college level. She notices that the number of points she scores in a game goes up in response to the number of hours she practices her jump shot each week. She records the following data:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th><em>X<\/em> (hours practicing jump shot)<\/th>\r\n<th><em>Y<\/em> (points scored in a game)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>5<\/td>\r\n<td>15<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7<\/td>\r\n<td>22<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9<\/td>\r\n<td>28<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10<\/td>\r\n<td>31<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>11<\/td>\r\n<td>33<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>12<\/td>\r\n<td>36<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConstruct a scatter plot and state if what Amelia thinks appears to be true.\r\n[reveal-answer q=\"212678\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"212678\"]\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/03l5-fu4g757i#fixme#fixme#fixme\" alt=\"This is a scatter plot for the data provided. The x-axis is labeled in increments of 2 from 0 - 16. The y-axis is labeled in increments of 5 from 0 - 35.\" \/>\r\n\r\nYes, Amelia's assumption appears to be correct. The number of points Amelia scores per game goes up when she practices her jump shot more.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Scatter Plots<\/h3>\r\nA scatter plot is a graph of plotted points that may show a relationship between two sets of data.\r\n\r\n<\/div>\r\nA scatter plot shows the\u00a0<strong>direction<\/strong> of a relationship between the variables. A clear direction happens when there are either: High values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable.\r\n\r\nYou can determine the\u00a0<strong>strength<\/strong> of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or some other type of function. For a linear relationship, there is an exception. Consider a scatter plot where all the points fall on a horizontal line providing a \"perfect fit.\" The horizontal line would in fact show no relationship.\r\n\r\nWhen you look at a scatterplot, you want to notice the\u00a0<strong>overall pattern<\/strong> and any <strong>deviations<\/strong> from the pattern. The following scatterplot examples illustrate these concepts.\r\n\r\n<img class=\"aligncenter wp-image-2294 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145834\/8a9a215df5f5bba8343f5fb4062e86014afb98ba.jpeg\" alt=\"The first graph is a scatter plot with 6 points plotted. The points form a pattern that moves upward to the right, almost in a straight line. The second graph is a scatter plot with the same 6 points as the first graph. A 7th point is plotted in the top left corner of the quadrant. It falls outside the general pattern set by the other 6 points.\" width=\"731\" height=\"250\" \/>\r\n\r\n<img class=\"aligncenter wp-image-2295 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145900\/4103f33f9b0281d0681324f3f731639f7b88f9cf.jpeg\" alt=\"The first graph is a scatter plot with 6 points plotted. The points form a pattern that moves downward to the right, almost in a straight line. The second graph is a scatter plot of 8 points. These points form a general downward pattern, but the point do not align in a tight pattern.\" width=\"731\" height=\"250\" \/>\r\n\r\n<img class=\"aligncenter wp-image-2296 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145948\/4f16ab3a180645fe110f56e827a61ce73bcc8404.jpeg\" alt=\"The first graph is a scatter plot of 7 points in an exponential pattern. The pattern of the points begins along the x-axis and curves steeply upward to the right side of the quadrant. The second graph shows a scatter plot with many points scattered everywhere, exhibiting no pattern.\" width=\"731\" height=\"250\" \/>\r\n\r\nIn this module, we are interested in scatter plots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line, except in the case of a horizontal line where there is no relationship. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called\u00a0<strong>linear regression<\/strong>. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If <em>x<\/em> is the independent variable and <em>y<\/em>\u00a0is the dependent variable, then we can use a regression line to predict\u00a0<em>y<\/em> for a given value of <em>x.<\/em>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning outcomes<\/h3>\n<ul>\n<li>Create a scatterplot for bivariate data<\/li>\n<li>Given a scatterplot, describe the strength, direction, and form of the relationship between the variables<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall: Ordered Pairs as Data Points<\/h3>\n<p>When expressing pairs of inputs and outputs on a graph, they take the form of (<em>input<\/em>, <em>output<\/em>). In scatter plots, the two variables relate to create each data point,\u00a0(<em>variable 1<\/em>, <em>variable 2<\/em>), but it is often not necessary to declare that one is dependent on the other. In the example below, each\u00a0<em>Age<\/em>\u00a0coordinate corresponds to a\u00a0<em>Final Exam Score <\/em>in the form (<em>age<\/em>,\u00a0<em>score<\/em>). Each corresponding pair is plotted on the graph.<\/p>\n<\/div>\n<p>Before we take up the discussion of linear regression and correlation, we need to examine a way to display the relationship between two variables\u00a0<em>x<\/em> and <em>y<\/em>. The most common and easiest way is a <strong>scatter plot<\/strong>. The following example illustrates a scatter plot.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example 1<\/h3>\n<p>In Europe and Asia, m-commerce is popular. M-commerce users have special mobile phones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or soda from a machine to banking to checking sports scores on the Internet. For the years 2000 through 2004, was there a relationship between the year and the number of m-commerce users? Construct a scatter plot. Let\u00a0<em>x<\/em> = the year and let <em>y<\/em> = the number of m-commerce users, in millions.<\/p>\n<table>\n<thead>\n<tr>\n<th>(year)<\/th>\n<th>(# of users)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2000<\/td>\n<td>0.5<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>20.0<\/td>\n<\/tr>\n<tr>\n<td>2003<\/td>\n<td>33.0<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>47.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Table showing the number of m-commerce users (in millions) by year.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2293 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145614\/d663df2918f84749b9d6e2700a905415c10a8a20.jpeg\" alt=\"This is a scatter plot for the data provided. The x-axis represents the year and the y-axis represents the number of m-commerce users in millions. There are four points plotted, at (2000, 0.5), (2002, 20.0), (2003, 33.0), (2004, 47.0).\" width=\"487\" height=\"323\" \/><\/p>\n<p>Scatter plot showing the number of m-commerce users (in millions) by year.<\/p>\n<\/div>\n<h2>Creating a Scatter Plot USING THE TI-83, 83+, 84, 84+ CALCULATOR<\/h2>\n<ol>\n<li>Enter your X data into list L1 and your Y data into list L2.<\/li>\n<li>Press 2nd STATPLOT ENTER to use Plot 1. On the input screen for PLOT 1, highlight On and press ENTER. (Make sure the other plots are OFF).<\/li>\n<li>For TYPE: highlight the very first icon, which is the scatter plot, and press ENTER.<\/li>\n<li>For Xlist:, enter L1 ENTER and for Ylist: L2 ENTER.<\/li>\n<li>For Mark: it does not matter which symbol you highlight, but the square is the easiest to see. Press ENTER.<\/li>\n<li>Make sure there are no other equations that could be plotted. Press Y = and clear any equations out.<\/li>\n<li>Press the ZOOM key and then the number 9 (for menu item &#8220;ZoomStat&#8221;); the calculator will fit the window to the data. You can press WINDOW to see the scaling of the axes.<\/li>\n<\/ol>\n<div class=\"textbox key-takeaways\">\n<h3>try it 1<\/h3>\n<p>Amelia plays basketball for her high school. She wants to improve to play at the college level. She notices that the number of points she scores in a game goes up in response to the number of hours she practices her jump shot each week. She records the following data:<\/p>\n<table>\n<thead>\n<tr>\n<th><em>X<\/em> (hours practicing jump shot)<\/th>\n<th><em>Y<\/em> (points scored in a game)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>5<\/td>\n<td>15<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>22<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>28<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>31<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>33<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>36<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Construct a scatter plot and state if what Amelia thinks appears to be true.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q212678\">Show Answer<\/span><\/p>\n<div id=\"q212678\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/03l5-fu4g757i#fixme#fixme#fixme\" alt=\"This is a scatter plot for the data provided. The x-axis is labeled in increments of 2 from 0 - 16. The y-axis is labeled in increments of 5 from 0 - 35.\" \/><\/p>\n<p>Yes, Amelia&#8217;s assumption appears to be correct. The number of points Amelia scores per game goes up when she practices her jump shot more.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall: Scatter Plots<\/h3>\n<p>A scatter plot is a graph of plotted points that may show a relationship between two sets of data.<\/p>\n<\/div>\n<p>A scatter plot shows the\u00a0<strong>direction<\/strong> of a relationship between the variables. A clear direction happens when there are either: High values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable.<\/p>\n<p>You can determine the\u00a0<strong>strength<\/strong> of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or some other type of function. For a linear relationship, there is an exception. Consider a scatter plot where all the points fall on a horizontal line providing a &#8220;perfect fit.&#8221; The horizontal line would in fact show no relationship.<\/p>\n<p>When you look at a scatterplot, you want to notice the\u00a0<strong>overall pattern<\/strong> and any <strong>deviations<\/strong> from the pattern. The following scatterplot examples illustrate these concepts.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2294 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145834\/8a9a215df5f5bba8343f5fb4062e86014afb98ba.jpeg\" alt=\"The first graph is a scatter plot with 6 points plotted. The points form a pattern that moves upward to the right, almost in a straight line. The second graph is a scatter plot with the same 6 points as the first graph. A 7th point is plotted in the top left corner of the quadrant. It falls outside the general pattern set by the other 6 points.\" width=\"731\" height=\"250\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2295 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145900\/4103f33f9b0281d0681324f3f731639f7b88f9cf.jpeg\" alt=\"The first graph is a scatter plot with 6 points plotted. The points form a pattern that moves downward to the right, almost in a straight line. The second graph is a scatter plot of 8 points. These points form a general downward pattern, but the point do not align in a tight pattern.\" width=\"731\" height=\"250\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2296 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145948\/4f16ab3a180645fe110f56e827a61ce73bcc8404.jpeg\" alt=\"The first graph is a scatter plot of 7 points in an exponential pattern. The pattern of the points begins along the x-axis and curves steeply upward to the right side of the quadrant. The second graph shows a scatter plot with many points scattered everywhere, exhibiting no pattern.\" width=\"731\" height=\"250\" \/><\/p>\n<p>In this module, we are interested in scatter plots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line, except in the case of a horizontal line where there is no relationship. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called\u00a0<strong>linear regression<\/strong>. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If <em>x<\/em> is the independent variable and <em>y<\/em>\u00a0is the dependent variable, then we can use a regression line to predict\u00a0<em>y<\/em> for a given value of <em>x.<\/em><\/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-306\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Scatter Plots. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-2-scatter-plots\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-2-scatter-plots<\/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>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Precalculus. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions\">https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions<\/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>: Access for free at https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\">https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/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>: Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/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":169134,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Scatter Plots\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-2-scatter-plots\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at 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