{"id":3809,"date":"2022-03-15T22:55:44","date_gmt":"2022-03-15T22:55:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=3809"},"modified":"2022-04-28T14:26:37","modified_gmt":"2022-04-28T14:26:37","slug":"what-to-know-about-5-a","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/what-to-know-about-5-a\/","title":{"raw":"What to Know About 5.A: The Correlation Coefficient","rendered":"What to Know About 5.A: The Correlation Coefficient"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>learning goals<\/h3>\r\nAt the end of this page, you should feel comfortable performing these skills:\r\n<ul>\r\n \t<li>Read a scatterplot.<\/li>\r\n \t<li>Describe the trend in a scatterplot.<\/li>\r\n \t<li>Describe the relationship in a scatterplot as linear or nonlinear.<\/li>\r\n \t<li>Recognize outliers in a scatterplot.<\/li>\r\n \t<li>Use technology to locate the correlation coefficient .<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the upcoming <em>Forming Connections\u00a0<\/em>activity, you will need to read a scatterplot and use it to determine whether a relationship between two variables has a positive or negative trend (or no trend). You'll also need to use a scatterplot to identify linear and non-linear relationships, and outliers. We'll prepare you for that by practicing those skills as we exploring greenhouse gases and our carbon footprint on the planet.\r\n<h2>Greenhouse Gases<\/h2>\r\nRead the following paragraph then\u00a0answer Questions 1 - 3 below to assess your comprehension of the paragraph. Be ready to discuss the ideas from your reading.\r\n\r\n<img class=\"aligncenter wp-image-4608 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/03\/18183853\/969px-RHSGlasshouse-300x223.jpg\" alt=\"A large, commercial greenhouse sits at the edge of water, with a path shown crossing the image in the lower foreground.\" width=\"300\" height=\"223\" \/>\r\n<h3>Greenhouse Effect and Carbon Footprint<\/h3>\r\nCarbon dioxide is a\u00a0greenhouse gas, which means it absorbs and radiates heat. Warmed by sunlight, Earth\u2019s land and ocean surfaces continuously radiate thermal infrared energy (heat). Unlike oxygen or nitrogen (which make up most of our atmosphere),\u00a0greenhouse gases absorb that heat and release it gradually over time, like bricks in a fireplace after the fire goes out. Without this natural\u00a0greenhouse effect, Earth\u2019s average annual temperature would be below freezing instead of close to 60\u00b0F. But increases in greenhouse gases have tipped the Earth's energy budget out of balance, trapping additional heat and raising Earth's average temperature.[footnote]Lindsey, R. (2020, August 14). Climate change: Atmospheric carbon dioxide. NOAA Climate.gov. https:\/\/www.climate.gov\/news-features\/understanding-climate\/climate-change-atmospheric-carbon- dioxide[\/footnote]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 1<\/h3>\r\nWhat is a greenhouse gas?\r\n\r\n[reveal-answer q=\"333582\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"333582\"]See the paragraph above. Hint: how does a greenhouse behave?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 2<\/h3>\r\nWhich greenhouse gas is mentioned?\r\n\r\n[reveal-answer q=\"619302\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"619302\"]See the paragraph above.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\nWhat would happen to the Earth without the greenhouse effect?\r\n\r\n[reveal-answer q=\"753076\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"753076\"]See the paragraph above.[\/hidden-answer]\r\n\r\n<\/div>\r\nNow let's look at a measure associated with greenhouse gases and the\u00a0products, foods, and services we consume.\r\n\r\nA carbon footprint is the total amount of greenhouse gas (GHG) emissions caused directly and indirectly by an individual, organization, event, or product. It is calculated by summing the emissions (in carbon dioxide equivalent units) resulting from every stage of a product or service\u2019s lifetime (material production, manufacturing, use, and end of life). A typical U.S. household has a carbon footprint of 48 metric tons of carbon dioxide equivalent per year (CO<sub>2<\/sub>e\/yr).[footnote]Center for Sustainable Systems, University of Michigan. (2020). Carbon footprint factsheet. http:\/\/css.umich.edu\/factsheets\/carbon-footprint-factsheet[\/footnote]\r\n\r\nThe food you eat has a carbon footprint. Energy is involved in producing the food, transporting the food, preparing the food, eating the food, and disposing of any waste from the food. We can analyze the energy content and carbon footprint of your food using a scatterplot.\r\n<h3>Scatterplot<\/h3>\r\n<div class=\"textbox\" style=\"padding-left: 30px;\">\r\n\r\nGo to the data analysis tool <em>Relationship Between Two Quantitative Variables: The Correlation<\/em> at\u00a0<a href=\"https:\/\/dcmathpathways.shinyapps.io\/Association_Quantitative\/\">https:\/\/dcmathpathways.shinyapps.io\/Association_Quantitative\/<\/a>\r\n<p style=\"padding-left: 30px;\">Step 1) Locate the drop-down menu under\u00a0<strong>Choose Dataset<\/strong>,<\/p>\r\n<p style=\"padding-left: 30px;\">Step 2) Select\u00a0<strong>Carbon Footprint<\/strong>.<\/p>\r\n\r\n<\/div>\r\nThe graph that you'll see\u00a0<span style=\"font-size: 1rem; text-align: initial;\">generated in the tool is called a <\/span><strong style=\"font-size: 1rem; text-align: initial;\">scatterplot<\/strong><span style=\"font-size: 1rem; text-align: initial;\">, and it is used to visualize the relationship between two quantitative variables, known as <\/span><strong style=\"font-size: 1rem; text-align: initial;\">bivariate data<\/strong><span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n<div class=\"textbox tryit\">\r\n<h3>Video Placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Perspective Video: a three-instructor video explaining how to read a scatterplot -- how each point on the graph gives a value for two variables, and why we use the word \"bivariate\" to describe the data in a scatterplot.]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\n<span style=\"background-color: #ffff99;\">[This could be a good place for a choose-your-own-dataset example offering two or three choices of scatterplots.]<\/span>\r\n\r\nSuppose a researcher wanted to explore whether a trend appeared to exist in the relationship between the number of hours a person exercises per week and that person's average resting heart rate. The scatterplot below offers an example of what a graph of these two variables might look like. Use it to answer the questions below.\r\n\r\n<img class=\"aligncenter size-full wp-image-4611\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/03\/22123542\/Scatterplot_Heartrate.png\" alt=\"A scatterplot labeled Minutes Spent Exercising Per Week along the horizontal axis and Resting Heart Rate along the vertical axis. Points are located on the graph at (45, 82), (47, 79), (48, 77), (49, 78 ), (55, 69 ), (60, 72, ), (62,67), (78,73), (120,65), (132, 64), (148, 60), (150, 58), (160, 65), (179, 57), (190, 60), (200, 55), (220, 54), (280, 56), (300, 53), (412, 51)\" width=\"906\" height=\"420\" \/>\r\n\r\nAnswer the following questions this scatterplot.\r\n<ol>\r\n \t<li>What does the word\u00a0<strong>bivariate<em>\u00a0<\/em><\/strong>mean in the context of a scatterplot?<\/li>\r\n \t<li>The values of what two bivariate variables are associated with each point on the scatterplot?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"959497\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"959497\"]\r\n<ol>\r\n \t<li><strong>Bivariate<em>\u00a0<\/em><\/strong>means two related variables.<\/li>\r\n \t<li>In this case, the bivariate data includes the number of minutes per week each person spent exercising and the resting heartrate of the person.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow return to the scatterplot you created in the <a href=\"https:\/\/dcmathpathways.shinyapps.io\/Association_Quantitative\/\">data analysis tool<\/a> using the dataset Carbon Footprint. Use it to answer Questions 4 - 6.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\nThe values of what two bivariate variables are associated with each point on the scatterplot?\r\n\r\n[reveal-answer q=\"767399\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"767399\"]Examine the axis labels.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\nWhich sandwich has the lowest carbon footprint? What is the energy content of this sandwich?\r\n\r\n[reveal-answer q=\"823623\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"823623\"]Hover over the point to view the name and energy content value, or use the data table.[\/hidden-answer]\r\n\r\n<\/div>\r\nQuestion 6 asks you to locate a value in the tool called the Pearson Correlation Coefficient,\u00a0<em>r<\/em>, which is a measure of the strength of the linear relationship between two variables. Please make note of this value and have it available when you begin the\u00a0<em>Forming Connections\u00a0<\/em>activity.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\nA statistic for measuring the strength of a bivariate linear relationship is provided in the table below the scatterplot. Write down the Pearson Correlation Coefficient, [latex]r[\/latex] and set it aside for use in the\u00a0<em>Forming Connections\u00a0<\/em>activity.\r\n\r\n[reveal-answer q=\"33400\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"33400\"]This value is located in the tool, under the plot. [\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Trends in the Data<\/h3>\r\nIdentifying patterns in scatterplots can help us determine if a relationship exists between two variables. Scatterplots might show positive trends, negative trends, or no trends at all.\r\n\r\nA scatterplot shows a <strong>positive trend<\/strong> if the response variable (represented on the vertical axis) tends to increase as the explanatory variable (represented on the horizontal axis) increases. Conversely, if the response variable tends to decrease as the explanatory variable increases, then the scatterplot shows a <strong>negative trend.<\/strong>\r\n<div class=\"textbox tryit\">\r\n<h3>Video Placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Worked example video: a 3-instructor video showing a worked example like Questions 7 but that also provides perspective on how to identify and outlier in a scatterplot (as the example below Question 7) and how to identify non-linear data (as the example below that one)]<\/span>\r\n\r\n<\/div>\r\nAnswer Question 7 by identifying what trend (if any) appears to be present in the given scatterplots.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 7<\/h3>\r\nDecide if each scatterplot shows a positive trend, a negative trend, or no trend.\r\n\r\na)<img class=\"alignnone wp-image-1135\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021313\/Picture82-300x130.png\" alt=\"A scatterplot of alcohol content vs calories. The points on the graph are grouped relatively close together in a roughly linear pattern. The higher x-values generally also have higher y-values.\" width=\"1246\" height=\"540\" \/>\r\n\r\n[reveal-answer q=\"541908\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"541908\"]Do the y-values tend to be rising, falling, or neither as the x-value increase?[\/hidden-answer]\r\n\r\nb)<img class=\"alignnone wp-image-1136\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021322\/Picture83-300x130.png\" alt=\"A scatterplot with points that are loosely grouped together in a linear fashion. As the x-values of the points increase, y-values generally decrease.\" width=\"1103\" height=\"478\" \/>\r\n\r\n[reveal-answer q=\"396615\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"396615\"]Do the y-values tend to be rising, falling, or neither as the x-value increase?[\/hidden-answer]\r\n\r\nc)<img class=\"alignnone wp-image-1137\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021329\/Picture84-300x130.png\" alt=\"A scatterplot of performance versus attractiveness. The points are somewhat clustered together in various areas, but there is no general pattern to them.\" width=\"1246\" height=\"540\" \/>\r\n\r\n[reveal-answer q=\"926408\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"926408\"]Do the y-values tend to be rising, falling, or neither as the x-value increase?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Outliers<\/h3>\r\nScatterplots can be used to identify shapes and patterns. The relationship between two variables is said to be <strong>linear<\/strong> when the points on the scatterplot resemble a straight line. The following scatterplot could be described as being linear.\r\n\r\n<img class=\"alignnone wp-image-1138\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021334\/Picture85-300x130.png\" alt=\"A scatterplot labeled &quot;Vote Totals for Reform Party Candidate.&quot; Most of the points have low y-values, but there is one point with a much higher y-value that is circled.\" width=\"939\" height=\"407\" \/>\r\n\r\nThe circled point in the upper right-hand corner of the scatterplot represents an outlier (Palm Beach). Outliers appear as departures from the general trend. Scatterplots can be used to identify outliers or extreme observations in the bivariate data.\r\n<h3>Nonlinear Data<\/h3>\r\nScatterplots are also useful for identifying non-linear relationships. The following scatterplot illustrates a non-linear relationship between driving speed and fuel efficiency. Do you see how the points in the graph appear to roughly describe an arc that opens downward? When the driving speed is very slow or very fast, the fuel efficiency is low. There appears to be an optimal fuel efficiency of about 44 mpg reached at about 40 mph, which tapers off to the left and right.\r\n\r\n<img class=\"alignnone wp-image-1139\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021339\/Picture86-300x130.png\" alt=\"A scatterplot of fuel efficiency versus steady driving speed. Going from left to right, the first two points have similar and low y-values, the second two points have similar and moderate y-values, the next three points have similar and high y-values and the rest of the points are arranged in an approximately linear fashion with decreasing y-values as the x-values increase.\" width=\"1244\" height=\"539\" \/>\r\n\r\nQuestion 8 asks you to consider the shape of a trend, linear or non-linear, and how outliers might appear in the data, by creating a sketch of your own. To answer this question, just make one up with new axis labels or take your inspiration from a scatterplot you've seen on this page.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 8<\/h3>\r\n<span style=\"background-color: #ffff99;\">[Note that two parts have been added (parts A and C) to this question in the style of the DC original to align with the learning objectives. It may be appropriate to replace this question with more formal Parts, as those found in Question 7 above]\u00a0\u00a0<\/span>\r\n\r\na) Think of a pattern that is linear (or do an Internet search for examples of linear graphs).\u00a0Sketch the scatterplot using an appropriate title and axis labels.\r\n\r\n[reveal-answer q=\"204659\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"204659\"]It's okay to make one up or to find one online. Upload an image or describe what it would look like.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nb) Think of a pattern that is non-linear (or do an Internet search for examples of non-linear graphs). Sketch the scatterplot using an appropriate title.\r\n\r\n[reveal-answer q=\"312928\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"312928\"]It's okay to make one up or to find one online. Upload an image or describe what it would look like.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nc) Using your sketch from either Part A or Part B, give an example of an outlier in the data, describing where in the plot it is located and how it departs from the general trend.\r\n\r\n[reveal-answer q=\"98823\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"98823\"]It's okay to make one up or to find one online. Upload an image or describe what it would look like.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nIn this <em>What to Know\u00a0<\/em>page,\u00a0you learned to read and interpret features of scatterplots, a kind of graphical display that represents bivariate data. You saw that scatterplots may indicate linear trends that are increasing or decreasing, or that they may suggest a non-linear trend, or even no trend at all, and can even suggest outliers in the data. Take a moment to review these features of scatterplots before beginning the activity.\r\n<ul>\r\n \t<li>In Questions 1 \u2013 3, you practiced gathering information from written text.<\/li>\r\n \t<li>In Questions 4 - 6, you used a scatterplot to\u00a0determine\u00a0whether a relationship between\u00a0two variables has a positive trend,\u00a0negative trend, or no trend.<\/li>\r\n \t<li>In Question 8, you used\u00a0a scatterplot to identify linear\u00a0relationships, non-linear\u00a0relationships, and outliers.<\/li>\r\n<\/ul>\r\nIf you feel comfortable with these ideas, move on to the Forming Connections activity to apply them.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>learning goals<\/h3>\n<p>At the end of this page, you should feel comfortable performing these skills:<\/p>\n<ul>\n<li>Read a scatterplot.<\/li>\n<li>Describe the trend in a scatterplot.<\/li>\n<li>Describe the relationship in a scatterplot as linear or nonlinear.<\/li>\n<li>Recognize outliers in a scatterplot.<\/li>\n<li>Use technology to locate the correlation coefficient .<\/li>\n<\/ul>\n<\/div>\n<p>In the upcoming <em>Forming Connections\u00a0<\/em>activity, you will need to read a scatterplot and use it to determine whether a relationship between two variables has a positive or negative trend (or no trend). You&#8217;ll also need to use a scatterplot to identify linear and non-linear relationships, and outliers. We&#8217;ll prepare you for that by practicing those skills as we exploring greenhouse gases and our carbon footprint on the planet.<\/p>\n<h2>Greenhouse Gases<\/h2>\n<p>Read the following paragraph then\u00a0answer Questions 1 &#8211; 3 below to assess your comprehension of the paragraph. Be ready to discuss the ideas from your reading.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4608 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/03\/18183853\/969px-RHSGlasshouse-300x223.jpg\" alt=\"A large, commercial greenhouse sits at the edge of water, with a path shown crossing the image in the lower foreground.\" width=\"300\" height=\"223\" \/><\/p>\n<h3>Greenhouse Effect and Carbon Footprint<\/h3>\n<p>Carbon dioxide is a\u00a0greenhouse gas, which means it absorbs and radiates heat. Warmed by sunlight, Earth\u2019s land and ocean surfaces continuously radiate thermal infrared energy (heat). Unlike oxygen or nitrogen (which make up most of our atmosphere),\u00a0greenhouse gases absorb that heat and release it gradually over time, like bricks in a fireplace after the fire goes out. Without this natural\u00a0greenhouse effect, Earth\u2019s average annual temperature would be below freezing instead of close to 60\u00b0F. But increases in greenhouse gases have tipped the Earth&#8217;s energy budget out of balance, trapping additional heat and raising Earth&#8217;s average temperature.<a class=\"footnote\" title=\"Lindsey, R. (2020, August 14). Climate change: Atmospheric carbon dioxide. NOAA Climate.gov. https:\/\/www.climate.gov\/news-features\/understanding-climate\/climate-change-atmospheric-carbon- dioxide\" id=\"return-footnote-3809-1\" href=\"#footnote-3809-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 1<\/h3>\n<p>What is a greenhouse gas?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q333582\">Hint<\/span><\/p>\n<div id=\"q333582\" class=\"hidden-answer\" style=\"display: none\">See the paragraph above. Hint: how does a greenhouse behave?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 2<\/h3>\n<p>Which greenhouse gas is mentioned?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q619302\">Hint<\/span><\/p>\n<div id=\"q619302\" class=\"hidden-answer\" style=\"display: none\">See the paragraph above.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>What would happen to the Earth without the greenhouse effect?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q753076\">Hint<\/span><\/p>\n<div id=\"q753076\" class=\"hidden-answer\" style=\"display: none\">See the paragraph above.<\/div>\n<\/div>\n<\/div>\n<p>Now let&#8217;s look at a measure associated with greenhouse gases and the\u00a0products, foods, and services we consume.<\/p>\n<p>A carbon footprint is the total amount of greenhouse gas (GHG) emissions caused directly and indirectly by an individual, organization, event, or product. It is calculated by summing the emissions (in carbon dioxide equivalent units) resulting from every stage of a product or service\u2019s lifetime (material production, manufacturing, use, and end of life). A typical U.S. household has a carbon footprint of 48 metric tons of carbon dioxide equivalent per year (CO<sub>2<\/sub>e\/yr).<a class=\"footnote\" title=\"Center for Sustainable Systems, University of Michigan. (2020). Carbon footprint factsheet. http:\/\/css.umich.edu\/factsheets\/carbon-footprint-factsheet\" id=\"return-footnote-3809-2\" href=\"#footnote-3809-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<p>The food you eat has a carbon footprint. Energy is involved in producing the food, transporting the food, preparing the food, eating the food, and disposing of any waste from the food. We can analyze the energy content and carbon footprint of your food using a scatterplot.<\/p>\n<h3>Scatterplot<\/h3>\n<div class=\"textbox\" style=\"padding-left: 30px;\">\n<p>Go to the data analysis tool <em>Relationship Between Two Quantitative Variables: The Correlation<\/em> at\u00a0<a href=\"https:\/\/dcmathpathways.shinyapps.io\/Association_Quantitative\/\">https:\/\/dcmathpathways.shinyapps.io\/Association_Quantitative\/<\/a><\/p>\n<p style=\"padding-left: 30px;\">Step 1) Locate the drop-down menu under\u00a0<strong>Choose Dataset<\/strong>,<\/p>\n<p style=\"padding-left: 30px;\">Step 2) Select\u00a0<strong>Carbon Footprint<\/strong>.<\/p>\n<\/div>\n<p>The graph that you&#8217;ll see\u00a0<span style=\"font-size: 1rem; text-align: initial;\">generated in the tool is called a <\/span><strong style=\"font-size: 1rem; text-align: initial;\">scatterplot<\/strong><span style=\"font-size: 1rem; text-align: initial;\">, and it is used to visualize the relationship between two quantitative variables, known as <\/span><strong style=\"font-size: 1rem; text-align: initial;\">bivariate data<\/strong><span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<div class=\"textbox tryit\">\n<h3>Video Placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Perspective Video: a three-instructor video explaining how to read a scatterplot &#8212; how each point on the graph gives a value for two variables, and why we use the word &#8220;bivariate&#8221; to describe the data in a scatterplot.]<\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p><span style=\"background-color: #ffff99;\">[This could be a good place for a choose-your-own-dataset example offering two or three choices of scatterplots.]<\/span><\/p>\n<p>Suppose a researcher wanted to explore whether a trend appeared to exist in the relationship between the number of hours a person exercises per week and that person&#8217;s average resting heart rate. The scatterplot below offers an example of what a graph of these two variables might look like. Use it to answer the questions below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4611\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/03\/22123542\/Scatterplot_Heartrate.png\" alt=\"A scatterplot labeled Minutes Spent Exercising Per Week along the horizontal axis and Resting Heart Rate along the vertical axis. Points are located on the graph at (45, 82), (47, 79), (48, 77), (49, 78 ), (55, 69 ), (60, 72, ), (62,67), (78,73), (120,65), (132, 64), (148, 60), (150, 58), (160, 65), (179, 57), (190, 60), (200, 55), (220, 54), (280, 56), (300, 53), (412, 51)\" width=\"906\" height=\"420\" \/><\/p>\n<p>Answer the following questions this scatterplot.<\/p>\n<ol>\n<li>What does the word\u00a0<strong>bivariate<em>\u00a0<\/em><\/strong>mean in the context of a scatterplot?<\/li>\n<li>The values of what two bivariate variables are associated with each point on the scatterplot?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q959497\">Show Answer<\/span><\/p>\n<div id=\"q959497\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li><strong>Bivariate<em>\u00a0<\/em><\/strong>means two related variables.<\/li>\n<li>In this case, the bivariate data includes the number of minutes per week each person spent exercising and the resting heartrate of the person.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Now return to the scatterplot you created in the <a href=\"https:\/\/dcmathpathways.shinyapps.io\/Association_Quantitative\/\">data analysis tool<\/a> using the dataset Carbon Footprint. Use it to answer Questions 4 &#8211; 6.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p>The values of what two bivariate variables are associated with each point on the scatterplot?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q767399\">Hint<\/span><\/p>\n<div id=\"q767399\" class=\"hidden-answer\" style=\"display: none\">Examine the axis labels.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p>Which sandwich has the lowest carbon footprint? What is the energy content of this sandwich?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q823623\">Hint<\/span><\/p>\n<div id=\"q823623\" class=\"hidden-answer\" style=\"display: none\">Hover over the point to view the name and energy content value, or use the data table.<\/div>\n<\/div>\n<\/div>\n<p>Question 6 asks you to locate a value in the tool called the Pearson Correlation Coefficient,\u00a0<em>r<\/em>, which is a measure of the strength of the linear relationship between two variables. Please make note of this value and have it available when you begin the\u00a0<em>Forming Connections\u00a0<\/em>activity.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p>A statistic for measuring the strength of a bivariate linear relationship is provided in the table below the scatterplot. Write down the Pearson Correlation Coefficient, [latex]r[\/latex] and set it aside for use in the\u00a0<em>Forming Connections\u00a0<\/em>activity.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q33400\">Hint<\/span><\/p>\n<div id=\"q33400\" class=\"hidden-answer\" style=\"display: none\">This value is located in the tool, under the plot. <\/div>\n<\/div>\n<\/div>\n<h3>Trends in the Data<\/h3>\n<p>Identifying patterns in scatterplots can help us determine if a relationship exists between two variables. Scatterplots might show positive trends, negative trends, or no trends at all.<\/p>\n<p>A scatterplot shows a <strong>positive trend<\/strong> if the response variable (represented on the vertical axis) tends to increase as the explanatory variable (represented on the horizontal axis) increases. Conversely, if the response variable tends to decrease as the explanatory variable increases, then the scatterplot shows a <strong>negative trend.<\/strong><\/p>\n<div class=\"textbox tryit\">\n<h3>Video Placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Worked example video: a 3-instructor video showing a worked example like Questions 7 but that also provides perspective on how to identify and outlier in a scatterplot (as the example below Question 7) and how to identify non-linear data (as the example below that one)]<\/span><\/p>\n<\/div>\n<p>Answer Question 7 by identifying what trend (if any) appears to be present in the given scatterplots.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 7<\/h3>\n<p>Decide if each scatterplot shows a positive trend, a negative trend, or no trend.<\/p>\n<p>a)<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1135\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021313\/Picture82-300x130.png\" alt=\"A scatterplot of alcohol content vs calories. The points on the graph are grouped relatively close together in a roughly linear pattern. The higher x-values generally also have higher y-values.\" width=\"1246\" height=\"540\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q541908\">Hint<\/span><\/p>\n<div id=\"q541908\" class=\"hidden-answer\" style=\"display: none\">Do the y-values tend to be rising, falling, or neither as the x-value increase?<\/div>\n<\/div>\n<p>b)<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1136\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021322\/Picture83-300x130.png\" alt=\"A scatterplot with points that are loosely grouped together in a linear fashion. As the x-values of the points increase, y-values generally decrease.\" width=\"1103\" height=\"478\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396615\">Hint<\/span><\/p>\n<div id=\"q396615\" class=\"hidden-answer\" style=\"display: none\">Do the y-values tend to be rising, falling, or neither as the x-value increase?<\/div>\n<\/div>\n<p>c)<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1137\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021329\/Picture84-300x130.png\" alt=\"A scatterplot of performance versus attractiveness. The points are somewhat clustered together in various areas, but there is no general pattern to them.\" width=\"1246\" height=\"540\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q926408\">Hint<\/span><\/p>\n<div id=\"q926408\" class=\"hidden-answer\" style=\"display: none\">Do the y-values tend to be rising, falling, or neither as the x-value increase?<\/div>\n<\/div>\n<\/div>\n<h3>Outliers<\/h3>\n<p>Scatterplots can be used to identify shapes and patterns. The relationship between two variables is said to be <strong>linear<\/strong> when the points on the scatterplot resemble a straight line. The following scatterplot could be described as being linear.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1138\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021334\/Picture85-300x130.png\" alt=\"A scatterplot labeled &quot;Vote Totals for Reform Party Candidate.&quot; Most of the points have low y-values, but there is one point with a much higher y-value that is circled.\" width=\"939\" height=\"407\" \/><\/p>\n<p>The circled point in the upper right-hand corner of the scatterplot represents an outlier (Palm Beach). Outliers appear as departures from the general trend. Scatterplots can be used to identify outliers or extreme observations in the bivariate data.<\/p>\n<h3>Nonlinear Data<\/h3>\n<p>Scatterplots are also useful for identifying non-linear relationships. The following scatterplot illustrates a non-linear relationship between driving speed and fuel efficiency. Do you see how the points in the graph appear to roughly describe an arc that opens downward? When the driving speed is very slow or very fast, the fuel efficiency is low. There appears to be an optimal fuel efficiency of about 44 mpg reached at about 40 mph, which tapers off to the left and right.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1139\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12021339\/Picture86-300x130.png\" alt=\"A scatterplot of fuel efficiency versus steady driving speed. Going from left to right, the first two points have similar and low y-values, the second two points have similar and moderate y-values, the next three points have similar and high y-values and the rest of the points are arranged in an approximately linear fashion with decreasing y-values as the x-values increase.\" width=\"1244\" height=\"539\" \/><\/p>\n<p>Question 8 asks you to consider the shape of a trend, linear or non-linear, and how outliers might appear in the data, by creating a sketch of your own. To answer this question, just make one up with new axis labels or take your inspiration from a scatterplot you&#8217;ve seen on this page.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 8<\/h3>\n<p><span style=\"background-color: #ffff99;\">[Note that two parts have been added (parts A and C) to this question in the style of the DC original to align with the learning objectives. It may be appropriate to replace this question with more formal Parts, as those found in Question 7 above]\u00a0\u00a0<\/span><\/p>\n<p>a) Think of a pattern that is linear (or do an Internet search for examples of linear graphs).\u00a0Sketch the scatterplot using an appropriate title and axis labels.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q204659\">Hint<\/span><\/p>\n<div id=\"q204659\" class=\"hidden-answer\" style=\"display: none\">It&#8217;s okay to make one up or to find one online. Upload an image or describe what it would look like.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>b) Think of a pattern that is non-linear (or do an Internet search for examples of non-linear graphs). Sketch the scatterplot using an appropriate title.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q312928\">Hint<\/span><\/p>\n<div id=\"q312928\" class=\"hidden-answer\" style=\"display: none\">It&#8217;s okay to make one up or to find one online. Upload an image or describe what it would look like.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>c) Using your sketch from either Part A or Part B, give an example of an outlier in the data, describing where in the plot it is located and how it departs from the general trend.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q98823\">Hint<\/span><\/p>\n<div id=\"q98823\" class=\"hidden-answer\" style=\"display: none\">It&#8217;s okay to make one up or to find one online. Upload an image or describe what it would look like.<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>In this <em>What to Know\u00a0<\/em>page,\u00a0you learned to read and interpret features of scatterplots, a kind of graphical display that represents bivariate data. You saw that scatterplots may indicate linear trends that are increasing or decreasing, or that they may suggest a non-linear trend, or even no trend at all, and can even suggest outliers in the data. Take a moment to review these features of scatterplots before beginning the activity.<\/p>\n<ul>\n<li>In Questions 1 \u2013 3, you practiced gathering information from written text.<\/li>\n<li>In Questions 4 &#8211; 6, you used a scatterplot to\u00a0determine\u00a0whether a relationship between\u00a0two variables has a positive trend,\u00a0negative trend, or no trend.<\/li>\n<li>In Question 8, you used\u00a0a scatterplot to identify linear\u00a0relationships, non-linear\u00a0relationships, and outliers.<\/li>\n<\/ul>\n<p>If you feel comfortable with these ideas, move on to the Forming Connections activity to apply them.<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-3809-1\">Lindsey, R. (2020, August 14). Climate change: Atmospheric carbon dioxide. NOAA Climate.gov. https:\/\/www.climate.gov\/news-features\/understanding-climate\/climate-change-atmospheric-carbon- dioxide <a href=\"#return-footnote-3809-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-3809-2\">Center for Sustainable Systems, University of Michigan. (2020). Carbon footprint factsheet. http:\/\/css.umich.edu\/factsheets\/carbon-footprint-factsheet <a href=\"#return-footnote-3809-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":428269,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3809","chapter","type-chapter","status-publish","hentry"],"part":3807,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3809","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/428269"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3809\/revisions"}],"predecessor-version":[{"id":4678,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3809\/revisions\/4678"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/3807"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3809\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=3809"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=3809"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=3809"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=3809"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}