{"id":358,"date":"2022-02-21T17:58:41","date_gmt":"2022-02-21T17:58:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&#038;p=358"},"modified":"2022-05-20T16:48:20","modified_gmt":"2022-05-20T16:48:20","slug":"five-number-summary-in-box-plots-and-datasets-forming-connections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/five-number-summary-in-box-plots-and-datasets-forming-connections\/","title":{"raw":"Five Number Summary in Boxplots and Data Sets: Forming Connections","rendered":"Five Number Summary in Boxplots and Data Sets: Forming Connections"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>objectives for this activity<\/h3>\r\nDuring this activity, you will:\r\n<ul>\r\n \t<li><a href=\"#bxplot_compare\">Use boxplots to compare the distributions of multiple populations<\/a><\/li>\r\n \t<li><a href=\"#bxplot_infer\">Use boxplots to draw inferences<\/a><\/li>\r\n<\/ul>\r\nClick on a skill above to jump to its location in this activity.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>video placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Intro: The data set used in this activity comes from the independent Tax Policy Center, a non-partisan group. We'll look at the Tax Policy Center's distributional tables of tax cuts by income levels, which were constructed in February 2018, in order to apply public policy analysis to a tax reform bill signed into law in 2017. Specifically, we'll use this situation to explore how boxplots can be used to compare distributions across several populations and draw inferences about them. As you read the two quotes at the start of the activity, consider the differences in them and ways in which you might investigate the accuracy of their claims.]<\/span>\r\n\r\n<\/div>\r\n<h2>Good Tax Policy?<\/h2>\r\n<strong><img class=\"wp-image-1026 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11223253\/Picture52-300x199.jpg\" alt=\"A calculator that has a display reading &quot;tax&quot; sitting on top of 100 dollar bills.\" width=\"489\" height=\"324\" \/><\/strong>\r\n\r\nRead these two quotes about the U.S. 2017 Tax Reform bill:[footnote] Data and lesson context adapted from Skew the Script, www.skewthescript.org [\/footnote]\r\n\r\n\u201cWe're doing everything we can to reduce the tax burden on you and your family. By eliminating tax breaks and loopholes, we will ensure that the benefits are focused on the middle class, the working men and women, not the highest income earners. Our framework includes our explicit commitment that tax reform will protect low-income and middle-income households, not the wealthy and well connected.\u201d \u2013 President Donald Trump in 2017, prior to the bill\u2019s enactment[footnote] Administration of Donald J. Trump. (2017, September 27).\u00a0<em>Remarks in Indianapolis, Indiana.\u00a0<\/em>Govinfo.gov. https:\/\/www.govinfo.gov\/content\/pkg\/DCPD-201700693\/html\/DCPD-201700693.htm\u00a0[\/footnote]\r\n\r\n\u201cYou remember just a few years ago when Trump and my Republican colleagues voted for almost $[latex]2[\/latex] trillion in tax breaks for the wealthiest people in this country and the largest corporations.\u201d \u2013 Bernie Sanders in 2021, after the bill was enacted[footnote] Cooper, A. (Interviewer). (2021, January 29.).\u00a0<em>CNN.<\/em> [\/footnote]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\n[ohm_question hide_question_numbers=1]241160[\/ohm_question]\r\n\r\n[reveal-answer q=\"589434\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"589434\"]What kind of data would help you to evaluate these claims?[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this activity, you'll see how to use a boxplot to provide a visual summary of a quantitative variable, and how boxplots can be used to compare the distributions of multiple populations. Since we'll be using data from the Tax Policy Center to analyze the claims made in the two quotes above, let's first investigate a statement the Center made about the\u00a0<em>average<\/em> tax cut experienced from the law.\r\n\r\nAccording to the independent Tax Policy Center, the average income tax cut as a result of the law was $[latex]1,260[\/latex].[footnote] Tax Policy Center. (2018, February 16).\u00a0<em>T18-0025 - The Tax Cuts and Jobs Act (TCJA): All provisions and individual income tax provisions; distribution of federal tax change by expanded cash income percentile, 2018.\u00a0<\/em>htpps:\/\/www.taxpolicycenter.org\/model-estimates\/individual-income-tax-provisions-tax-cuts-and-jobs-act-tcja-february-2019\/t18-0025[\/footnote] However, as we\u2019ve seen before, averages can be misleading and uninformative. Two boxplots are provided below, each displaying a <em>hypothetical<\/em> distribution of\u00a0[latex]5,000[\/latex] tax cuts that would result in a mean tax cut of $[latex]1,260[\/latex].\r\n\r\n<strong><img class=\"alignnone wp-image-1027\" style=\"font-size: 1em;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11223300\/Picture53-300x71.png\" alt=\"Two box plots. The horizontal axis is labeled &quot;Household Tax Costs ($ thousands) and is numbered in increments of one. The top is labeled A and has its lowest point at zero and its highest at approximately 2.3. The lower end of the box is at approximately 0.5 while the upper end is at approximately 2. The center line is at approximately 1.4. For plot B, the lowest point is at 0 and the highest point is at approximately 1.25. The low end of the box is at approximately 0.05 while the upper end is at approximately 0.5. The middle line is at approximately 0.1. Above the high point of plot B, there are lots of individual dots very close together all the way up the rest of the box plot.\" width=\"1132\" height=\"268\" \/><\/strong>\r\n<h3 id=\"bxplot_compare\">Comparing Multiple Distributions<\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\n[ohm_question hide_question_numbers=1]241161[\/ohm_question]\r\n\r\n[reveal-answer q=\"223883\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"223883\"]Remember that the scale of the graph shows tax cuts in thousands of dollars. Where is the value $[latex]500[\/latex] located on the graph?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\n[ohm_question hide_question_numbers=1]241162[\/ohm_question]\r\n\r\n[reveal-answer q=\"539176\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"539176\"]Recall where to find Q1, Median, and Q3 on the boxplot. Which of these is located at the $[latex]500[\/latex] mark on the graph?[\/hidden-answer]\r\n\r\n<\/div>\r\nNow, let's compare these distributions.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\n[ohm_question hide_question_numbers=1]241163[\/ohm_question]\r\n\r\n[reveal-answer q=\"562548\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"562548\"]What measure of center is displayed in a boxplot?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\n[ohm_question hide_question_numbers=1]241164[\/ohm_question]\r\n\r\n[reveal-answer q=\"39682\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"39682\"]Which IQR is wider? Which range is larger?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]241165 [\/ohm_question]\r\n\r\n[reveal-answer q=\"348809\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"348809\"]Use the presence (or lack of) outliers to help identify the shapes.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]241166[\/ohm_question]\r\n\r\n[reveal-answer q=\"515365\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"515365\"]What do <em>you<\/em> think? Which distribution appears more to have covered a broad swath of households?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\n[ohm_question hide_question_numbers=1]241167[\/ohm_question]\r\n\r\n[reveal-answer q=\"781497\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"781497\"]What do <em>you<\/em> think? Use what you know about percentiles and outliers to guide your reasoning.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"bxplot_infer\">Drawing inferences from boxplots<\/h3>\r\nNow let's look at the data from the 2017 Tax Reform bill.\r\n\r\nThe data from the Tax Policy Center in the table below separates income groups into quintiles. Quintiles divide data sets into five parts, from the lowest\u00a0[latex]20[\/latex]% of values up to the highest\u00a0[latex]20[\/latex]% of values.\r\n\r\nThe Tax Policy Center released the following data about the actual tax cuts households experienced because of the 2017 Tax Reform bill.[footnote]\u00a0Tax Policy Center. (2018, February 16). <em>T18-0025 - The Tax Cuts and Jobs Act (TCJA): All provisions\u00a0<\/em><em>and individual income tax provisions; distribution of federal tax change by expanded cash income\u00a0<\/em><em>percentile, 2018.<\/em> https:\/\/www.taxpolicycenter.org\/model-estimates\/individual-income-tax-provisions-tax-cuts-and-jobs-act-tcja-february-2018\/t18-0025[\/footnote]\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\" colspan=\"2\"><strong>Household Tax Cuts<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\"><strong>Income Group<\/strong><\/td>\r\n<td style=\"width: 162.125px; text-align: center;\"><strong>Mean Tax Cut<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\"><strong>Lowest Quintile<\/strong><\/td>\r\n<td style=\"width: 162.125px; text-align: center;\">$[latex]40[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\"><strong>Second Quintile<\/strong><\/td>\r\n<td style=\"width: 162.125px; text-align: center;\">$[latex]320[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\"><strong>Middle Quintile<\/strong><\/td>\r\n<td style=\"width: 162.125px; text-align: center;\">$[latex]780[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\"><strong>Fourth Quintile<\/strong><\/td>\r\n<td style=\"width: 162.125px; text-align: center;\">$[latex]1,480[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\"><strong>Top Quintile<\/strong><\/td>\r\n<td style=\"width: 162.125px; text-align: center;\">$[latex]5,790[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\"><strong>Top 1 Percent<\/strong><\/td>\r\n<td style=\"width: 162.125px; text-align: center;\">$[latex]32,650[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 184.188px; text-align: center;\"><strong>Top 0.1 Percent<\/strong><\/td>\r\n<td style=\"width: 162.125px; text-align: center;\">$[latex]89,060[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\n[ohm_question hide_question_numbers=1]241184[\/ohm_question]\r\n\r\n[reveal-answer q=\"566117\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"566117\"]What pattern do you notice in the tax cut amounts as the incomes tend to climb? Which quintile do you assume represents the median of the five quintiles? [\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\n[ohm_question hide_question_numbers=1]241185[\/ohm_question]\r\n\r\n[reveal-answer q=\"359334\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"359334\"]Compare the maximum listed in the data table to the maximum shown on the graph for Boxplot B.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\n[ohm_question hide_question_numbers=1]241186[\/ohm_question]\r\n\r\n[reveal-answer q=\"942338\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"942338\"]In other words, is the statement factual?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 12<\/h3>\r\n[ohm_question hide_question_numbers=1]241187[\/ohm_question]\r\n\r\n[reveal-answer q=\"194601\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"194601\"]If you consider possible skew due to outliers in the data set, how do you think the value $[latex]1,260[\/latex] was generated? How does that number compare with what you understand from the data table to be the \"typical\" American household?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>video placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">Wrap up: \"There are many perspectives to take and questions to answer when trying to sort out as complicated a situation as this one. We are only presented a small amount of information in this activity so we limited ourselves to questions we could answer using the data we had. It is of interest, though, to compare these tax cuts given in dollar amounts to their equivalencies in terms of <em>percent of peoples' incomes<\/em>. [voice over the table given in the hidden text below]. The Tax Policy Center also provides this information. The top income groups received both the largest cuts in dollar amounts but also the largest percent growth in after-tax income as a result of the bill (the top 0.1 percent don't follow this trend, however). But hopefully you've been able to understand from this activity how boxplots provide a quick glance, a summary, of the data to make comparisons based on median, skew, outliers, and percentiles. Let's take a look at a few general five-number summaries to make sure you feel good about what you've learned here.\" -- Provide a few five-number summaries and ask students to sketch boxplots of them. Then show the answers for comparison. -- \"You've seen that boxplots provide visual summaries of quantitative variables that can be used to compare the distributions of multiple populations. Hopefully, you feel that you are now comfortable using boxplots to compare and draw inferences.\"<\/span>\r\n\r\n<\/div>\r\n[reveal-answer q=\"825052\"]Percent Change in After-Tax Income by Income Group[\/reveal-answer]\r\n[hidden-answer a=\"825052\"]\r\n<table style=\"border-collapse: collapse; width: 49.3841%; height: 112px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\" colspan=\"2\"><strong>After-Tax Income by Income Group<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Income Group<\/strong><\/td>\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Percent Change in After-Tax Income<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Lowest Quintile<\/strong><\/td>\r\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]0.3[\/latex]%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Second Quintile<\/strong><\/td>\r\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]1.0[\/latex]%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Middle Quintile<\/strong><\/td>\r\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]1.4[\/latex]%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Fourth Quintile<\/strong><\/td>\r\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]1.6[\/latex]%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Top Quintile<\/strong><\/td>\r\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]2.2[\/latex]%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Top 1 Percent<\/strong><\/td>\r\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]2.2[\/latex]%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Top 0.1 Percent<\/strong><\/td>\r\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]1.3[\/latex]%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>objectives for this activity<\/h3>\n<p>During this activity, you will:<\/p>\n<ul>\n<li><a href=\"#bxplot_compare\">Use boxplots to compare the distributions of multiple populations<\/a><\/li>\n<li><a href=\"#bxplot_infer\">Use boxplots to draw inferences<\/a><\/li>\n<\/ul>\n<p>Click on a skill above to jump to its location in this activity.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Intro: The data set used in this activity comes from the independent Tax Policy Center, a non-partisan group. We&#8217;ll look at the Tax Policy Center&#8217;s distributional tables of tax cuts by income levels, which were constructed in February 2018, in order to apply public policy analysis to a tax reform bill signed into law in 2017. Specifically, we&#8217;ll use this situation to explore how boxplots can be used to compare distributions across several populations and draw inferences about them. As you read the two quotes at the start of the activity, consider the differences in them and ways in which you might investigate the accuracy of their claims.]<\/span><\/p>\n<\/div>\n<h2>Good Tax Policy?<\/h2>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1026 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11223253\/Picture52-300x199.jpg\" alt=\"A calculator that has a display reading &quot;tax&quot; sitting on top of 100 dollar bills.\" width=\"489\" height=\"324\" \/><\/strong><\/p>\n<p>Read these two quotes about the U.S. 2017 Tax Reform bill:<a class=\"footnote\" title=\"Data and lesson context adapted from Skew the Script, www.skewthescript.org\" id=\"return-footnote-358-1\" href=\"#footnote-358-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p>\u201cWe&#8217;re doing everything we can to reduce the tax burden on you and your family. By eliminating tax breaks and loopholes, we will ensure that the benefits are focused on the middle class, the working men and women, not the highest income earners. Our framework includes our explicit commitment that tax reform will protect low-income and middle-income households, not the wealthy and well connected.\u201d \u2013 President Donald Trump in 2017, prior to the bill\u2019s enactment<a class=\"footnote\" title=\"Administration of Donald J. Trump. (2017, September 27).\u00a0Remarks in Indianapolis, Indiana.\u00a0Govinfo.gov. https:\/\/www.govinfo.gov\/content\/pkg\/DCPD-201700693\/html\/DCPD-201700693.htm\u00a0\" id=\"return-footnote-358-2\" href=\"#footnote-358-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<p>\u201cYou remember just a few years ago when Trump and my Republican colleagues voted for almost $[latex]2[\/latex] trillion in tax breaks for the wealthiest people in this country and the largest corporations.\u201d \u2013 Bernie Sanders in 2021, after the bill was enacted<a class=\"footnote\" title=\"Cooper, A. (Interviewer). (2021, January 29.).\u00a0CNN.\" id=\"return-footnote-358-3\" href=\"#footnote-358-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241160\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241160&theme=oea&iframe_resize_id=ohm241160\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q589434\">Hint<\/span><\/p>\n<div id=\"q589434\" class=\"hidden-answer\" style=\"display: none\">What kind of data would help you to evaluate these claims?<\/div>\n<\/div>\n<\/div>\n<p>In this activity, you&#8217;ll see how to use a boxplot to provide a visual summary of a quantitative variable, and how boxplots can be used to compare the distributions of multiple populations. Since we&#8217;ll be using data from the Tax Policy Center to analyze the claims made in the two quotes above, let&#8217;s first investigate a statement the Center made about the\u00a0<em>average<\/em> tax cut experienced from the law.<\/p>\n<p>According to the independent Tax Policy Center, the average income tax cut as a result of the law was $[latex]1,260[\/latex].<a class=\"footnote\" title=\"Tax Policy Center. (2018, February 16).\u00a0T18-0025 - The Tax Cuts and Jobs Act (TCJA): All provisions and individual income tax provisions; distribution of federal tax change by expanded cash income percentile, 2018.\u00a0htpps:\/\/www.taxpolicycenter.org\/model-estimates\/individual-income-tax-provisions-tax-cuts-and-jobs-act-tcja-february-2019\/t18-0025\" id=\"return-footnote-358-4\" href=\"#footnote-358-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a> However, as we\u2019ve seen before, averages can be misleading and uninformative. Two boxplots are provided below, each displaying a <em>hypothetical<\/em> distribution of\u00a0[latex]5,000[\/latex] tax cuts that would result in a mean tax cut of $[latex]1,260[\/latex].<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1027\" style=\"font-size: 1em;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11223300\/Picture53-300x71.png\" alt=\"Two box plots. The horizontal axis is labeled &quot;Household Tax Costs ($ thousands) and is numbered in increments of one. The top is labeled A and has its lowest point at zero and its highest at approximately 2.3. The lower end of the box is at approximately 0.5 while the upper end is at approximately 2. The center line is at approximately 1.4. For plot B, the lowest point is at 0 and the highest point is at approximately 1.25. The low end of the box is at approximately 0.05 while the upper end is at approximately 0.5. The middle line is at approximately 0.1. Above the high point of plot B, there are lots of individual dots very close together all the way up the rest of the box plot.\" width=\"1132\" height=\"268\" \/><\/strong><\/p>\n<h3 id=\"bxplot_compare\">Comparing Multiple Distributions<\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241161\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241161&theme=oea&iframe_resize_id=ohm241161\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223883\">Hint<\/span><\/p>\n<div id=\"q223883\" class=\"hidden-answer\" style=\"display: none\">Remember that the scale of the graph shows tax cuts in thousands of dollars. Where is the value $[latex]500[\/latex] located on the graph?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241162\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241162&theme=oea&iframe_resize_id=ohm241162\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q539176\">Hint<\/span><\/p>\n<div id=\"q539176\" class=\"hidden-answer\" style=\"display: none\">Recall where to find Q1, Median, and Q3 on the boxplot. Which of these is located at the $[latex]500[\/latex] mark on the graph?<\/div>\n<\/div>\n<\/div>\n<p>Now, let&#8217;s compare these distributions.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241163\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241163&theme=oea&iframe_resize_id=ohm241163\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q562548\">Hint<\/span><\/p>\n<div id=\"q562548\" class=\"hidden-answer\" style=\"display: none\">What measure of center is displayed in a boxplot?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241164\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241164&theme=oea&iframe_resize_id=ohm241164\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q39682\">Hint<\/span><\/p>\n<div id=\"q39682\" class=\"hidden-answer\" style=\"display: none\">Which IQR is wider? Which range is larger?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241165\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241165&theme=oea&iframe_resize_id=ohm241165\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q348809\">Hint<\/span><\/p>\n<div id=\"q348809\" class=\"hidden-answer\" style=\"display: none\">Use the presence (or lack of) outliers to help identify the shapes.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241166\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241166&theme=oea&iframe_resize_id=ohm241166\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q515365\">Hint<\/span><\/p>\n<div id=\"q515365\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think? Which distribution appears more to have covered a broad swath of households?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241167\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241167&theme=oea&iframe_resize_id=ohm241167\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q781497\">Hint<\/span><\/p>\n<div id=\"q781497\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think? Use what you know about percentiles and outliers to guide your reasoning.<\/div>\n<\/div>\n<\/div>\n<h3 id=\"bxplot_infer\">Drawing inferences from boxplots<\/h3>\n<p>Now let&#8217;s look at the data from the 2017 Tax Reform bill.<\/p>\n<p>The data from the Tax Policy Center in the table below separates income groups into quintiles. Quintiles divide data sets into five parts, from the lowest\u00a0[latex]20[\/latex]% of values up to the highest\u00a0[latex]20[\/latex]% of values.<\/p>\n<p>The Tax Policy Center released the following data about the actual tax cuts households experienced because of the 2017 Tax Reform bill.<a class=\"footnote\" title=\"\u00a0Tax Policy Center. (2018, February 16). T18-0025 - The Tax Cuts and Jobs Act (TCJA): All provisions\u00a0and individual income tax provisions; distribution of federal tax change by expanded cash income\u00a0percentile, 2018. https:\/\/www.taxpolicycenter.org\/model-estimates\/individual-income-tax-provisions-tax-cuts-and-jobs-act-tcja-february-2018\/t18-0025\" id=\"return-footnote-358-5\" href=\"#footnote-358-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a><\/p>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\" colspan=\"2\"><strong>Household Tax Cuts<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\"><strong>Income Group<\/strong><\/td>\n<td style=\"width: 162.125px; text-align: center;\"><strong>Mean Tax Cut<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\"><strong>Lowest Quintile<\/strong><\/td>\n<td style=\"width: 162.125px; text-align: center;\">$[latex]40[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\"><strong>Second Quintile<\/strong><\/td>\n<td style=\"width: 162.125px; text-align: center;\">$[latex]320[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\"><strong>Middle Quintile<\/strong><\/td>\n<td style=\"width: 162.125px; text-align: center;\">$[latex]780[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\"><strong>Fourth Quintile<\/strong><\/td>\n<td style=\"width: 162.125px; text-align: center;\">$[latex]1,480[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\"><strong>Top Quintile<\/strong><\/td>\n<td style=\"width: 162.125px; text-align: center;\">$[latex]5,790[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\"><strong>Top 1 Percent<\/strong><\/td>\n<td style=\"width: 162.125px; text-align: center;\">$[latex]32,650[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 184.188px; text-align: center;\"><strong>Top 0.1 Percent<\/strong><\/td>\n<td style=\"width: 162.125px; text-align: center;\">$[latex]89,060[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241184\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241184&theme=oea&iframe_resize_id=ohm241184\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q566117\">Hint<\/span><\/p>\n<div id=\"q566117\" class=\"hidden-answer\" style=\"display: none\">What pattern do you notice in the tax cut amounts as the incomes tend to climb? Which quintile do you assume represents the median of the five quintiles? <\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241185\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241185&theme=oea&iframe_resize_id=ohm241185\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q359334\">Hint<\/span><\/p>\n<div id=\"q359334\" class=\"hidden-answer\" style=\"display: none\">Compare the maximum listed in the data table to the maximum shown on the graph for Boxplot B.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241186\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241186&theme=oea&iframe_resize_id=ohm241186\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q942338\">Hint<\/span><\/p>\n<div id=\"q942338\" class=\"hidden-answer\" style=\"display: none\">In other words, is the statement factual?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 12<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241187\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241187&theme=oea&iframe_resize_id=ohm241187\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q194601\">Hint<\/span><\/p>\n<div id=\"q194601\" class=\"hidden-answer\" style=\"display: none\">If you consider possible skew due to outliers in the data set, how do you think the value $[latex]1,260[\/latex] was generated? How does that number compare with what you understand from the data table to be the &#8220;typical&#8221; American household?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">Wrap up: &#8220;There are many perspectives to take and questions to answer when trying to sort out as complicated a situation as this one. We are only presented a small amount of information in this activity so we limited ourselves to questions we could answer using the data we had. It is of interest, though, to compare these tax cuts given in dollar amounts to their equivalencies in terms of <em>percent of peoples&#8217; incomes<\/em>. [voice over the table given in the hidden text below]. The Tax Policy Center also provides this information. The top income groups received both the largest cuts in dollar amounts but also the largest percent growth in after-tax income as a result of the bill (the top 0.1 percent don&#8217;t follow this trend, however). But hopefully you&#8217;ve been able to understand from this activity how boxplots provide a quick glance, a summary, of the data to make comparisons based on median, skew, outliers, and percentiles. Let&#8217;s take a look at a few general five-number summaries to make sure you feel good about what you&#8217;ve learned here.&#8221; &#8212; Provide a few five-number summaries and ask students to sketch boxplots of them. Then show the answers for comparison. &#8212; &#8220;You&#8217;ve seen that boxplots provide visual summaries of quantitative variables that can be used to compare the distributions of multiple populations. Hopefully, you feel that you are now comfortable using boxplots to compare and draw inferences.&#8221;<\/span><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q825052\">Percent Change in After-Tax Income by Income Group<\/span><\/p>\n<div id=\"q825052\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 49.3841%; height: 112px;\">\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\" colspan=\"2\"><strong>After-Tax Income by Income Group<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Income Group<\/strong><\/td>\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Percent Change in After-Tax Income<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Lowest Quintile<\/strong><\/td>\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]0.3[\/latex]%<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Second Quintile<\/strong><\/td>\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]1.0[\/latex]%<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Middle Quintile<\/strong><\/td>\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]1.4[\/latex]%<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Fourth Quintile<\/strong><\/td>\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]1.6[\/latex]%<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Top Quintile<\/strong><\/td>\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]2.2[\/latex]%<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Top 1 Percent<\/strong><\/td>\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]2.2[\/latex]%<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px; text-align: center;\"><strong>Top 0.1 Percent<\/strong><\/td>\n<td style=\"width: 50%; height: 14px; text-align: center;\">[latex]1.3[\/latex]%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-358-1\"> Data and lesson context adapted from Skew the Script, www.skewthescript.org  <a href=\"#return-footnote-358-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-358-2\"> Administration of Donald J. Trump. (2017, September 27).\u00a0<em>Remarks in Indianapolis, Indiana.\u00a0<\/em>Govinfo.gov. https:\/\/www.govinfo.gov\/content\/pkg\/DCPD-201700693\/html\/DCPD-201700693.htm\u00a0 <a href=\"#return-footnote-358-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-358-3\"> Cooper, A. (Interviewer). (2021, January 29.).\u00a0<em>CNN.<\/em>  <a href=\"#return-footnote-358-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-358-4\"> Tax Policy Center. (2018, February 16).\u00a0<em>T18-0025 - The Tax Cuts and Jobs Act (TCJA): All provisions and individual income tax provisions; distribution of federal tax change by expanded cash income percentile, 2018.\u00a0<\/em>htpps:\/\/www.taxpolicycenter.org\/model-estimates\/individual-income-tax-provisions-tax-cuts-and-jobs-act-tcja-february-2019\/t18-0025 <a href=\"#return-footnote-358-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-358-5\">\u00a0Tax Policy Center. (2018, February 16). <em>T18-0025 - The Tax Cuts and Jobs Act (TCJA): All provisions\u00a0<\/em><em>and individual income tax provisions; distribution of federal tax change by expanded cash income\u00a0<\/em><em>percentile, 2018.<\/em> https:\/\/www.taxpolicycenter.org\/model-estimates\/individual-income-tax-provisions-tax-cuts-and-jobs-act-tcja-february-2018\/t18-0025 <a href=\"#return-footnote-358-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":175116,"menu_order":53,"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-358","chapter","type-chapter","status-publish","hentry"],"part":1252,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/358","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/users\/175116"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/358\/revisions"}],"predecessor-version":[{"id":1264,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/358\/revisions\/1264"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/parts\/1252"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/358\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/media?parent=358"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapter-type?post=358"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/contributor?post=358"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/license?post=358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}