{"id":875,"date":"2023-03-20T19:17:50","date_gmt":"2023-03-20T19:17:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-boxplot-data-and-displays-fresh-take-1\/"},"modified":"2025-05-08T03:04:35","modified_gmt":"2025-05-08T03:04:35","slug":"boxplot-data-and-displays-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/boxplot-data-and-displays-fresh-take\/","title":{"raw":"Boxplot Data and Displays: Fresh Take","rendered":"Boxplot Data and Displays: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Read information from a boxplot and make conclusions<\/li>\r\n\t<li>Compare boxplots<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Additional Example of Five-Number Summary &amp; Outliers<\/h2>\r\n<p>Recall that when we describe the distribution of a quantitative variable, we describe the overall pattern (shape, center, and spread) in the data and deviations from the pattern (outliers).<\/p>\r\n\r\n[caption id=\"attachment_5614\" align=\"aligncenter\" width=\"591\"]<img class=\"wp-image-5614 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29235006\/4.3.L.Diagram-2.png\" alt=\"A flow chart beginning with Graph the distribution of a quantitative variable. Describe the following: with one arrow pointing to Overall pattern and another arrow pointing to Deviations from the pattern. The overall pattern box points to shape, center, and spread, with the latter being highlighted. The deviations from the pattern box points to outliers.\" width=\"591\" height=\"505\" \/> Figure 1. When analyzing a graph, describe the overall pattern (shape, center, spread) and look for deviations, or outliers, that don\u2019t follow the pattern.[\/caption]\r\n\r\n<section class=\"textbox example\"><strong>Two sets of exam scores<br \/>\r\n<br \/>\r\n<\/strong>Consider the following two distributions of exam scores:<br \/>\r\n[caption id=\"attachment_5616\" align=\"aligncenter\" width=\"427\"]<img class=\"wp-image-5616 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29235404\/picture.png\" alt=\"A boxplot graphic comparing exam scores for two classes, A and B. Each boxplot shows five number summaries. For class A, the minimum score is approximately 40, the first quartile is just above 70, the median is near 75, the third quartile is just below 80, and the maximum is around 95. There are two low outliers of 40 and 55 and two upper outliers of 90 and 95. For class B, the minimum is around 40, the first quartile is just above 60, the median is about 75, the third quartile is just below 90, with the maximum score close to 95. \" width=\"427\" height=\"186\" \/> Figure 2. Two boxplots of exam scores.[\/caption]\r\n<br \/>\r\nBoth distributions have a median of approximately [latex]74.5[\/latex].<br \/>\r\n[reveal-answer q=\"894713\"]Which distribution has more variability?[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"894713\"]The answer to this question depends on how we measure variability. Both distributions have the same range. The range is the distance spanned by the data. We calculate the range by subtracting the minimum value from the maximum value.\r\n\r\n<ul>\r\n\t<li>Range = Maximum value \u2013 minimum value<\/li>\r\n<\/ul>\r\n<p>For both of these data sets, the range is [latex]55[\/latex] (here is how we calculated the range: [latex]95 \u2013 40 = 55[\/latex]). If we use the range to measure variability, we say the distributions have the same amount of variability.<\/p>\r\n<p>But the variability in the distributions differs when we look at how the data is distributed about the median. Class A has a large portion of its data close to the median. This is not true for Class B. From this viewpoint, Class A has less variability about the median.<\/p>\r\n<p><strong>Therefore, Class B has more variability about the median.<\/strong>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3 id=\"Q1\">[latex]Q1[\/latex], [latex]Q3[\/latex], and IQR<\/h3>\r\n<p>Now we can develop a way to measure the variability about the median. To do so, we use <em>quartiles<\/em>. Quartile marks divide the data set into four groups with equal counts.<\/p>\r\n<p>To find the first and third quartiles ([latex]Q1[\/latex] and [latex]Q3[\/latex] respectively), first determine the list of values that lie both above and below the median. Then, take the medians of those lists.<\/p>\r\n<section class=\"textbox recall\">A boxplot is constructed from five values: <strong>the minimum value, the first quartile, the median, the third quartile, and the maximum value.<\/strong><br \/>\r\nThis set of five numbers is called the <strong>five-number summary.\u00a0<\/strong><br \/>\r\n[caption id=\"attachment_3078\" align=\"aligncenter\" width=\"433\"]<img class=\"wp-image-3078\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/12\/11181025\/Screen-Shot-2022-02-11-at-1.09.31-PM.png\" alt=\"A general horizontal boxplot displaying the following features from left to right: lower outliers, minimum, Q1, median, Q3, maximum, and upper outliers. The Interquartile Range (IQR) is shown at the top of the boxplot.\" width=\"433\" height=\"240\" \/> Figure 3. A boxplot creates a visual summary of a data set using five important values: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. It also shows the outliers in the data.[\/caption]\r\n<\/section>\r\n<section class=\"textbox example\"><strong>Two sets of exam scores: <\/strong>Consider the following two distributions of exam scores:\r\n[caption id=\"attachment_5616\" align=\"aligncenter\" width=\"427\"]<img class=\"wp-image-5616 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29235404\/picture.png\" alt=\"A boxplot graphic comparing exam scores for two classes, A and B. Each boxplot shows five number summaries. For class A, the minimum score is approximately 40, the first quartile is just above 70, the median is near 75, the third quartile is just below 80, and the maximum is around 95. There are two low outliers of 40 and 55 and two upper outliers of 90 and 95. For class B, the minimum is around 40, the first quartile is just above 60, the median is about 75, the third quartile is just below 90, with the maximum score close to 95. \" width=\"427\" height=\"186\" \/> Figure 2. Two boxplots of exam scores.[\/caption]\r\n<br \/>\r\n[reveal-answer q=\"835596\"]Using IQR, which distribution has more variability?[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"835596\"]The quartiles, together with the minimum and maximum score,s give the five-number summary:\r\n\r\n<ul>\r\n\t<li>Class A: Min: [latex]40[\/latex] [latex]Q1: 71[\/latex] [latex]Q2: 74.5[\/latex] [latex]Q3: 78.5[\/latex] Max: [latex]95[\/latex]<\/li>\r\n\t<li>Class B: Min: [latex]40[\/latex] [latex]Q1: 61[\/latex] [latex]Q2: 74.5[\/latex] [latex]Q3: 89[\/latex] Max: [latex]95[\/latex]<\/li>\r\n<\/ul>\r\n<p>Notice: The second quartile mark ([latex]Q2[\/latex]) is the median.<\/p>\r\n<p>Notice: Some quartiles exhibit more variability in the data, even though each quartile contains the same amount of data.<\/p>\r\n<ul>\r\n\t<li>For example, [latex]25[\/latex]% of the scores in Class A are between [latex]40[\/latex] and [latex]71[\/latex]. There is a lot of variability in this first quartile ([latex]Q1[\/latex]). The eight scores in [latex]Q1[\/latex] vary by [latex]30[\/latex] points.<\/li>\r\n\t<li>Compare this to the third quartile ([latex]Q3[\/latex]) for Class A: [latex]25[\/latex]% of the scores in Class A are between [latex]74.5[\/latex] and [latex]78.5[\/latex]. There is not much variability in [latex]Q3[\/latex]. The eight scores in [latex]Q3[\/latex] vary by only [latex]4[\/latex] points.<\/li>\r\n<\/ul>\r\n<p>How are quartiles used to measure variability about the median? The <em>interquartile range <\/em>(IQR) is the distance between the first and third quartile marks. The IQR is a measurement of the variability about the median. More specifically, the IQR tells us the range of the middle half of the data.<\/p>\r\n<p>Here is the IQR for these two distributions:<\/p>\r\n<ul>\r\n\t<li>Class A: IQR [latex]= Q3 \u2013 Q1 = 78.5 \u2013 71 = 7.5[\/latex]<\/li>\r\n\t<li>Class B: IQR [latex]= Q3 \u2013 Q1 = 89 \u2013 61 = 28[\/latex]<\/li>\r\n<\/ul>\r\n<p>As we observed earlier, Class A has less variability about its median. Its IQR is much smaller. The middle [latex]50[\/latex]% of exam scores for Class A vary by only [latex]7.5[\/latex] points. The middle [latex]50[\/latex]% of exam scores for Class B vary by [latex]28[\/latex] points.<\/p>\r\n<p><strong>Using IQR, Class B has more variability.<\/strong>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2744[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2745[\/ohm2_question]<\/section>\r\n<h2>Using the IQR to Identify Outliers<\/h2>\r\n<section class=\"textbox recall\">A value is an outlier when:\r\n\r\n<ul>\r\n\t<li>Upper outlier: The value is greater than [latex]Q3 + (1.5 *[\/latex]IQR[latex])[\/latex]<\/li>\r\n\t<li>Lower outlier: The value is less than [latex]Q1 - (1.5 *[\/latex]IQR[latex])[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>To make more sense of this rule, let\u2019s look at a visual example.<\/p>\r\n<section class=\"textbox tryIt\"><strong>Two sets of exam scores: <\/strong><br \/>\r\n<br \/>\r\nConsider the following two distributions of exam scores:\r\n[caption id=\"attachment_5616\" align=\"aligncenter\" width=\"427\"]<img class=\"wp-image-5616 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29235404\/picture.png\" alt=\"A boxplot graphic comparing exam scores for two classes, A and B. Each boxplot shows five number summaries. For class A, the minimum score is approximately 40, the first quartile is just above 70, the median is near 75, the third quartile is just below 80, and the maximum is around 95. There are two low outliers of 40 and 55 and two upper outliers of 90 and 95. For class B, the minimum is around 40, the first quartile is just above 60, the median is about 75, the third quartile is just below 90, with the maximum score close to 95. \" width=\"427\" height=\"186\" \/> Figure 2. Two boxplots of exam scores.[\/caption]\r\n[reveal-answer q=\"785698\"]Use IQR and identify the outliers of the data set.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"785698\"]For the Class A data set in the dotplot, [latex]Q1 = 71[\/latex] and [latex]Q3 = 78.25[\/latex], so the IQR [latex]= 7.5[\/latex].\r\n\r\n<ul>\r\n\t<li>[latex]Q1 - (1.5 *[\/latex]IQR[latex]) = 71 \u2013 1.5 * 7.5 = 71 \u2013 11.25 = 59.75[\/latex]\r\n\r\n<ul>\r\n\t<li>The data point at [latex]40[\/latex] and [latex]55[\/latex] is considered an outlier because it is below [latex]59.75[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>[latex]Q3 + (1.5 *[\/latex]IQR[latex]) = 71 + 1.5 * 7.5 = 71 + 11.25 = 82.25[\/latex]\r\n\r\n<ul>\r\n\t<li>The data points at [latex]85, 90[\/latex], and [latex]95[\/latex] are considered outliers because they are above [latex]82.25[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n\r\n[caption id=\"attachment_360\" align=\"aligncenter\" width=\"532\"]<img class=\"wp-image-360\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/08\/09215145\/Screen-Shot-2022-08-09-at-2.51.38-PM.png\" alt=\"\" width=\"532\" height=\"193\" \/> Figure 4. Dot plots comparing exam scores for two classes, A and B, with both having the same mean score of 74.5; with various outliers.[\/caption]\r\n\r\n<p>Now do the same for Class B. Do you find any outliers for Class B? [\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Read information from a boxplot and make conclusions<\/li>\n<li>Compare boxplots<\/li>\n<\/ul>\n<\/section>\n<h2>Additional Example of Five-Number Summary &amp; Outliers<\/h2>\n<p>Recall that when we describe the distribution of a quantitative variable, we describe the overall pattern (shape, center, and spread) in the data and deviations from the pattern (outliers).<\/p>\n<figure id=\"attachment_5614\" aria-describedby=\"caption-attachment-5614\" style=\"width: 591px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5614 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29235006\/4.3.L.Diagram-2.png\" alt=\"A flow chart beginning with Graph the distribution of a quantitative variable. Describe the following: with one arrow pointing to Overall pattern and another arrow pointing to Deviations from the pattern. The overall pattern box points to shape, center, and spread, with the latter being highlighted. The deviations from the pattern box points to outliers.\" width=\"591\" height=\"505\" \/><figcaption id=\"caption-attachment-5614\" class=\"wp-caption-text\">Figure 1. When analyzing a graph, describe the overall pattern (shape, center, spread) and look for deviations, or outliers, that don\u2019t follow the pattern.<\/figcaption><\/figure>\n<section class=\"textbox example\"><strong>Two sets of exam scores<\/p>\n<p><\/strong>Consider the following two distributions of exam scores:<\/p>\n<figure id=\"attachment_5616\" aria-describedby=\"caption-attachment-5616\" style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5616 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29235404\/picture.png\" alt=\"A boxplot graphic comparing exam scores for two classes, A and B. Each boxplot shows five number summaries. For class A, the minimum score is approximately 40, the first quartile is just above 70, the median is near 75, the third quartile is just below 80, and the maximum is around 95. There are two low outliers of 40 and 55 and two upper outliers of 90 and 95. For class B, the minimum is around 40, the first quartile is just above 60, the median is about 75, the third quartile is just below 90, with the maximum score close to 95.\" width=\"427\" height=\"186\" \/><figcaption id=\"caption-attachment-5616\" class=\"wp-caption-text\">Figure 2. Two boxplots of exam scores.<\/figcaption><\/figure>\n<p>\nBoth distributions have a median of approximately [latex]74.5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q894713\">Which distribution has more variability?<\/button><\/p>\n<div id=\"q894713\" class=\"hidden-answer\" style=\"display: none\">The answer to this question depends on how we measure variability. Both distributions have the same range. The range is the distance spanned by the data. We calculate the range by subtracting the minimum value from the maximum value.<\/p>\n<ul>\n<li>Range = Maximum value \u2013 minimum value<\/li>\n<\/ul>\n<p>For both of these data sets, the range is [latex]55[\/latex] (here is how we calculated the range: [latex]95 \u2013 40 = 55[\/latex]). If we use the range to measure variability, we say the distributions have the same amount of variability.<\/p>\n<p>But the variability in the distributions differs when we look at how the data is distributed about the median. Class A has a large portion of its data close to the median. This is not true for Class B. From this viewpoint, Class A has less variability about the median.<\/p>\n<p><strong>Therefore, Class B has more variability about the median.<\/strong><\/div>\n<\/div>\n<\/section>\n<h3 id=\"Q1\">[latex]Q1[\/latex], [latex]Q3[\/latex], and IQR<\/h3>\n<p>Now we can develop a way to measure the variability about the median. To do so, we use <em>quartiles<\/em>. Quartile marks divide the data set into four groups with equal counts.<\/p>\n<p>To find the first and third quartiles ([latex]Q1[\/latex] and [latex]Q3[\/latex] respectively), first determine the list of values that lie both above and below the median. Then, take the medians of those lists.<\/p>\n<section class=\"textbox recall\">A boxplot is constructed from five values: <strong>the minimum value, the first quartile, the median, the third quartile, and the maximum value.<\/strong><br \/>\nThis set of five numbers is called the <strong>five-number summary.\u00a0<\/strong><\/p>\n<figure id=\"attachment_3078\" aria-describedby=\"caption-attachment-3078\" style=\"width: 433px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3078\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/12\/11181025\/Screen-Shot-2022-02-11-at-1.09.31-PM.png\" alt=\"A general horizontal boxplot displaying the following features from left to right: lower outliers, minimum, Q1, median, Q3, maximum, and upper outliers. The Interquartile Range (IQR) is shown at the top of the boxplot.\" width=\"433\" height=\"240\" \/><figcaption id=\"caption-attachment-3078\" class=\"wp-caption-text\">Figure 3. A boxplot creates a visual summary of a data set using five important values: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. It also shows the outliers in the data.<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox example\"><strong>Two sets of exam scores: <\/strong>Consider the following two distributions of exam scores:<\/p>\n<figure id=\"attachment_5616\" aria-describedby=\"caption-attachment-5616\" style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5616 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29235404\/picture.png\" alt=\"A boxplot graphic comparing exam scores for two classes, A and B. Each boxplot shows five number summaries. For class A, the minimum score is approximately 40, the first quartile is just above 70, the median is near 75, the third quartile is just below 80, and the maximum is around 95. There are two low outliers of 40 and 55 and two upper outliers of 90 and 95. For class B, the minimum is around 40, the first quartile is just above 60, the median is about 75, the third quartile is just below 90, with the maximum score close to 95.\" width=\"427\" height=\"186\" \/><figcaption id=\"caption-attachment-5616\" class=\"wp-caption-text\">Figure 2. Two boxplots of exam scores.<\/figcaption><\/figure>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q835596\">Using IQR, which distribution has more variability?<\/button><\/p>\n<div id=\"q835596\" class=\"hidden-answer\" style=\"display: none\">The quartiles, together with the minimum and maximum score,s give the five-number summary:<\/p>\n<ul>\n<li>Class A: Min: [latex]40[\/latex] [latex]Q1: 71[\/latex] [latex]Q2: 74.5[\/latex] [latex]Q3: 78.5[\/latex] Max: [latex]95[\/latex]<\/li>\n<li>Class B: Min: [latex]40[\/latex] [latex]Q1: 61[\/latex] [latex]Q2: 74.5[\/latex] [latex]Q3: 89[\/latex] Max: [latex]95[\/latex]<\/li>\n<\/ul>\n<p>Notice: The second quartile mark ([latex]Q2[\/latex]) is the median.<\/p>\n<p>Notice: Some quartiles exhibit more variability in the data, even though each quartile contains the same amount of data.<\/p>\n<ul>\n<li>For example, [latex]25[\/latex]% of the scores in Class A are between [latex]40[\/latex] and [latex]71[\/latex]. There is a lot of variability in this first quartile ([latex]Q1[\/latex]). The eight scores in [latex]Q1[\/latex] vary by [latex]30[\/latex] points.<\/li>\n<li>Compare this to the third quartile ([latex]Q3[\/latex]) for Class A: [latex]25[\/latex]% of the scores in Class A are between [latex]74.5[\/latex] and [latex]78.5[\/latex]. There is not much variability in [latex]Q3[\/latex]. The eight scores in [latex]Q3[\/latex] vary by only [latex]4[\/latex] points.<\/li>\n<\/ul>\n<p>How are quartiles used to measure variability about the median? The <em>interquartile range <\/em>(IQR) is the distance between the first and third quartile marks. The IQR is a measurement of the variability about the median. More specifically, the IQR tells us the range of the middle half of the data.<\/p>\n<p>Here is the IQR for these two distributions:<\/p>\n<ul>\n<li>Class A: IQR [latex]= Q3 \u2013 Q1 = 78.5 \u2013 71 = 7.5[\/latex]<\/li>\n<li>Class B: IQR [latex]= Q3 \u2013 Q1 = 89 \u2013 61 = 28[\/latex]<\/li>\n<\/ul>\n<p>As we observed earlier, Class A has less variability about its median. Its IQR is much smaller. The middle [latex]50[\/latex]% of exam scores for Class A vary by only [latex]7.5[\/latex] points. The middle [latex]50[\/latex]% of exam scores for Class B vary by [latex]28[\/latex] points.<\/p>\n<p><strong>Using IQR, Class B has more variability.<\/strong><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2744\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2744&theme=lumen&iframe_resize_id=ohm2744&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2745\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2745&theme=lumen&iframe_resize_id=ohm2745&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Using the IQR to Identify Outliers<\/h2>\n<section class=\"textbox recall\">A value is an outlier when:<\/p>\n<ul>\n<li>Upper outlier: The value is greater than [latex]Q3 + (1.5 *[\/latex]IQR[latex])[\/latex]<\/li>\n<li>Lower outlier: The value is less than [latex]Q1 - (1.5 *[\/latex]IQR[latex])[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p>To make more sense of this rule, let\u2019s look at a visual example.<\/p>\n<section class=\"textbox tryIt\"><strong>Two sets of exam scores: <\/strong><\/p>\n<p>Consider the following two distributions of exam scores:<\/p>\n<figure id=\"attachment_5616\" aria-describedby=\"caption-attachment-5616\" style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5616 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29235404\/picture.png\" alt=\"A boxplot graphic comparing exam scores for two classes, A and B. Each boxplot shows five number summaries. For class A, the minimum score is approximately 40, the first quartile is just above 70, the median is near 75, the third quartile is just below 80, and the maximum is around 95. There are two low outliers of 40 and 55 and two upper outliers of 90 and 95. For class B, the minimum is around 40, the first quartile is just above 60, the median is about 75, the third quartile is just below 90, with the maximum score close to 95.\" width=\"427\" height=\"186\" \/><figcaption id=\"caption-attachment-5616\" class=\"wp-caption-text\">Figure 2. Two boxplots of exam scores.<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q785698\">Use IQR and identify the outliers of the data set.<\/button><\/p>\n<div id=\"q785698\" class=\"hidden-answer\" style=\"display: none\">For the Class A data set in the dotplot, [latex]Q1 = 71[\/latex] and [latex]Q3 = 78.25[\/latex], so the IQR [latex]= 7.5[\/latex].<\/p>\n<ul>\n<li>[latex]Q1 - (1.5 *[\/latex]IQR[latex]) = 71 \u2013 1.5 * 7.5 = 71 \u2013 11.25 = 59.75[\/latex]\n<ul>\n<li>The data point at [latex]40[\/latex] and [latex]55[\/latex] is considered an outlier because it is below [latex]59.75[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>[latex]Q3 + (1.5 *[\/latex]IQR[latex]) = 71 + 1.5 * 7.5 = 71 + 11.25 = 82.25[\/latex]\n<ul>\n<li>The data points at [latex]85, 90[\/latex], and [latex]95[\/latex] are considered outliers because they are above [latex]82.25[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<figure id=\"attachment_360\" aria-describedby=\"caption-attachment-360\" style=\"width: 532px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-360\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/08\/09215145\/Screen-Shot-2022-08-09-at-2.51.38-PM.png\" alt=\"\" width=\"532\" height=\"193\" \/><figcaption id=\"caption-attachment-360\" class=\"wp-caption-text\">Figure 4. Dot plots comparing exam scores for two classes, A and B, with both having the same mean score of 74.5; with various outliers.<\/figcaption><\/figure>\n<p>Now do the same for Class B. 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