{"id":69,"date":"2023-01-31T00:46:22","date_gmt":"2023-01-31T00:46:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/distribution-of-quantitative-variables-dig-deeper\/"},"modified":"2025-05-11T19:43:20","modified_gmt":"2025-05-11T19:43:20","slug":"distribution-of-quantitative-variables-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/distribution-of-quantitative-variables-fresh-take\/","title":{"raw":"Distribution of Quantitative Variables: Fresh Take","rendered":"Distribution of Quantitative Variables: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Describe the graph of a data set using shape, center, spread, and outliers&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:0,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Describe the graph of a data set using its shape, center, spread, and outliers<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Shape, Center, and Spread<\/h2>\r\n<p>When the observations of a quantitative variable are displayed in a graph, we call the display the distribution of a <strong>quantitative variable<\/strong>. We display the distribution to read information about the dataset from its graph. The first three characteristics of the distribution are shape, center, and spread.<\/p>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<h3><strong>The Main Idea<\/strong><\/h3>\r\n<p><strong>Shape:<\/strong> Is the data symmetrically distributed, or does it \u201cbunch up\u201d on one side or the other?<\/p>\r\n<ul>\r\n\t<li>If the majority of the data lies to the left with a long tail of lower values to the right, we say it is <strong>right-skewed.<\/strong><\/li>\r\n\t<li>If the majority of the data lies to the right with a long tail of lower values to the left, we say it is <strong>left-skewed.<\/strong><\/li>\r\n\t<li>If the data is centered and falls out evenly on both sides, we say it is <strong>symmetric.<\/strong><\/li>\r\n\t<li><strong>Unimodal data<\/strong> has one mound (cluster) of data. <strong>Bimodal<\/strong> has two mounds of data. <strong>Multimodal<\/strong> has more than two mounds of data.<\/li>\r\n<\/ul>\r\n<p><strong>Center:<\/strong> Where does the center of the data appear to be? Center can be measured as the <strong>median<\/strong> or the <strong>mean<\/strong> of the dataset. When looking at the distribution, you should consider where the heaviest \u201cweight\u201d of the data lies.<\/p>\r\n<p><strong>Spread:<\/strong> The spread of a data distribution measures the <strong>range<\/strong> of the data (from least to greatest, found by subtracting the smallest value from the largest). Spread is also concerned with <strong>gaps<\/strong> in the data and with <strong>outliers<\/strong>, which are rare values far to the left (lower outliers) or to the right (upper outliers) of the bulk of the data. Outliers extend the range beyond what the bulk of the data indicates it should be. Extreme outliers can affect the mean of the dataset, pulling it in the direction of the outlier.<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<p>The following video will help you visually examine a quantitative data distribution for shape, center, spread, and the presence of outliers.<\/p>\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/2Y2l9QJCe6M[\/embed]<\/p>\r\n<\/section>\r\n<section class=\"textbox recall\">A\u00a0<strong>histogram<\/strong> looks somewhat like a bar graph. But while a bar graph displays categorical data and shows counts of observations within categories, a histogram displays quantitative data by showing frequencies of a quantitative variable. The bars of a histogram are each of the same width and meet smoothly together over the horizontal axis. The width of each bar covers a range of values along the axis called a bin<em>.\u00a0<\/em>A\u00a0<strong>bin\u00a0<\/strong>is a range of values that the quantitative variable can take.\u00a0A bin can be defined by its\u00a0<strong>end points<\/strong>, the smallest and largest values of the quantitative variable represented in the bin.\u00a0The\u00a0<strong>width<\/strong>\u00a0of the bin, called\u00a0<strong>binwidth<\/strong>, is calculated by taking the difference between the values of the end points.<\/section>\r\n<section><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" width=\"100%\" height=\"850\"><\/iframe> <br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]795[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]796[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]797[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]798[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]799[\/ohm2_question]\u00a0<\/section>\r\n<h3>Distribution of a Quantitative Variable<\/h3>\r\n<p>When we describe patterns in data, we use descriptions of <strong>shape<\/strong>,\u00a0<strong>center<\/strong>, and\u00a0<strong>spread<\/strong>. We also describe exceptions to the pattern. We call these exceptions\u00a0<strong>outliers<\/strong>.<\/p>\r\n\r\n[caption id=\"attachment_3061\" align=\"aligncenter\" width=\"450\"]<img class=\"wp-image-3061\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29172703\/3.4L.png\" alt=\"Flow chart with three levels. The first level is &quot;Graph the distribution of a quantitiative variable&quot; which points to two different boxes on the second level, &quot;Overall pattern&quot; and &quot;Deviations from the pattern&quot;. Overall pattern points to three options, &quot;Shape&quot;, &quot;Center&quot;, and &quot;Spread.&quot; &quot;Deviations from the pattern&quot; points to one option, &quot;Outliers.&quot;\" width=\"450\" height=\"377\" \/> 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<div>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]805[\/ohm2_question]<\/section>\r\n<\/div>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]815[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]816[\/ohm2_question]<\/section>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]817[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]818[\/ohm2_question]<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Describe the graph of a data set using shape, center, spread, and outliers&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:0,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Describe the graph of a data set using its shape, center, spread, and outliers<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Shape, Center, and Spread<\/h2>\n<p>When the observations of a quantitative variable are displayed in a graph, we call the display the distribution of a <strong>quantitative variable<\/strong>. We display the distribution to read information about the dataset from its graph. The first three characteristics of the distribution are shape, center, and spread.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<h3><strong>The Main Idea<\/strong><\/h3>\n<p><strong>Shape:<\/strong> Is the data symmetrically distributed, or does it \u201cbunch up\u201d on one side or the other?<\/p>\n<ul>\n<li>If the majority of the data lies to the left with a long tail of lower values to the right, we say it is <strong>right-skewed.<\/strong><\/li>\n<li>If the majority of the data lies to the right with a long tail of lower values to the left, we say it is <strong>left-skewed.<\/strong><\/li>\n<li>If the data is centered and falls out evenly on both sides, we say it is <strong>symmetric.<\/strong><\/li>\n<li><strong>Unimodal data<\/strong> has one mound (cluster) of data. <strong>Bimodal<\/strong> has two mounds of data. <strong>Multimodal<\/strong> has more than two mounds of data.<\/li>\n<\/ul>\n<p><strong>Center:<\/strong> Where does the center of the data appear to be? Center can be measured as the <strong>median<\/strong> or the <strong>mean<\/strong> of the dataset. When looking at the distribution, you should consider where the heaviest \u201cweight\u201d of the data lies.<\/p>\n<p><strong>Spread:<\/strong> The spread of a data distribution measures the <strong>range<\/strong> of the data (from least to greatest, found by subtracting the smallest value from the largest). Spread is also concerned with <strong>gaps<\/strong> in the data and with <strong>outliers<\/strong>, which are rare values far to the left (lower outliers) or to the right (upper outliers) of the bulk of the data. Outliers extend the range beyond what the bulk of the data indicates it should be. Extreme outliers can affect the mean of the dataset, pulling it in the direction of the outlier.<\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<p>The following video will help you visually examine a quantitative data distribution for shape, center, spread, and the presence of outliers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Describing Distributions: Center, Spread &amp; Shape | Statistics Tutorial | MarinStatsLectures\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2Y2l9QJCe6M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox recall\">A\u00a0<strong>histogram<\/strong> looks somewhat like a bar graph. But while a bar graph displays categorical data and shows counts of observations within categories, a histogram displays quantitative data by showing frequencies of a quantitative variable. The bars of a histogram are each of the same width and meet smoothly together over the horizontal axis. The width of each bar covers a range of values along the axis called a bin<em>.\u00a0<\/em>A\u00a0<strong>bin\u00a0<\/strong>is a range of values that the quantitative variable can take.\u00a0A bin can be defined by its\u00a0<strong>end points<\/strong>, the smallest and largest values of the quantitative variable represented in the bin.\u00a0The\u00a0<strong>width<\/strong>\u00a0of the bin, called\u00a0<strong>binwidth<\/strong>, is calculated by taking the difference between the values of the end points.<\/section>\n<section><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" width=\"100%\" height=\"850\"><\/iframe> <br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm795\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=795&theme=lumen&iframe_resize_id=ohm795&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm796\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=796&theme=lumen&iframe_resize_id=ohm796&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm797\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=797&theme=lumen&iframe_resize_id=ohm797&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm798\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=798&theme=lumen&iframe_resize_id=ohm798&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm799\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=799&theme=lumen&iframe_resize_id=ohm799&source=tnh\" width=\"100%\" height=\"150\"><\/iframe>\u00a0<\/section>\n<h3>Distribution of a Quantitative Variable<\/h3>\n<p>When we describe patterns in data, we use descriptions of <strong>shape<\/strong>,\u00a0<strong>center<\/strong>, and\u00a0<strong>spread<\/strong>. We also describe exceptions to the pattern. We call these exceptions\u00a0<strong>outliers<\/strong>.<\/p>\n<figure id=\"attachment_3061\" aria-describedby=\"caption-attachment-3061\" style=\"width: 450px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3061\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/29172703\/3.4L.png\" alt=\"Flow chart with three levels. The first level is &quot;Graph the distribution of a quantitiative variable&quot; which points to two different boxes on the second level, &quot;Overall pattern&quot; and &quot;Deviations from the pattern&quot;. Overall pattern points to three options, &quot;Shape&quot;, &quot;Center&quot;, and &quot;Spread.&quot; &quot;Deviations from the pattern&quot; points to one option, &quot;Outliers.&quot;\" width=\"450\" height=\"377\" \/><figcaption id=\"caption-attachment-3061\" 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<div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm805\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=805&theme=lumen&iframe_resize_id=ohm805&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm815\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=815&theme=lumen&iframe_resize_id=ohm815&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm816\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=816&theme=lumen&iframe_resize_id=ohm816&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm817\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=817&theme=lumen&iframe_resize_id=ohm817&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm818\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=818&theme=lumen&iframe_resize_id=ohm818&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n","protected":false},"author":6,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":20,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/69"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/69\/revisions"}],"predecessor-version":[{"id":6614,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/69\/revisions\/6614"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/69\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=69"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=69"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=69"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=69"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}