{"id":60,"date":"2023-01-31T00:46:13","date_gmt":"2023-01-31T00:46:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/distribution-of-quantitative-variables-learn-it-1\/"},"modified":"2025-05-11T19:40:31","modified_gmt":"2025-05-11T19:40:31","slug":"distribution-of-quantitative-variables-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/distribution-of-quantitative-variables-learn-it-1\/","title":{"raw":"Distribution of Quantitative Variables: Learn It 1","rendered":"Distribution of Quantitative Variables: Learn It 1"},"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>Describing Distribution<\/h2>\r\n<p>We can use different graphs, such as dotplots and histograms, to summarize the distribution of a quantitative variable. By displaying the data in such graphs, we can describe features of the distribution of the quantitative variable. The features used to describe the distribution of a quantitative variable are: Shape, center, spread, and presence of outliers. Let's look at the species and size measurements of [latex]342[\/latex] penguins found foraging near Palmer Station, Antarctica.<\/p>\r\n\r\n[caption id=\"attachment_1863\" align=\"aligncenter\" width=\"600\"]<img class=\"wp-image-1863\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/01\/31004611\/pexels-david-dibert-1299391.jpg\" alt=\"Black and white penguins in a group walking across sand away from waves of water. \" width=\"600\" height=\"400\" \/> Figure 1. Field data on penguins, like body size and species type, can be graphed and analyzed to explore the shape, center, and spread of distributions.[\/caption]\r\n\r\n<h3>Shape<\/h3>\r\n<p>To describe the shape of a distribution, imagine sketching the outline of the data to emphasize the general trend. The description of shape includes two parts: the overall pattern and the number of peaks.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>overall shape pattern<br \/>\r\n(left-skewed, symmetric, right-skewed)<\/h3>\r\n<p><strong>Left-skewed<\/strong>: A cluster of data on the right with a tail of data tapering off to the left. A left-skewed distribution has a lot of data at higher variable values with smaller amounts of data at lower variable values.<\/p>\r\n<p><strong>Symmetric <\/strong>(also called <strong>bell-shaped<\/strong>): A cluster of data with a central peak where the left and right sides of the distribution <em>closely<\/em> <em>mirror<\/em> each other. If you drew a vertical line down the center of the distribution and folded it in half, the left and right sides would basically match. A bell-shaped distribution has a lot of data in the center, with smaller amounts of data tapering off in each direction.<\/p>\r\n<p><strong>Right-skewed<\/strong>:\u00a0A cluster of data on the left with a\u00a0tail\u00a0of data tapering off to the right. A right-skewed distribution has a lot of data at lower variable values with smaller amounts of data at higher variable values.<\/p>\r\n\r\n[caption id=\"attachment_905\" align=\"aligncenter\" width=\"972\"]<img class=\"wp-image-905 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/12191051\/Picture12.png\" alt=\"Three clusters of data. On the left side there is an image of left-skewed data, in the center there is a cluster of symmetric data, and on the right side there is right-skewed data.\" width=\"972\" height=\"374\" \/> Figure 2. These three histograms show common distribution shapes\u2014left-skewed, symmetric, and right-skewed\u2014each with different patterns in how the data is spread.[\/caption]\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>number of peaks<br \/>\r\n(unimodal, bimodal, multimodal, uniform)<\/h3>\r\n<p><strong>Unimodal<\/strong>: There is one prominent peak.<\/p>\r\n<p><strong>Bimodal<\/strong>: There are two prominent peaks.<\/p>\r\n<p><strong>Multimodal<\/strong>: There are three or more prominent peaks.<\/p>\r\n<p><strong>Uniform<\/strong>: There are no prominent peaks. A rectangular shape, with the same amount of data for each variable value.<\/p>\r\n\r\n[caption id=\"attachment_906\" align=\"aligncenter\" width=\"936\"]<img class=\"wp-image-906 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/12191227\/Picture13.png\" alt=\"Four different graphs showing different peaks. From the top left to right there is unimodal and bimodal peaks shown, and on the bottom left to write shows multimodal and uniform data.\" width=\"936\" height=\"554\" \/> Figure 3. These histograms show four types of distribution shapes based on the number of peaks: unimodal, bimodal, multimodal, and uniform.[\/caption]\r\n<\/section>\r\n<section class=\"textbox interact\">Create a histogram for the data set \"<strong>Penguins \u2013 Body Mass\"<\/strong> using the Describing and Exploring Quantitative Variables tool.[reveal-answer q=\"902344\"]Steps to create a histogram.[\/reveal-answer][hidden-answer a=\"902344\"]\r\n\r\n<p style=\"padding-left: 30px;\"><strong>STEP 1:<\/strong> Select the \"Single Group\" tab at the top of the page.<\/p>\r\n<p style=\"padding-left: 30px;\"><strong>STEP 2:<\/strong> Choose the data set \"Penguins \u2013 Body Mass.\"<\/p>\r\n<p style=\"padding-left: 30px;\"><strong>STEP 3:<\/strong> Unselect everything but \u201cHistogram.\u201d<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p><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>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]800[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]801[\/ohm2_question]<\/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>Describing Distribution<\/h2>\n<p>We can use different graphs, such as dotplots and histograms, to summarize the distribution of a quantitative variable. By displaying the data in such graphs, we can describe features of the distribution of the quantitative variable. The features used to describe the distribution of a quantitative variable are: Shape, center, spread, and presence of outliers. Let&#8217;s look at the species and size measurements of [latex]342[\/latex] penguins found foraging near Palmer Station, Antarctica.<\/p>\n<figure id=\"attachment_1863\" aria-describedby=\"caption-attachment-1863\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1863\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/01\/31004611\/pexels-david-dibert-1299391.jpg\" alt=\"Black and white penguins in a group walking across sand away from waves of water.\" width=\"600\" height=\"400\" \/><figcaption id=\"caption-attachment-1863\" class=\"wp-caption-text\">Figure 1. Field data on penguins, like body size and species type, can be graphed and analyzed to explore the shape, center, and spread of distributions.<\/figcaption><\/figure>\n<h3>Shape<\/h3>\n<p>To describe the shape of a distribution, imagine sketching the outline of the data to emphasize the general trend. The description of shape includes two parts: the overall pattern and the number of peaks.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>overall shape pattern<br \/>\n(left-skewed, symmetric, right-skewed)<\/h3>\n<p><strong>Left-skewed<\/strong>: A cluster of data on the right with a tail of data tapering off to the left. A left-skewed distribution has a lot of data at higher variable values with smaller amounts of data at lower variable values.<\/p>\n<p><strong>Symmetric <\/strong>(also called <strong>bell-shaped<\/strong>): A cluster of data with a central peak where the left and right sides of the distribution <em>closely<\/em> <em>mirror<\/em> each other. If you drew a vertical line down the center of the distribution and folded it in half, the left and right sides would basically match. A bell-shaped distribution has a lot of data in the center, with smaller amounts of data tapering off in each direction.<\/p>\n<p><strong>Right-skewed<\/strong>:\u00a0A cluster of data on the left with a\u00a0tail\u00a0of data tapering off to the right. A right-skewed distribution has a lot of data at lower variable values with smaller amounts of data at higher variable values.<\/p>\n<figure id=\"attachment_905\" aria-describedby=\"caption-attachment-905\" style=\"width: 972px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-905 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/12191051\/Picture12.png\" alt=\"Three clusters of data. On the left side there is an image of left-skewed data, in the center there is a cluster of symmetric data, and on the right side there is right-skewed data.\" width=\"972\" height=\"374\" \/><figcaption id=\"caption-attachment-905\" class=\"wp-caption-text\">Figure 2. These three histograms show common distribution shapes\u2014left-skewed, symmetric, and right-skewed\u2014each with different patterns in how the data is spread.<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>number of peaks<br \/>\n(unimodal, bimodal, multimodal, uniform)<\/h3>\n<p><strong>Unimodal<\/strong>: There is one prominent peak.<\/p>\n<p><strong>Bimodal<\/strong>: There are two prominent peaks.<\/p>\n<p><strong>Multimodal<\/strong>: There are three or more prominent peaks.<\/p>\n<p><strong>Uniform<\/strong>: There are no prominent peaks. A rectangular shape, with the same amount of data for each variable value.<\/p>\n<figure id=\"attachment_906\" aria-describedby=\"caption-attachment-906\" style=\"width: 936px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-906 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/12191227\/Picture13.png\" alt=\"Four different graphs showing different peaks. From the top left to right there is unimodal and bimodal peaks shown, and on the bottom left to write shows multimodal and uniform data.\" width=\"936\" height=\"554\" \/><figcaption id=\"caption-attachment-906\" class=\"wp-caption-text\">Figure 3. These histograms show four types of distribution shapes based on the number of peaks: unimodal, bimodal, multimodal, and uniform.<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox interact\">Create a histogram for the data set &#8220;<strong>Penguins \u2013 Body Mass&#8221;<\/strong> using the Describing and Exploring Quantitative Variables tool.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q902344\">Steps to create a histogram.<\/button><\/p>\n<div id=\"q902344\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"padding-left: 30px;\"><strong>STEP 1:<\/strong> Select the &#8220;Single Group&#8221; tab at the top of the page.<\/p>\n<p style=\"padding-left: 30px;\"><strong>STEP 2:<\/strong> Choose the data set &#8220;Penguins \u2013 Body Mass.&#8221;<\/p>\n<p style=\"padding-left: 30px;\"><strong>STEP 3:<\/strong> Unselect everything but \u201cHistogram.\u201d<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm800\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=800&theme=lumen&iframe_resize_id=ohm800&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm801\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=801&theme=lumen&iframe_resize_id=ohm801&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":6,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Penguins\",\"author\":\"David Dibert\",\"organization\":\"Pexels\",\"url\":\"https:\/\/www.pexels.com\/photo\/flock-of-penguins-near-sea-1299391\/\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":20,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Penguins","author":"David Dibert","organization":"Pexels","url":"https:\/\/www.pexels.com\/photo\/flock-of-penguins-near-sea-1299391\/","project":"","license":"cc0","license_terms":""}],"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\/60"}],"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":13,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/60\/revisions"}],"predecessor-version":[{"id":6608,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/60\/revisions\/6608"}],"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\/60\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=60"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=60"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=60"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=60"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}