{"id":2079,"date":"2021-08-03T17:33:01","date_gmt":"2021-08-03T21:33:01","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=2079"},"modified":"2025-05-07T17:35:57","modified_gmt":"2025-05-07T21:35:57","slug":"1-5-shape-of-a-distribution","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/1-5-shape-of-a-distribution\/","title":{"raw":"1.5 Shape of a Distribution","rendered":"1.5 Shape of a Distribution"},"content":{"raw":"A histogram shows the shape of the distribution of a quantitative variable. The shape of a distribution includes the following three aspects:\r\n<ul>\r\n \t<li>Overall shape: what the distribution looks like, e.g., bell-shaped, J-shaped, triangular, and uniform. (See examples in the figures below.)<\/li>\r\n \t<li>Modality: number of peaks (highest points). A distribution is called <strong>unimodal<\/strong> if it has only one peak, <strong>bimodal<\/strong> if it has two peaks, and multimodal if it has more than two peaks.<\/li>\r\n \t<li>Symmetry and skewness. If you fold a distribution in the middle and the two parts can match, the distribution is called <strong>symmetric<\/strong>. If it has a longer left tail, it is called skewed to the left (or left skewed), and skewed to the right (or right skewed) if it has a longer right tail.<\/li>\r\n<\/ul>\r\nThe following figure shows some special shapes of distributions.<a id=\"retfig1.8\"><\/a>\r\n\r\n[caption id=\"attachment_3030\" align=\"aligncenter\" width=\"800\"]<img class=\"wp-image-3030 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/shape.png\" alt=\"Nine special shapes of distributions presented in three rows and three columns. Image description available.\" width=\"800\" height=\"600\" \/> <strong>Figure 1.8<\/strong>: Some Special Shapes of Distributions [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig1.8\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig1.8\">(See Appendix D Figure 1.8)<\/a>][\/caption]Distributions in Figure 1.8 can be described respectively as follows:\r\n\r\n(a) bell-shaped, unimodal, and symmetric\r\n\r\n(b) triangular, unimodal, and symmetric\r\n\r\n(c) rectangular, no peak, and symmetric\r\n\r\n(d) unimodal and right-skewed\r\n\r\n(e) unimodal and left-skewed\r\n\r\n(f) J-shaped, unimodal, and left skewed\r\n\r\n(g) reversed J-shaped, unimodal, and right-skewed\r\n\r\n(h) bimodal and symmetric\r\n\r\n(i) multimodal and asymmetric\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example: Shape of a Distribution<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nComment on the overall shape and modality of the following histogram.\r\n\r\n<img class=\"wp-image-68 aligncenter\" style=\"margin-bottom: 4.44444em; color: #333333;\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/m01_example_Grades_Histogram.png\" alt=\"Histogram of grade, the y-axis is frequency and x-axis is grades. Image description available\" width=\"380\" height=\"363\" \/><a id=\"retfig1.8.1\"><\/a>Figure 1.8.1: The histogram of the grades data above shows its distribution is bell-shaped, unimodal, and symmetric. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig1.8.1\">Image Description (See Appendix D Figure 1.8.1)<\/a>]\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div style=\"height: 55px; margin-top: 2.1428571429em;\">\r\n\r\n<img class=\"size-full wp-image-99 alignleft\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png\" alt=\"\" width=\"250\" height=\"50\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">Exercise: Shape of a Distribution<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFigure 1.9 is the histogram of survival time in years after cancer diagnosis. Comment on the overall shape and modality of the histogram.<a id=\"retfig1.9\"><\/a>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"336\"]<img style=\"color: #373d3f; font-weight: bold; font-size: 1em;\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/m02_FiveNumber_Historgram-300x267.png\" alt=\"Histogram of survival time after diagnosis of cancer, the y-axis is frequency and x-axis is survival time in years. Image description available\" width=\"336\" height=\"299\" \/> <strong>Figure 1.9<\/strong>: Histogram of Survival Time (in years) [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig1.9\">Image Description (See Appendix D Figure 1.9)<\/a>][\/caption]<details><summary>Show\/Hide Answer<\/summary>We can see the distribution of the survival time is unimodal and right-skewed.\r\n\r\n<\/details><\/div>\r\n<\/div>","rendered":"<p>A histogram shows the shape of the distribution of a quantitative variable. The shape of a distribution includes the following three aspects:<\/p>\n<ul>\n<li>Overall shape: what the distribution looks like, e.g., bell-shaped, J-shaped, triangular, and uniform. (See examples in the figures below.)<\/li>\n<li>Modality: number of peaks (highest points). A distribution is called <strong>unimodal<\/strong> if it has only one peak, <strong>bimodal<\/strong> if it has two peaks, and multimodal if it has more than two peaks.<\/li>\n<li>Symmetry and skewness. If you fold a distribution in the middle and the two parts can match, the distribution is called <strong>symmetric<\/strong>. If it has a longer left tail, it is called skewed to the left (or left skewed), and skewed to the right (or right skewed) if it has a longer right tail.<\/li>\n<\/ul>\n<p>The following figure shows some special shapes of distributions.<a id=\"retfig1.8\"><\/a><\/p>\n<figure id=\"attachment_3030\" aria-describedby=\"caption-attachment-3030\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3030 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/shape.png\" alt=\"Nine special shapes of distributions presented in three rows and three columns. Image description available.\" width=\"800\" height=\"600\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/shape.png 800w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/shape-300x225.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/shape-768x576.png 768w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/shape-65x49.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/shape-225x169.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/shape-350x263.png 350w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><figcaption id=\"caption-attachment-3030\" class=\"wp-caption-text\"><strong>Figure 1.8<\/strong>: Some Special Shapes of Distributions [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig1.8\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig1.8\">(See Appendix D Figure 1.8)<\/a>]<\/figcaption><\/figure>\n<p>Distributions in Figure 1.8 can be described respectively as follows:<\/p>\n<p>(a) bell-shaped, unimodal, and symmetric<\/p>\n<p>(b) triangular, unimodal, and symmetric<\/p>\n<p>(c) rectangular, no peak, and symmetric<\/p>\n<p>(d) unimodal and right-skewed<\/p>\n<p>(e) unimodal and left-skewed<\/p>\n<p>(f) J-shaped, unimodal, and left skewed<\/p>\n<p>(g) reversed J-shaped, unimodal, and right-skewed<\/p>\n<p>(h) bimodal and symmetric<\/p>\n<p>(i) multimodal and asymmetric<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Shape of a Distribution<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Comment on the overall shape and modality of the following histogram.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-68 aligncenter\" style=\"margin-bottom: 4.44444em; color: #333333;\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/m01_example_Grades_Histogram.png\" alt=\"Histogram of grade, the y-axis is frequency and x-axis is grades. Image description available\" width=\"380\" height=\"363\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/m01_example_Grades_Histogram.png 351w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/m01_example_Grades_Histogram-65x62.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/m01_example_Grades_Histogram-225x214.png 225w\" sizes=\"auto, (max-width: 380px) 100vw, 380px\" \/><a id=\"retfig1.8.1\"><\/a>Figure 1.8.1: The histogram of the grades data above shows its distribution is bell-shaped, unimodal, and symmetric. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig1.8.1\">Image Description (See Appendix D Figure 1.8.1)<\/a>]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"height: 55px; margin-top: 2.1428571429em;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-99 alignleft\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png\" alt=\"\" width=\"250\" height=\"50\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png 250w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity-65x13.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity-225x45.png 225w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/p>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">Exercise: Shape of a Distribution<\/header>\n<div class=\"textbox__content\">\n<p>Figure 1.9 is the histogram of survival time in years after cancer diagnosis. Comment on the overall shape and modality of the histogram.<a id=\"retfig1.9\"><\/a><\/p>\n<figure style=\"width: 336px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" style=\"color: #373d3f; font-weight: bold; font-size: 1em;\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/m02_FiveNumber_Historgram-300x267.png\" alt=\"Histogram of survival time after diagnosis of cancer, the y-axis is frequency and x-axis is survival time in years. Image description available\" width=\"336\" height=\"299\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.9<\/strong>: Histogram of Survival Time (in years) [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig1.9\">Image Description (See Appendix D Figure 1.9)<\/a>]<\/figcaption><\/figure>\n<details>\n<summary>Show\/Hide Answer<\/summary>\n<p>We can see the distribution of the survival time is unimodal and right-skewed.<\/p>\n<\/details>\n<\/div>\n<\/div>\n","protected":false},"author":19,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2079","chapter","type-chapter","status-publish","hentry"],"part":34,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2079","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":32,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2079\/revisions"}],"predecessor-version":[{"id":5218,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2079\/revisions\/5218"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/34"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2079\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=2079"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=2079"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=2079"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=2079"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}