{"id":261,"date":"2020-06-29T11:32:59","date_gmt":"2020-06-29T15:32:59","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=261"},"modified":"2025-06-19T15:58:19","modified_gmt":"2025-06-19T19:58:19","slug":"2-2-quartiles-and-percentiles","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/2-2-quartiles-and-percentiles\/","title":{"raw":"2.2 Quartiles and Percentiles","rendered":"2.2 Quartiles and Percentiles"},"content":{"raw":"In addition to the measures of centre, some other measures can be used to describe a distribution such as quartiles and percentiles. Recall that the median is the middle value that divides the sorted data into two halves with an equal number of observations; that is, 50% of the observations are below the median, and another 50% are above the median. Similarly, quartiles and percentiles of a distribution are defined as follows:\r\n<ul>\r\n \t<li><strong>Quartiles<\/strong> are the three values that divide the sorted data into four parts with an equal number of observations, denoted as [latex]Q_1, Q_2, Q_3[\/latex]. Each part contains 25% of the data. Actually, the second quartile [latex]Q_2[\/latex] is the median of the entire data set; the first quartile [latex]Q_1[\/latex] is the median of the bottom 50% (first half) and the third quartile [latex]Q_3[\/latex] is the median of the top 50% (second half). <strong>Note that when the number of observations [latex]n[\/latex] is odd, we include the median in both the first half and the second half when calculating [latex]Q_1[\/latex] and [latex]Q_3[\/latex].<\/strong><\/li>\r\n \t<li><strong>Percentiles<\/strong> are those 99 values that divide the sorted data into 100 parts with an equal number of observations. Each part contains 1% of the data. The first quartile [latex]Q_1[\/latex] is the 25th percentile, the second quartile [latex]Q_2[\/latex] (median) is the 50th percentile, and the third quartile [latex]Q_3[\/latex] is the 75th percentile. In this course, we will not calculate percentiles by hand, except for the important special cases of quartiles. The software can calculate any arbitrary percentiles for us.<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example: Find the Quartiles<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the quartiles for 3, 1, 9, 7, 5, 11, 21\r\n\r\nSteps:\r\n<ol>\r\n \t<li>Sort into 1, 3, 5, <strong>7<\/strong>, 9, 11, 21.<\/li>\r\n \t<li>n = 7 is odd, [latex]Q_2[\/latex] = median = 7.<\/li>\r\n \t<li>The bottom half consists of the first three smallest observations and the median, i.e., 1, <strong>3<\/strong>, <strong>5<\/strong>, 7. [latex]Q_1[\/latex] is the median of the first half, i.e., [latex] Q_1 = \\frac{3+5}{2} = 4[\/latex].<\/li>\r\n \t<li>The top half consists of the three largest values and the median, i.e., 7, <strong>9<\/strong>, <strong>11<\/strong>, 21. [latex]Q_3[\/latex] is the median of the second half, [latex] Q_3 = \\frac{9+11}{2} = 10[\/latex].<\/li>\r\n<\/ol>\r\nTherefore, the quartiles are [latex]Q_1 = 4, Q_2=7, Q_3=10[\/latex].\r\n\r\n<strong>Note<\/strong>: since the number of observations [latex]n=7[\/latex] which is odd, we include the median [latex]Q_2=7[\/latex] in both the first half {1, 3, 5, <strong>7<\/strong>} and the second half {<strong>7<\/strong>, 9, 11, 21}.\r\n\r\n<\/div>\r\n<\/div>\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\">\r\n<p class=\"textbox__title\">Exercise: Find the Quartiles<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the quartiles for the data 3, 1, 9, 7, 5, 11, 21, 19.\r\n\r\n<details><summary>Show\/Hide Answer<\/summary><strong>Steps<\/strong>:\r\n<ol>\r\n \t<li>Sort into 1, 3, 5, 7, 9, 11, 19, 21.<\/li>\r\n \t<li>n=8 is even, [latex]Q_2[\/latex] = median = \u00a0[latex] \\frac{7+9}{2} = 8[\/latex].<\/li>\r\n \t<li>The bottom half is the first four observations in the sorted list 1, 3, 5, 7, and [latex]Q_1[\/latex] is the median of the first half, i.e., [latex] Q_1 = \\frac{3+5}{2} = 4[\/latex].<\/li>\r\n \t<li>The top half is the last four observations 9, 11, 19, 21, and [latex] Q_3 = \\frac{11+19}{2} = 15[\/latex] is the median of the second half.<\/li>\r\n<\/ol>\r\nTherefore, the quartiles are [latex]Q_1 =4, Q_2=8, Q_3=15[\/latex].\r\n\r\n<\/details><\/div>\r\n<\/div>","rendered":"<p>In addition to the measures of centre, some other measures can be used to describe a distribution such as quartiles and percentiles. Recall that the median is the middle value that divides the sorted data into two halves with an equal number of observations; that is, 50% of the observations are below the median, and another 50% are above the median. Similarly, quartiles and percentiles of a distribution are defined as follows:<\/p>\n<ul>\n<li><strong>Quartiles<\/strong> are the three values that divide the sorted data into four parts with an equal number of observations, denoted as [latex]Q_1, Q_2, Q_3[\/latex]. Each part contains 25% of the data. Actually, the second quartile [latex]Q_2[\/latex] is the median of the entire data set; the first quartile [latex]Q_1[\/latex] is the median of the bottom 50% (first half) and the third quartile [latex]Q_3[\/latex] is the median of the top 50% (second half). <strong>Note that when the number of observations [latex]n[\/latex] is odd, we include the median in both the first half and the second half when calculating [latex]Q_1[\/latex] and [latex]Q_3[\/latex].<\/strong><\/li>\n<li><strong>Percentiles<\/strong> are those 99 values that divide the sorted data into 100 parts with an equal number of observations. Each part contains 1% of the data. The first quartile [latex]Q_1[\/latex] is the 25th percentile, the second quartile [latex]Q_2[\/latex] (median) is the 50th percentile, and the third quartile [latex]Q_3[\/latex] is the 75th percentile. In this course, we will not calculate percentiles by hand, except for the important special cases of quartiles. The software can calculate any arbitrary percentiles for us.<\/li>\n<\/ul>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Find the Quartiles<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the quartiles for 3, 1, 9, 7, 5, 11, 21<\/p>\n<p>Steps:<\/p>\n<ol>\n<li>Sort into 1, 3, 5, <strong>7<\/strong>, 9, 11, 21.<\/li>\n<li>n = 7 is odd, [latex]Q_2[\/latex] = median = 7.<\/li>\n<li>The bottom half consists of the first three smallest observations and the median, i.e., 1, <strong>3<\/strong>, <strong>5<\/strong>, 7. [latex]Q_1[\/latex] is the median of the first half, i.e., [latex]Q_1 = \\frac{3+5}{2} = 4[\/latex].<\/li>\n<li>The top half consists of the three largest values and the median, i.e., 7, <strong>9<\/strong>, <strong>11<\/strong>, 21. [latex]Q_3[\/latex] is the median of the second half, [latex]Q_3 = \\frac{9+11}{2} = 10[\/latex].<\/li>\n<\/ol>\n<p>Therefore, the quartiles are [latex]Q_1 = 4, Q_2=7, Q_3=10[\/latex].<\/p>\n<p><strong>Note<\/strong>: since the number of observations [latex]n=7[\/latex] which is odd, we include the median [latex]Q_2=7[\/latex] in both the first half {1, 3, 5, <strong>7<\/strong>} and the second half {<strong>7<\/strong>, 9, 11, 21}.<\/p>\n<\/div>\n<\/div>\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\">\n<p class=\"textbox__title\">Exercise: Find the Quartiles<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the quartiles for the data 3, 1, 9, 7, 5, 11, 21, 19.<\/p>\n<details>\n<summary>Show\/Hide Answer<\/summary>\n<p><strong>Steps<\/strong>:<\/p>\n<ol>\n<li>Sort into 1, 3, 5, 7, 9, 11, 19, 21.<\/li>\n<li>n=8 is even, [latex]Q_2[\/latex] = median = \u00a0[latex]\\frac{7+9}{2} = 8[\/latex].<\/li>\n<li>The bottom half is the first four observations in the sorted list 1, 3, 5, 7, and [latex]Q_1[\/latex] is the median of the first half, i.e., [latex]Q_1 = \\frac{3+5}{2} = 4[\/latex].<\/li>\n<li>The top half is the last four observations 9, 11, 19, 21, and [latex]Q_3 = \\frac{11+19}{2} = 15[\/latex] is the median of the second half.<\/li>\n<\/ol>\n<p>Therefore, the quartiles are [latex]Q_1 =4, Q_2=8, Q_3=15[\/latex].<\/p>\n<\/details>\n<\/div>\n<\/div>\n","protected":false},"author":19,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-261","chapter","type-chapter","status-publish","hentry"],"part":209,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/261","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":23,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/261\/revisions"}],"predecessor-version":[{"id":5562,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/261\/revisions\/5562"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/209"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/261\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=261"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=261"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=261"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}