{"id":2665,"date":"2020-06-04T14:25:12","date_gmt":"2020-06-04T18:25:12","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/mycopy\/?post_type=chapter&#038;p=2665"},"modified":"2021-08-24T16:46:37","modified_gmt":"2021-08-24T20:46:37","slug":"8-1-percentiles-and-quartiles","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/mycopy\/chapter\/8-1-percentiles-and-quartiles\/","title":{"raw":"8.1 Percentiles and Quartiles","rendered":"8.1 Percentiles and Quartiles"},"content":{"raw":"<div class=\"part-title\">\r\n\r\n<img class=\"public domai aligncenter wp-image-5605 size-large\" title=\"https:\/\/commons.wikimedia.org\/wiki\/File:Ready_for_final_exam_at_Norwegian_University_of_Science_and_Technology.jpg\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-1024x768.jpg\" alt=\"\" width=\"1024\" height=\"768\" \/>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section it is expected that you will be able to:\r\n<ul>\r\n \t<li>Describe the measures of location:\u00a0 percentile and quartile<\/li>\r\n \t<li>Find the percentile represented by a given data value<\/li>\r\n \t<li>Determine the first, second and third quartiles for a set of data<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h6><strong>Measures of Central Tendency<\/strong><\/h6>\r\nThe mean, median and mode, as\u00a0 <strong>measures of central tendency,<\/strong> provide us with a point of comparison. As an example, consider Company ABC where the average (mean) salary is $55,000\/year. An employee earning $38,000\/year might feel unjustly treated or at the very least the employee might explore the reasons for the substantial difference. If in the process the employee learns that the\u00a0[pb_glossary id=\"4755\"]median[\/pb_glossary] salary at his workplace\u00a0 is $26,000\/year the employee would learn that relative to everyone else this employee's\u00a0 salary is in the upper half of the employee group.\r\n\r\nTo provide additional comparison the employee could consider other measures of position or location. Two such measures are percentiles and quartiles.\r\n<h6><strong>Percentiles<\/strong><\/h6>\r\n<\/div>\r\nPercentiles are useful for comparing values. If a data item is in the 75th percentile then three-quarters of the values are less than this value. This is not to be confused with a score of 75%, which is something very different. A student could score 35% on an exam but be in the 75th percentile. This means that relative to the rest of the class the student had a score that was higher than 75% of the students.\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Percentiles<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nPercentiles divide ordered data into hundredths. A data item is said to be in the k<sup>th<\/sup> <strong>percentile<\/strong> of a data set if k% of the data items are less than the item.\r\n\r\n<\/div>\r\n<\/div>\r\nThe notation\u00a0 P<sub>k<\/sub> can be used to represent the k<sup>th<\/sup> percentile. A data set can be divided into one hundred equal parts by ninety-nine percentiles P<sub>1<\/sub> , P<sub>2<\/sub> , P<sub>3<\/sub> , ... P<sub>99<\/sub> . The 60<sup>th<\/sup> percentile would be denoted P<sub>60<\/sub> . If an item is in the 60th percentile, then 60 percent of the data items are less than this item.\r\n\r\nConsider a set of math exam scores. A student scoring in the 60th percentile achieved a score equal to or higher than 60 percent of the other students. This does not mean that the student scored 60% on the exam. Perhaps the student's score was 78%, which would mean that 60 percent of the other students in the class had exam scores less than (or equal to) 78%.\r\n\r\nIt is important to note that since percentiles divide a data set into one hundred equal parts, percentiles are best used with large data sets. Percentiles are mostly used with very large populations. For a specified percentile P<sub>k<\/sub> if you were to say that k percent of the data values are less (and not the same or less) than a specified data value, it would be acceptable because removing one particular data value is not significant.\r\n\r\nRefer again to the employee earning $38,000\/year at Company ABC. If the employee learns that their salary is in the 90th percentile then 90 percent of the other\u00a0 employees at Company ABC have a salary less than (or possibly equal to) this salary. In relation to the other employees this salary ranks among the upper portion of the employee group.\r\n\r\nPercentiles are useful for comparing values. For this reason, universities and colleges use percentiles on entrance exams. Rather than set one value as an acceptance score, a university may set a percentile target. Perhaps all students scoring\u00a0 in the the 80th percentile or above will receive an acceptance letter. Every year there is likely to be a different acceptance score. Students will be accepted based on their score relative to all other applicants.\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Determining Percentiles<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nTo determine the k<sup>th<\/sup> percentile that is represented by a particular data item <strong><em>x<\/em>,<\/strong> the following formula can be used.\r\n\r\n<img class=\"aligncenter wp-image-2744 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula.png\" alt=\"\" width=\"364\" height=\"62\" \/>\r\n\r\nStep 1: If necessary order the data values from smallest to largest.\r\n\r\nStep 2: Determine the total number of data values, n. This will be the denominator in the formula.\r\n\r\nStep 3: Count the number of data values that are less than the value <em><strong>x.<\/strong><\/em> This will be the value in the numerator of the formula.\r\n\r\nStep 4: Calculate the percentile, <em>k<\/em>, that is associated <span style=\"font-size: 0.9em;\">with a score of\u00a0 <em><strong>x<\/strong><\/em> using the formula.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div class=\"textbox__content\">\r\n\r\nA class set of exam scores for 48 students are ranked from lowest to highest. Determine the percentiles associated with the scores of\u00a0 a) 39%\u00a0 b) 60% c) 94%.\r\n\r\n<\/div>\r\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 100%; height: 159px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 12.5%; height: 16px;\">39<\/td>\r\n<td style=\"width: 12.5%; height: 16px;\">54<\/td>\r\n<td style=\"width: 12.5%; height: 16px;\">59<\/td>\r\n<td style=\"width: 12.5%; height: 16px;\">65<\/td>\r\n<td style=\"width: 12.5%; height: 16px;\">75<\/td>\r\n<td style=\"width: 12.5%; height: 16px;\">79<\/td>\r\n<td style=\"width: 12.5%; height: 16px;\">84<\/td>\r\n<td style=\"width: 12.5%; height: 16px;\">92<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 12.5%; height: 15px;\">42<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">54<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">60<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">67<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">76<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">80<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">86<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">92<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 12.5%; height: 15px;\">43<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">55<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">60<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">69<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">76<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">80<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">88<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">94<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 12.5%; height: 15px;\">48<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">57<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">60<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">69<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">77<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">82<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">88<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">95<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 12.5%; height: 15px;\">51<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">57<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">63<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">72<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">77<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">83<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">89<\/td>\r\n<td style=\"width: 12.5%; height: 15px;\">96<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 12.5%; height: 10px;\">51<\/td>\r\n<td style=\"width: 12.5%; height: 10px;\">59<\/td>\r\n<td style=\"width: 12.5%; height: 10px;\">65<\/td>\r\n<td style=\"width: 12.5%; height: 10px;\">72<\/td>\r\n<td style=\"width: 12.5%; height: 10px;\">78<\/td>\r\n<td style=\"width: 12.5%; height: 10px;\">83<\/td>\r\n<td style=\"width: 12.5%; height: 10px;\">91<\/td>\r\n<td style=\"width: 12.5%; height: 10px;\">97<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Solution<\/strong>\r\n\r\n<span style=\"font-size: 0.9em;\">a) For a score of 39%:\u00a0<\/span>\r\n\r\nStep 1: The data values are already ordered from smallest to largest.\r\n\r\nStep 2: Determine the number of data values.\u00a0Since there are 48 students n = 48.\r\n\r\nStep 3: We count\u00a0 0\u00a0 data values that are less than 39\r\n\r\nStep 4: Calculate the percentile, <em>k<\/em>, that is associated <span style=\"font-size: 0.9em;\">with a score of\u00a0 x using the formula<\/span>\r\n<p style=\"text-align: center;\"><span style=\"font-size: 0.9em;\"> <img class=\"aligncenter\" style=\"background-color: #ffffff; color: #333333; font-size: 14pt;\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula-300x51.png\" width=\"229\" height=\"39\" \/><\/span><\/p>\r\n<p style=\"text-align: center;\"><span style=\"font-size: 0.9em;\"> k = (0\/48)*100% = 0%. <\/span><\/p>\r\n<span style=\"font-size: 0.9em;\">This means that the student who scored 39% is in the 0 percentile. A score of 39% is not higher than any other score. <\/span>\r\n\r\n<span style=\"font-size: 0.9em;\">b) For a score of 60%: <\/span>\r\n\r\n<span style=\"font-size: 0.9em;\">There are 13 scores lower than 60%\u00a0 so\u00a0 \u00a0 k = (13\/48)*100% = 27%. A score of\u00a0 60% is in the 27th percentile which means that 27% (or just over one-fourth)\u00a0 of the test scores are less than 60%.\u00a0 <\/span>\r\n\r\n<span style=\"font-size: 0.9em;\">c) For a score of 94%:\u00a0 <\/span>\r\n\r\n<span style=\"font-size: 0.9em;\">There are 44 scores less than 94%\u00a0 so\u00a0 \u00a0 \u00a0k = (44\/48)*100% = 92%. A score of\u00a0 94% is in the 92nd percentile which means that 92%\u00a0 of the test scores are less than 94%.\u00a0<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA set of assignment scores for a class of 32 students are provided in the table below. Determine the percentiles associated with the scores of\u00a0 a) 61%\u00a0 b) 79%\u00a0 c) 98%.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">72<\/td>\r\n<td style=\"width: 14.2857%;\">65<\/td>\r\n<td style=\"width: 14.2857%;\">85<\/td>\r\n<td style=\"width: 14.2857%;\">52<\/td>\r\n<td style=\"width: 14.2857%;\">61<\/td>\r\n<td style=\"width: 14.2857%;\">49<\/td>\r\n<td style=\"width: 7.14285%;\">65<\/td>\r\n<td style=\"width: 7.14285%;\">82<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">55<\/td>\r\n<td style=\"width: 14.2857%;\">99<\/td>\r\n<td style=\"width: 14.2857%;\">58<\/td>\r\n<td style=\"width: 14.2857%;\">79<\/td>\r\n<td style=\"width: 14.2857%;\">98<\/td>\r\n<td style=\"width: 14.2857%;\">79<\/td>\r\n<td style=\"width: 7.14285%;\">58<\/td>\r\n<td style=\"width: 7.14285%;\">93<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">88<\/td>\r\n<td style=\"width: 14.2857%;\">48<\/td>\r\n<td style=\"width: 14.2857%;\">97<\/td>\r\n<td style=\"width: 14.2857%;\">74<\/td>\r\n<td style=\"width: 14.2857%;\">65<\/td>\r\n<td style=\"width: 14.2857%;\">85<\/td>\r\n<td style=\"width: 7.14285%;\">71<\/td>\r\n<td style=\"width: 7.14285%;\">75<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">99<\/td>\r\n<td style=\"width: 14.2857%;\">39<\/td>\r\n<td style=\"width: 14.2857%;\">60<\/td>\r\n<td style=\"width: 14.2857%;\">96<\/td>\r\n<td style=\"width: 14.2857%;\">80<\/td>\r\n<td style=\"width: 14.2857%;\">70<\/td>\r\n<td style=\"width: 7.14285%;\">54<\/td>\r\n<td style=\"width: 7.14285%;\">77<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<details><summary>Show answer<\/summary>a) 61% is 28th percentile\u00a0 \u00a0b) 79% is 59th percentile\u00a0 \u00a0 c) 98% is 91st percentile\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<strong style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1em;\">Quartiles<\/strong>\r\n<p id=\"fs-idp16986528\">Quartiles divide ordered data into quarters. Quartiles are special percentiles. The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the same as the 25<sup>th<\/sup> percentile, and the third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the same as the 75<sup>th<\/sup> percentile.\u00a0The <span data-type=\"term\">median<\/span> is a number that separates ordered data into halves. Half the values are the same as or smaller than the median, and half the values are the same as or larger than the median. The median can be called both the second quartile Q<sub>2<\/sub> and the 50<sup>th<\/sup> percentile.<\/p>\r\n\r\n<div class=\"part-title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Quartiles<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nQuartiles divide the data set into <strong>four<\/strong> equal parts.\r\n\r\nThe first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the same as the 25<sup>th<\/sup> percentile, and the third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the same as the 75<sup>th<\/sup> percentile.\u00a0The median can be called both the second quartile, Q<sub>2 <\/sub>, and the 50<sup>th<\/sup> percentile.\r\n\r\nAs with the median, the quartiles may or may not be part of the data set.\r\n\r\n<\/div>\r\n<\/div>\r\nAs indicated in <a href=\"#figure1\">Figure 1<\/a> each quartile divides a data set into four equal parts so that one-fourth of the data set is located in each part.<a id=\"#figure1\"><\/a>\r\n\r\n<\/div>\r\n\r\n[caption id=\"attachment_2736\" align=\"aligncenter\" width=\"636\"]<img class=\"wp-image-2736 size-full\" title=\"created by Kim Moshenko\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles.png\" alt=\"\" width=\"636\" height=\"226\" \/> Fig. 1[\/caption]\r\n\r\n<div class=\"part-title\">\r\n<h6><strong>Determining Quartiles<\/strong><\/h6>\r\nWe will consider two methods for determining quartiles. As with percentiles, the data values must first be ordered from smallest to largest. The first method involves dividing the data set into four equal parts. The second method involves the use of formulas.\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Determining Quartiles: Method 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nStep 1: Order the data from smallest to largest.\r\n\r\nStep 2: Determine the number of data values <strong>n<\/strong>.\r\n\r\nStep 3: Determine the median (Q<sub>2<\/sub>) of the data set. This will divide the data set into two equal parts.\r\n\r\nStep 4: Determine Q<sub>1<\/sub>. This will divide the first half of the data set into two equal parts.\r\n\r\nStep 5: Determine Q<sub>3<\/sub>. This will divide the second half of the data set into two equal parts.\r\n\r\n<strong>Note:<\/strong> The median and the quartiles may not be actual observations from the data set.\r\n\r\n<\/div>\r\n<\/div>\r\n<h6><strong>Method 1<\/strong><\/h6>\r\nConsider the following data set:\r\n<p style=\"text-align: center;\">15\u00a0 \u00a0 \u00a0 \u00a04\u00a0 \u00a0 \u00a0 \u00a020\u00a0 \u00a0 \u00a0 \u00a08\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0 \u00a012\u00a0 \u00a0 \u00a0 14\u00a0 \u00a0 \u00a0 11\u00a0 \u00a0 \u00a0 7\u00a0 \u00a0 \u00a02\u00a0 \u00a0 \u00a06\u00a0 \u00a0 \u00a023\u00a0 \u00a0 \u00a016<\/p>\r\nStep 1: To determine the \u00a0quartiles, order the data values from smallest to largest:\r\n<p style=\"text-align: center;\">2\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a04\u00a0 \u00a0 \u00a06\u00a0 \u00a0 \u00a07\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a08\u00a0\u00a0 \u00a0\u00a0\u00a011\u00a0\u00a0\u00a0\u00a0 12\u00a0\u00a0\u00a0 \u00a0\u00a014\u00a0 \u00a0\u00a0\u00a0\u00a015\u00a0 \u00a0 16\u00a0 \u00a0 \u00a020\u00a0 \u00a0 \u00a023<\/p>\r\nStep 2: The number of data values is 13.\r\n\r\nStep 3: Determine the [pb_glossary id=\"4755\"]median[\/pb_glossary]<span data-type=\"term\">, which<\/span> measures the \"centre\" of the data. It is the number that separates ordered data into halves. Half the observations are the same number or smaller than the median, and half the observations are the same number or larger.\r\n\r\n<img class=\"aligncenter wp-image-5562 \" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1.png\" alt=\"\" width=\"469\" height=\"66\" \/>\r\n\r\n&nbsp;\r\n<p id=\"element-546\">Since there are 13 observations, the median will be in the seventhh position. The median, and therefore the 2nd quartile Q<sub>2<\/sub> , is eleven. The median is often referred to as\u00a0 the \"middle observation,\" but it is important to note that it does not actually have to be one of the observed values.<\/p>\r\n<p id=\"element-308\" style=\"text-align: left;\">Step 4: The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the <strong>middle value of the lower half<\/strong> of the data.<\/p>\r\nTo determine the<strong> first quartile<\/strong>, Q<sub>1<\/sub>, consider the lower half of the data observations:\r\n<p style=\"text-align: center;\">2\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a04\u00a0 \u00a0 \u00a0 6\u00a0 \u00a0 \u00a0 7\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a08<\/p>\r\nSince there are six observations, the middle observation will be the average of the third and fourth data values\u00a0 or\u00a0 (4 + 6)\/2 = 5\u00a0 therefore\u00a0 \u00a0Q<sub>1<\/sub>\u00a0 is 5\r\n<p id=\"element-308\" style=\"text-align: left;\">Step 5: The third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the <strong>middle value of the upper half<\/strong> of the data.<\/p>\r\nTo determine the<strong> third quartile<\/strong>, Q<sub>3<\/sub>, consider the upper half of the data observations:\r\n<p style=\"text-align: center;\">12\u00a0\u00a0\u00a0 \u00a0\u00a014\u00a0 \u00a0\u00a0\u00a0\u00a015\u00a0 \u00a0 \u00a016\u00a0 \u00a0 20\u00a0 \u00a0 23<\/p>\r\n<span data-type=\"newline\">Since there are six observations, the middle observation will be the average of 15 and 16 , or 15.5 therefore\u00a0 \u00a0 Q<sub>3<\/sub>\u00a0 is 15.5.<\/span>\r\n\r\n<a href=\"#figure2\">Figure 2<\/a> illustrates the three quartiles, which divide the data set into four equal parts.<a id=\"#figure2\"><\/a>\r\n\r\n[caption id=\"attachment_5558\" align=\"aligncenter\" width=\"610\"]<img class=\"wp-image-5558\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno.png\" alt=\"\" width=\"610\" height=\"149\" \/> Fig. 2[\/caption]\r\n<p id=\"element-227\">The number 4.5 is the <span data-type=\"term\">first quartile, Q<sub>1<\/sub><\/span>. One-fourth of the entire set of observations lie below 4.5 and\u00a0 three-fourths of the data observations lie above 4.5.<\/p>\r\nThe <span data-type=\"term\">third quartile<\/span>, <em data-effect=\"italics\">Q<sub>3<\/sub><\/em>, is 15.5. Three-fourths (75%) of the ordered data set lie below 15.5. One-fourth (25%) of the ordered data set lie above 15.5.\r\n\r\n<\/div>\r\nIt is important to note that a quartile may not be a data observation. Sometimes there may be a need to average or weight the data values when determining the quartiles.\r\n<div class=\"part-title\">\r\n\r\n<span style=\"text-align: initial; font-size: 14pt;\">A second method for determining quartiles is to use a formula to determine the position of each quartile. This is especially useful when there is a large number of data items.<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Determining Quartiles: Method 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Quartile Formula<\/strong>\r\n\r\nThe following formulas, where <strong>n<\/strong> is the <strong>number of data values<\/strong>,\u00a0 can be used to determine the <strong>position<\/strong> of the three quartiles.\r\n\r\n<img class=\"size-medium wp-image-2796 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_56_13-8.1-DA2-Images.docx-Word-300x73.png\" alt=\"\" width=\"300\" height=\"73\" \/>\r\n\r\nIt is important to note that these results indicate the <strong>positions<\/strong> of the quartiles, not the actual data obervations. If, for example, the calculation gives Q<sub>1<\/sub>=3, this indicates that the first quartile will be the data obervation in the 3rd <strong>position.<\/strong> If\u00a0 Q<sub>3<\/sub> = 32, this indicates that the third quartile will be the data observation in the 32nd <strong>position.<\/strong>\r\n\r\nStep 1: Order the data from smallest to largest.\r\n\r\nStep 2:\u00a0 Determine<strong> n.<\/strong>\r\n\r\nStep 3: Use the formula to determine the <strong>position<\/strong> for the median (Q<sub>2<\/sub>) of the data set. Count from left to right to determine the corresponding data value. If the position is a fraction then two data values will need to be weighted to determine the median value.\r\n\r\nStep 4: Use the formula to determine the <strong>position<\/strong> for the first quartile Q<sub>1<\/sub> of the data set. Count from left to right to determine the corresponding data value. If the position is a fraction then two data values will need to be weighted to determine the value of Q<sub>1<\/sub>.\r\n\r\nStep 5: Use the formula to determine the <strong>position<\/strong> for the third quartile\u00a0 Q<sub>3<\/sub> of the data set. Count from left to right to determine the corresponding data value. If the position is a fraction then two data values will need to be weighted to determine the value of Q<sub>3<\/sub>.\r\n\r\n<\/div>\r\n<\/div>\r\n<h6><strong>Method 2:<\/strong><\/h6>\r\nConsider the following data set:\r\n<p style=\"text-align: center;\">15\u00a0 \u00a0 \u00a0 \u00a04\u00a0 \u00a0 \u00a0 \u00a020\u00a0 \u00a0 \u00a0 \u00a08\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0 \u00a012\u00a0 \u00a0 \u00a0 14\u00a0 \u00a0 \u00a0 11\u00a0 \u00a0 \u00a0 7\u00a0 \u00a0 \u00a02\u00a0 \u00a0 \u00a06\u00a0 \u00a0 \u00a023\u00a0 \u00a0 \u00a016<\/p>\r\nStep 1: To determine the \u00a0quartiles, order the data values from smallest to largest:\r\n<p style=\"text-align: center;\">2\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a04\u00a0 \u00a0 \u00a06\u00a0 \u00a0 \u00a07\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a08\u00a0\u00a0 \u00a0\u00a0\u00a011\u00a0\u00a0\u00a0\u00a0 12\u00a0\u00a0\u00a0 \u00a0\u00a014\u00a0 \u00a0\u00a0\u00a0\u00a015\u00a0 \u00a0 16\u00a0 \u00a0 \u00a020\u00a0 \u00a0 \u00a023<\/p>\r\nStep 2: The number of data values is 13.\r\n<div class=\"part-title\">\r\n\r\nStep 3: Use the formula to determine the <strong>position<\/strong> for the median (Q<sub>2<\/sub>) of the data set.\r\n\r\n<img class=\"size-medium wp-image-5557 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno1-300x76.png\" alt=\"\" width=\"300\" height=\"76\" \/>\r\n\r\nCount from left to right to determine the corresponding data value in the 7th position. The corresponding value is 11.\r\n\r\n<img class=\"aligncenter wp-image-5562 \" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1.png\" alt=\"\" width=\"469\" height=\"66\" \/>\r\n<p style=\"text-align: left;\">Step 4:\u00a0Use the formula to determine the <strong>position<\/strong> for the first quartile (Q<sub>1<\/sub>) of the data set.<\/p>\r\n<img class=\"size-medium wp-image-5609 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1Q3-method2-300x76.png\" alt=\"\" width=\"300\" height=\"76\" \/>\r\n\r\nSince 3.5 is a fraction, the first quartile will be the average of the two data values that are in the 3rd and 4th positions. Count from left to right to determine the corresponding data values. The data value 4 is in the 3rd position and the data value 6 is in the 4th position so these will be averaged (4 + 6)\/2 = 5. The first quartile will be 5.\r\n<p style=\"text-align: left;\">Step 5:\u00a0Use the formula to determine the <strong>position<\/strong> for the third quartile (Q<sub>3<\/sub>) of the data set.<\/p>\r\n<img class=\"size-medium wp-image-5560 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Q3method1oddno-300x64.png\" alt=\"\" width=\"300\" height=\"64\" \/>\r\n\r\nSince 10.5 is a fraction, the third quartile will be the average of the two data values that are in the 10th and 11th positions. Count from left to right to determine the corresponding data values. The data value 15 is in the 10th position and the data value 16\u00a0 is in the 11th position so these will be averaged (15 + 16)\/2 = 15.5. The third quartile will be 15.5.\r\n\r\n<a href=\"#figure3\">Figure 3<\/a> illustrates the three quartiles, which divide the data set into four equal parts.<a id=\"#figure3\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_5558\" align=\"aligncenter\" width=\"610\"]<img class=\"wp-image-5558\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno.png\" alt=\"\" width=\"610\" height=\"149\" \/> Fig. 3[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div class=\"textbox__content\">\r\n\r\nA shoe store wanted to determine the popularity of different shoe sizes for women's tennis shoes. It planned to place its next order using this information. In\u00a0 a five day period it sold nine pairs of women's tennis shoes in the following sizes:\u00a0 \u00a0 7,\u00a0 8, 11,\u00a0 10,\u00a0 7,\u00a0 \u00a06,\u00a0 \u00a09, 10,\u00a0 7\r\n\r\n<strong>Solution<\/strong>\r\n\r\n<strong>Method 1:<\/strong>\r\n\r\nTo determine the quartiles:\r\n<ol>\r\n \t<li>Order the shoe sizes from smallest to largest:\u00a0 6,\u00a0 7,\u00a0 \u00a07,\u00a0 \u00a07,\u00a0 8,\u00a0 \u00a09,\u00a0 10,\u00a0 \u00a010,\u00a0 \u00a011<\/li>\r\n \t<li>Count the number of values: n = 9<\/li>\r\n \t<li>Determine Q<sub>2<\/sub>, the median, which is the middle observation. Since there are nine data observations (shoe sizes) t<span style=\"text-align: initial; font-size: 0.9em;\">he median, or second quartile, will be the 5th <\/span><span style=\"text-align: initial; font-size: 0.9em;\">data value. The 5th data value is 8.\u00a0<\/span><\/li>\r\n<\/ol>\r\n<img class=\"alignnone size-medium wp-image-4775 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-300x110.png\" alt=\"\" width=\"300\" height=\"110\" \/>\r\n\r\n4. Determine the first quartile Q<sub style=\"text-align: initial;\">1. <\/sub>It <span style=\"font-size: 0.9em; text-align: initial;\">will be the middle observation of the <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">lower half<\/strong><span style=\"font-size: 0.9em; text-align: initial;\"> of data values. This will be the average of the 2nd and 3rd data values\u00a0 so (7 +7)\/2 = 7.<\/span>\r\n\r\n<img class=\"aligncenter wp-image-2804 \" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"301\" height=\"123\" \/>\r\n\r\n<span style=\"font-size: 0.9em; text-align: initial;\">5. Determine the third quartile Q<\/span><sub style=\"text-align: initial;\">3<\/sub>.<span style=\"font-size: 0.9em; text-align: initial;\">\u00a0This will be the middle observation of the <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">upper half<\/strong><span style=\"font-size: 0.9em; text-align: initial;\">. This will be the average of the 7th and 8th data values\u00a0 so (10+10)\/2 = 10\u00a0<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong>Method 2:<\/strong>\r\n\r\nThe formulas can be used to determine the quartiles.\r\n<ol>\r\n \t<li>Order the shoe sizes from smallest to largest:\u00a0 6,\u00a0 7,\u00a0 \u00a07,\u00a0 \u00a07,\u00a0 8,\u00a0 \u00a09,\u00a0 10,\u00a0 \u00a010,\u00a0 \u00a011 .<\/li>\r\n \t<li>Determine the number of data values, n.\u00a0 \u00a0 \u00a0n = 9<\/li>\r\n \t<li>Use the formula to determine the median. The median , or second quartile,\u00a0 can be determined as follows:<\/li>\r\n<\/ol>\r\n<\/div>\r\n<img class=\"size-full wp-image-2794 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_54_42-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"259\" height=\"64\" \/>\r\n\r\nCounting from left to right, the 5<sup>th<\/sup> data value is 8. The median, or 2nd quartile Q<sub>2<\/sub>, is 8.\r\n\r\n<img class=\"alignnone size-medium wp-image-4775 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-300x110.png\" alt=\"\" width=\"300\" height=\"110\" \/>\r\n\r\n4 &amp; 5.\u00a0 The first and third quartiles can be determined as follows:\r\n\r\n<img class=\"aligncenter wp-image-2795 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_55_16-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"608\" height=\"80\" \/>\r\n\r\nThe first quartile is the 2.5th data value. To determine the 2.5<sup>th<\/sup> data value we must take the average of the 2nd and 3rd data values. The 2nd data value is 7 and the 3rd data value is 7 so\u00a0 (7+7)\/2 = 7.\r\n\r\nThe first quartile, Q<sub>1<\/sub> = 7\r\n\r\nThe third quartile is the 7.5th data value. This will be the average of the 7th and 8th data values. The 7th data value is 10 and the 8th data value is also 10\u00a0 so\u00a0 \u00a0(10+10)\/2 = 10.\r\n\r\nThe third quartile, Q<sub>3<\/sub> = 10\r\n\r\nWe can see that\u00a0 Q<sub>2 <\/sub>= 8\u00a0 splits the data set into two halves. <span style=\"text-align: initial; font-size: 0.9em;\">Q<\/span><sub style=\"text-align: initial;\">1<\/sub>= 7\u00a0<span style=\"text-align: initial; font-size: 0.9em;\"> is the middle value of the lower half of the data set and Q<\/span><sub style=\"text-align: initial;\">3<\/sub> = 10<span style=\"text-align: initial; font-size: 0.9em;\"> is the middle value of the upper half of the data set.\u00a0 \u00a0<\/span>\r\n\r\n<img class=\"aligncenter wp-image-2804 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"333\" height=\"136\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn Example 2 the number of data items was <strong>odd.<\/strong> When <em>n<\/em>\u00a0 is odd the median or Q<sub>2<\/sub> will be one of the data observations. When <em>n <\/em>is odd the formula for finding quartiles is straight forward.\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDetermine the quartiles for the following temperature data that was recorded over a 3-week period in May:\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter wp-image-2817 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_17_54-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"665\" height=\"161\" \/>\r\n\r\n&nbsp;\r\n\r\n<details><summary>Show answer<\/summary>Q<sub>2<\/sub> = 21; Q<sub>1<\/sub> = 18; Q<sub>3<\/sub> = 25\r\n\r\n<\/details><\/div>\r\n<\/div>\r\nIt is important to note that a quartile may <strong>not<\/strong> be a data observation. When the number of data values <em>n<\/em>\u00a0 is <strong>even<\/strong> the median or Q<sub>2<\/sub> will <strong>not<\/strong> be one of the actual data observations. As a result, when <em>n <\/em>is <strong>even<\/strong> an adjustment must be made to the value of <strong>n<\/strong> that is to be used in the formula to determine the <strong>first<\/strong> and <strong>third<\/strong> quartiles.\r\n\r\n<strong>Method 1<\/strong>:\r\n\r\nConsider the following data set:\r\n<p style=\"text-align: center;\">1;\u00a0 11.5;\u00a0 6;\u00a0 7.2;\u00a0 4;\u00a0 8;\u00a0 9;\u00a0 10;\u00a0 6.8;\u00a0 \u00a08.3;\u00a0 \u00a02;\u00a0 \u00a02;\u00a0 \u00a010;\u00a0 1<\/p>\r\nStep 1: To determine the \u00a0quartiles, order the data values from smallest to largest:\r\n<p style=\"text-align: center;\">1\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a02\u00a0 \u00a04\u00a0 \u00a06\u00a0 \u00a06.8\u00a0 \u00a07.2\u00a0 \u00a08\u00a0 \u00a0 8.3\u00a0 \u00a09\u00a0 \u00a0 10\u00a0 \u00a010\u00a0 \u00a011.5<\/p>\r\nStep 2:\u00a0 The number of data values is 14\r\n\r\nStep 3: Determine the [pb_glossary id=\"4755\"]median[\/pb_glossary]<span data-type=\"term\">, which<\/span> measures the \"centre\" of the data. It is the number that separates ordered data into halves. Half the observations are the same number or smaller than the median, and half the observations are the same number or larger.\r\n\r\n<img class=\"wp-image-2787 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_18_38-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"524\" height=\"78\" \/>\r\n<p id=\"element-546\">Since there are 14 observations, the median lies between the seventh observation, 6.8, and the eighth observation, 7.2. To find the median, add the two values together and divide by two.\u00a0 \u00a0Median = (6.8 + 7.2)\/2 = 7<\/p>\r\nThe median, and therefore the 2nd quartile Q<sub>2<\/sub> , is seven. It is important to note that the median is not actually one of the observed data values.\r\n<p id=\"element-308\" style=\"text-align: left;\">Step 4: The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the <strong>middle value of the lower half<\/strong> of the data.<\/p>\r\nTo determine the<strong> first quartile<\/strong>, Q<sub>1<\/sub>, consider the lower half of the data observations:\r\n<p style=\"text-align: center;\">1\u00a0 \u00a0 \u00a01\u00a0 \u00a0 \u00a02\u00a0 \u00a0 \u00a02\u00a0 \u00a0 4\u00a0 \u00a0 6\u00a0 \u00a0 6.8.<\/p>\r\nSince there are seven observations, the middle observation will be the 4th item. The middle or 4<sup>th<\/sup> item of these data observations\u00a0 is 2.\r\n<p id=\"element-308\" style=\"text-align: left;\">Step 5: The third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the <strong>middle value of the upper half<\/strong> of the data.<\/p>\r\nTo determine the<strong> third quartile<\/strong>, Q<sub>3<\/sub>, consider the upper half of the data observations:\r\n<p style=\"text-align: center;\">7.2\u00a0 \u00a0 \u00a08\u00a0 \u00a0 \u00a08.3\u00a0 \u00a0 9\u00a0 \u00a0 10\u00a0 \u00a0 10\u00a0 \u00a0 \u00a011.5.<\/p>\r\nSince there are seven observations, the middle observation will be the 4th item in the upper half. The middle item of these data observations is 9.<span data-type=\"newline\">\r\n<\/span>\r\n\r\n<a href=\"#figure4\">Figure 4<\/a> illustrates the three quartiles, which divide the data set into four equal parts.<a id=\"#figure4\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_2788\" align=\"aligncenter\" width=\"526\"]<img class=\"wp-image-2788 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"526\" height=\"114\" \/> Fig. 4[\/caption]\r\n<p id=\"element-227\">The number 2 is the <span data-type=\"term\">first quartile, Q<sub>1<\/sub><\/span>. One-fourth of the entire set of observations lie below 2 and\u00a0 three-fourths of the data observations lie above 2.<\/p>\r\nThe <span data-type=\"term\">third quartile<\/span>, <em data-effect=\"italics\">Q<sub>3<\/sub><\/em>, is 9. Three-fourths (75%) of the ordered data set lie below 9. One-fourth (25%) of the ordered data set lie above 9.\r\n\r\n<strong>Method 2<\/strong>:\r\n\r\nConsider the following data set:\r\n<p style=\"text-align: center;\">1;\u00a0 11.5;\u00a0 6;\u00a0 7.2;\u00a0 4;\u00a0 8;\u00a0 9;\u00a0 10;\u00a0 6.8;\u00a0 \u00a08.3;\u00a0 \u00a02;\u00a0 \u00a02;\u00a0 \u00a010;\u00a0 1<\/p>\r\nStep 1: To determine the \u00a0quartiles, order the data values from smallest to largest:\r\n<p style=\"text-align: center;\">1\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a02\u00a0 \u00a04\u00a0 \u00a06\u00a0 \u00a06.8\u00a0 \u00a07.2\u00a0 \u00a08\u00a0 \u00a0 8.3\u00a0 \u00a09\u00a0 \u00a0 10\u00a0 \u00a010\u00a0 \u00a011.5<\/p>\r\nStep 2:\u00a0 The number of data values is 14 so <strong>n is an even number.<\/strong>\r\n\r\nStep 3: Use the formula to determine the position of\u00a0 Q<sub>2<\/sub>, the median. The position will be (14 + 1)\/2 = 7.5. This means that the median,\u00a0 or Q<sub>2<\/sub>, will be in the 7.5th observation or halfway between the 7th and 8th position. The observation 6.8 is in the 7th position and the observation 7.2 is in the 8th position therefore the average of these (6.8 + 7.2)\/ 2 is the median or Q<sub>2<\/sub>.\r\n\r\nNote that the median is <strong>not<\/strong> an actual observation in the data set. If we use the formula to find Q<sub>1<\/sub> and Q<sub>3<\/sub> then we must adjust \"n\" to include this additional item so in effect \"n\" will be 15. <span style=\"text-align: initial; font-size: 0.9em;\">This is done <strong>only<\/strong> when determining the positions of Q<sub>1<\/sub> and Q<sub>3\u00a0<\/sub>(and not for determining the position of Q<sub>2<\/sub>)<\/span>\r\n\r\nStep 4: Use the formula to determine the position of\u00a0 Q<sub>1<\/sub>, the first quartile. Remember than<strong> n<\/strong> will now be 15, not 14. The position will be\u00a0(15 + 1)\/4 = 4 th. This means that\u00a0 Q<sub>1<\/sub> will be in the 4th position.\u00a0Counting from the left, the data value 2 is in the 4th position so Q<sub>1<\/sub>= 2.\r\n\r\nStep 5: Use the formula to determine the position of\u00a0 Q<sub>3<\/sub>, the third quartile. Remember than<strong> n<\/strong> will now be 15, not 14. The position will be 3(15 + 1)\/4 = 12th. This means that\u00a0 Q<sub>3<\/sub> will be in the 12th position. Refer to <a href=\"#figure5\">Figure 5<\/a>. Counting from the left, we include the median value of\u00a0 7,\u00a0 to determine that the data value in the 12th position. This value is 9\u00a0 so \u00a0Q<sub>3<\/sub> will be 9.<a id=\"#figure5\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_5513\" align=\"aligncenter\" width=\"544\"]<img class=\"wp-image-5513\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4-1024x214.png\" alt=\"\" width=\"544\" height=\"114\" \/> Fig. 5[\/caption]\r\n\r\nIt is also important to recognize that the median of 7 is not an actual data value in this set. It was included in <a href=\"#figure5\">Figure 5<\/a> to illustrate that its <strong>position<\/strong> must be counted when determing the position of the third quartile. It is not actually part of the data set. The actual data set is illustrated in <a href=\"#figure6\">Figure 6<\/a>\u00a0 (and <a href=\"#figure4\">Figure 4<\/a>).<a id=\"#figure6\"><\/a>\r\n\r\n[caption id=\"attachment_2788\" align=\"aligncenter\" width=\"526\"]<img class=\"wp-image-2788 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"526\" height=\"114\" \/> Fig. 6[\/caption]\r\n\r\n<span style=\"font-size: 14pt; text-align: initial;\">Consider <a href=\"#figure7\">Figure 7<\/a> where the data set that has an even number of data values:\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a0 4\u00a0 \u00a0 5\u00a0 <a id=\"#figure7\"><\/a><\/span>\r\n\r\n[caption id=\"attachment_3043\" align=\"aligncenter\" width=\"274\"]<img class=\"wp-image-3043 \" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-2-300x204.png\" alt=\"\" width=\"274\" height=\"186\" \/> Fig. 7[\/caption]\r\n\r\nIn this data set\u00a0 Q<sub>1<\/sub> = 1.5,\u00a0 Q<sub>2<\/sub> = 3,\u00a0 and Q<sub>3<\/sub> = 4.5\u00a0 \u00a0This illustrates that quartile values need not be actual values in the data set. The second quartile Q<sub>2<\/sub> is 3 which \u00a0is the average of the data values 2 and 4. Similarily the first quartile of 1.5 is the average of two data values 1 and 2 and the third quartile of 4.5 is the average of the two data values 4 and 5. Determining the quartile values can become complex as it may require different weightings of the data values but this is beyond the scope of this textbook.\r\n\r\nExample 3 illustrates two techniques for determining quartiles when the number of data observations is <strong>even<\/strong>.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div class=\"textbox__content\">\r\n\r\nConsider again the\u00a0 shoe store and a different week. Over a five day period it sold ten pairs of tennis shoes in the following sizes:\r\n<p style=\"text-align: center;\">6,\u00a0 8, 11,\u00a0 10,\u00a0 7,\u00a0 \u00a06,\u00a0 \u00a09, 10,\u00a0 8,\u00a0 \u00a09<\/p>\r\nNote that there is an <strong>even<\/strong> number of data values\u00a0 n = 10\r\n\r\n<strong>Solution<\/strong>\r\n\r\n<strong>Method 1:<\/strong>\r\n\r\nTo determine the quartiles:\r\n<ol>\r\n \t<li>Rank the sizes from smallest to largest:\u00a0 \u00a06,\u00a0 6,\u00a0 \u00a07,\u00a0 \u00a08,\u00a0 8,\u00a0 \u00a09,\u00a0 \u00a09,\u00a0 \u00a010,\u00a0 \u00a010,\u00a0 11 and divide the data set into four equal quarters.<\/li>\r\n \t<li>n = 10<\/li>\r\n \t<li>Start with the median which is the middle observation. <span style=\"text-align: initial; font-size: 0.9em;\">The median, or second quartile, will <\/span><span style=\"text-align: initial; font-size: 0.9em;\">lie between the 5th and 6th data values. The 5th data value is 8\u00a0 and the 6th data value is 9\u00a0 so the average of 8 and 9, or 8.5, is the median.<\/span><\/li>\r\n \t<li>Determine the first quartile Q<sub style=\"text-align: initial;\">1. <\/sub>It <span style=\"font-size: 0.9em; text-align: initial;\">will be the middle observation of the <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">lower half<\/strong><span style=\"font-size: 0.9em; text-align: initial;\"> of data values. This is the 3rd data value or the observation of\u00a0 7.<\/span><\/li>\r\n \t<li><span style=\"font-size: 0.9em; text-align: initial;\">Determine the third quartile Q<\/span><sub style=\"text-align: initial;\">3<\/sub>.<span style=\"font-size: 0.9em; text-align: initial;\">\u00a0This will be the middle observation of the <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">upper half<\/strong><span style=\"font-size: 0.9em; text-align: initial;\">. This will be the data observation of 10.<\/span><\/li>\r\n<\/ol>\r\n<img class=\"wp-image-2807 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_40_24-8.1-DA2-Images.docx-Word-300x106.png\" alt=\"\" width=\"308\" height=\"109\" \/>\r\n\r\n&nbsp;\r\n\r\nNote that each quartile divides the data values such that there are an equal number of data values in each of the four sections.\r\n\r\n<strong>Method 2:<\/strong>\r\n\r\n<strong style=\"font-size: 0.9em; text-align: initial;\">An alternative is to use the formulas <\/strong><span style=\"font-size: 0.9em; text-align: initial;\">\u00a0to determine the quartiles.\u00a0<\/span>\r\n\r\nTo determine the quartiles:\r\n<ol>\r\n \t<li>Rank the sizes from smallest to largest:\u00a0 \u00a06,\u00a0 6,\u00a0 \u00a07,\u00a0 \u00a08,\u00a0 8,\u00a0 \u00a09,\u00a0 \u00a09,\u00a0 \u00a010,\u00a0 \u00a010,\u00a0 11<\/li>\r\n \t<li>n = 10<\/li>\r\n \t<li>Determine the position of the median\u00a0 using the formula. <img class=\"size-full wp-image-2805 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_35-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"287\" height=\"72\" \/><\/li>\r\n<\/ol>\r\nThe 5.5th data value will be the average of the 5th and 6th data values. The 5th data value is 8\u00a0 and the 6th data value is 9\u00a0 so\r\n<p style=\"text-align: center;\">(8 + 9) \/2 = 8.5\u00a0 The median or Q<sub>2<\/sub>\u00a0 is 8.5.<\/p>\r\n<span style=\"font-size: 0.9em; text-align: initial;\"><strong>Note:<\/strong><\/span><span style=\"font-size: 0.9em; text-align: initial;\">\u00a0Q<sub style=\"text-align: initial;\">2<\/sub> is <strong>not<\/strong> one of the <strong>actual<\/strong> data values. <\/span><span style=\"text-align: initial; font-size: 0.9em;\">\u00a0In this example Q<sub>2<\/sub> is the 5.5th data value or 8.5. It is the data value that lies between the 5th and 6th data values but it is not one of the original data values.\u00a0<\/span>\r\n\r\n4 &amp; 5.\u00a0 \u00a0Determine the first quartile Q<sub style=\"text-align: initial;\">1\u00a0<\/sub>and the third quartile Q<sub>3<\/sub>.\r\n\r\n<span style=\"font-size: 0.9em; text-align: initial;\">Since the number of data values <strong><em>n<\/em><\/strong> is <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">even<\/strong><span style=\"font-size: 0.9em; text-align: initial;\"> the median or\u00a0 Q<sub style=\"text-align: initial;\">2<\/sub> is <strong>not<\/strong> one of the <strong>actual<\/strong> data values so when we use<\/span><span style=\"text-align: initial; font-size: 0.9em;\">\u00a0the formula to determine Q<\/span><sub style=\"text-align: initial;\">1<\/sub><span style=\"text-align: initial; font-size: 0.9em;\">\u00a0 and Q<\/span><sub style=\"text-align: initial;\">3<\/sub><span style=\"text-align: initial; font-size: 0.9em;\"> we must <strong>increase <\/strong><strong>the value of n by 1<\/strong>. In effect the number of data values has increased by one and therefore the value of <strong>n<\/strong> in the formula must be increased by 1. This is done only when determining the positions of Q<sub>1<\/sub> and Q<sub>3\u00a0<\/sub>(and not for determining the position of Q<sub>2<\/sub>)<\/span>\r\n\r\n<span style=\"text-align: initial; font-size: 0.9em;\">In this example, when determining Q<sub>1<\/sub> and Q<sub>3\u00a0<\/sub> the original value of\u00a0 <strong>n = 10<\/strong> will now be increased by 1. The new number for <strong>n<\/strong> to be used in the formula will be<strong> n = 11.<\/strong> Using the formula, the first and third quartile positions can be determined as follows:<\/span>\r\n<p style=\"text-align: center;\"><span style=\"text-align: initial; font-size: 0.9em;\"><img class=\"alignnone wp-image-4787\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles3i-300x97.png\" alt=\"\" width=\"204\" height=\"66\" \/><\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox__content\">\r\n<div class=\"textbox__content\">\r\n\r\n<img class=\"wp-image-2806 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_56-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"501\" height=\"61\" \/>\r\n\r\nUsing the results from the formula we count to get the 3rd and 9th data values. When determining these values\u00a0 be sure to include and count the <strong>\u00a0position occupied by the new median value<\/strong> of\u00a0 8.5. The 3rd data value is 7\u00a0 and the 9th data value is 10.\r\n\r\n<img class=\"size-medium wp-image-3045 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3-300x93.png\" alt=\"\" width=\"300\" height=\"93\" \/>\r\n\r\nNote that Method 1 and Method 2 yield the same results.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWe have seen that either of Method 1 or Method 2 will produce the same quartile values although the formula method can be less intuitive when <strong>n<\/strong> is <strong>even.<\/strong>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse either technique to determine the quartiles for the following temperature data that was recorded over the month of April:\r\n\r\n<img class=\"wp-image-2818 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_18_19-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"656\" height=\"213\" \/>\r\n\r\n<details><summary>Show answer<\/summary>Q<sub>2<\/sub> = 18.5 Q<sub>1<\/sub> = 15; Q<sub>3<\/sub> = 22\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nConsider the data set:\u00a0 3, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15.\u00a0 \u00a0Determine the three quartiles using either technique.\r\n\r\n<strong>Method 1:<\/strong>\r\n\r\nStep 1: Order the data values\u00a0 \u00a0 \u00a03, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15\r\n\r\nStep 2:\u00a0 n = 16\r\n\r\nStep 3:\u00a0 The median will be the average of 9 and 10, so 9.5. This is not one of the observed values.\r\n\r\nStep 4:\u00a0 Q<sub>1<\/sub> is the value that splits the lower half, which will be the average of 6 and 7, so 6.5.\r\n\r\nStep 5:\u00a0 Q<sub>3<\/sub> is the value that splits the upper half, which will be the average of 12 and 13, so 12.5.\r\n\r\n<strong>Method 2:<\/strong>\r\n\r\nStep 1: Order the data values\u00a0 \u00a0 \u00a03, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15\r\n\r\nStep 2:\u00a0 n = 16\r\n\r\nStep 3: Use the formula\u00a0 (16 + 1) \/2 = 8.5. The median will be in the 8.5th position. This is the average of the 8th value of\u00a0 9 and the 9th value of 10 so the median is 9.5\r\n\r\nStep 4 and 5: Since n is <strong>even,<\/strong> we will use a value of 17, not 16, in the formulas to determine Q<sub>1<\/sub> and Q<sub>3<\/sub>.\r\n\r\nQ<sub>1<\/sub> will be (17 + 1)\/4 = 4.5 th. This means that\u00a0 Q<sub>1<\/sub> will be in the 4.5th position or the average of the 4th and 5th data values. The 4th value is 6 and the 5th value is 7 so\u00a0 Q<sub>1<\/sub>= 6.5.\r\n\r\nQ<sub>3<\/sub> will be 3(17 + 1)\/4 = 13.5 th. This means that\u00a0 Q<sub>3<\/sub> will be in the 13.5th position or the average of the 13th and 14th data values. Including the median's position when we count, the 13th value is 12 and the 14th value is 13 so\u00a0 Q<sub>3<\/sub> = (12 + 13)\/2 = 12.5.\r\n\r\n<strong>Note<\/strong> that these identical results were obtained without using\u00a0 the formulas. It is also important to recognize that the median of 9.5 is not an actual data value in this set. It serves only to divide the data set into two equal halves and it is not actually part of the data set.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAn athlete was training for a race and logged the following distances (in km) over a 36 day period. Determine the three quartiles for the distances covered by the athlete.\r\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse; width: 100%; height: 60px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">22<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">34<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">38<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">42<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">14<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">22<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">0<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 11.1111%; height: 15px;\">21<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">30<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">41<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">56<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">11<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">30<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 11.1111%; height: 15px;\">24<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">52<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">11<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">16<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">28<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">36<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">25<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">25<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">11<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 11.1111%; height: 15px;\">0<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">24<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">20<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">46<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">38<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">40<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">27<\/td>\r\n<td style=\"width: 11.1111%; height: 15px;\">10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<details open=\"open\"><summary>Show answer<\/summary>Q<sub>1<\/sub>= 17,\u00a0 Q<sub>2<\/sub> = 23,\u00a0 Q<sub>3<\/sub>= 35\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<h1>Key Concepts<\/h1>\r\n<ul>\r\n \t<li>A data set can be divided into one hundred equal parts by ninety-nine percentiles P<sub>1<\/sub> , P<sub>2<\/sub> , P<sub>3<\/sub> , ... P<sub>99<\/sub> . Percentiles are best used with large sets of data.<\/li>\r\n \t<li>Quartiles divide the data set into <strong>four<\/strong> equal parts.\u00a0The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the same as the 25<sup>th<\/sup> percentile, and the third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the same as the 75<sup>th<\/sup> percentile.\u00a0The median can be called both the second quartile, Q<sub>2 <\/sub>, and the 50<sup>th<\/sup> percentile.<\/li>\r\n \t<li>Quartiles may or may not be actual observations within a set of data.<\/li>\r\n<\/ul>\r\n<h1><strong>Glossary<\/strong><\/h1>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>Percentiles<\/strong>\r\n\r\ndivide ordered data into hundredths.\r\n\r\n<strong>Quartiles<\/strong>\r\n\r\ndivide ordered data into four equal parts.\r\n\r\n<\/div>\r\n<h1>8.1 Exercise Set<\/h1>\r\n<ol>\r\n \t<li><span style=\"text-align: initial; font-size: 14pt;\">Your instructor announces to the class that anyone with a midterm exam score of 63% scored in the 80th percentile.\u00a0 If you received a score of 63% how did you do in relation to your classmates?<\/span><\/li>\r\n \t<li>A test consists of 40 marks. Fifty students wrote the test and their scores are in the table below. Determine the percentiles that are associated with scores of:\r\n<ol type=\"a\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"a\">\r\n \t<li>15<\/li>\r\n \t<li>25<\/li>\r\n \t<li>37<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter wp-image-3330 size-large\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table-1024x266.png\" alt=\"\" width=\"1024\" height=\"266\" \/>\r\n<div class=\"part-title\">\r\n<ol start=\"3\">\r\n \t<li>An employee at a large manufacturing company learns that their salary is in the 45th percentile. If the median salary at the company is $56,000\/year can we conclude that this employee\u2019s annual salary is more than $56,000?<\/li>\r\n \t<li><span style=\"text-align: initial; font-size: 14pt;\">Your instructor announces to the class that the third quartile for the midterm exam was a score of 88%.\u00a0 \u00a0If you received a score of 88% how did you do in relation to your classmates?<\/span><\/li>\r\n \t<li>A cell phone provider is trying to improve its service by reducing the amount of time that its help desk spends with each customer. It kept track of the average length of time (to the nearest minute) each of its 33\u00a0 employees spent with customers.<img class=\"aligncenter wp-image-3331 size-large\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table-1024x178.png\" alt=\"\" width=\"1024\" height=\"178\" \/>\r\n<ol type=\"a\">\r\n \t<li>Determine the mean, median and mode for average wait times.<\/li>\r\n \t<li>Determine the percentiles that correspond to help times of 14 minutes, 23 minutes and 47 minutes.<\/li>\r\n \t<li>Determine the 1st, 2nd and 3rd quartiles for average help times.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li value=\"6\">A test consists of 40 marks. Fifty students wrote the test and their scores are as recorded. Determine the 1st, 2nd and 3rd quartiles.<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter wp-image-3329 size-large\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table-1024x277.png\" alt=\"\" width=\"1024\" height=\"277\" \/>\r\n<ol start=\"7\">\r\n \t<li>Which of the following must be an actual data value:\u00a0 mean, median, mode, first quartile, third quartile?<\/li>\r\n \t<li>At a restaurant one evening the customers were asked to rate the service they received. Scores could range from 1 to 10. The following thirty responses (scores) were provided:1\u00a0 1\u00a0 1\u00a0 2\u00a0 2\u00a0 2\u00a0 3\u00a0 3\u00a0 3\u00a0 4\u00a0 4\u00a0 4\u00a0 5\u00a0 5\u00a0 5\u00a0 6\u00a0 6\u00a0 6\u00a0 7\u00a0 7\u00a0 7\u00a0 8\u00a0 8\u00a0 8\u00a0 9\u00a0 9\u00a0 9\u00a0 10\u00a0 10\u00a0 10\r\n<ol type=\"a\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"a\">\r\n \t<li>Determine the percentiles that correspond to scores of\u00a0 2 and 5. Explain what this means.<\/li>\r\n \t<li>Determine the first, second and third quartiles.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li value=\"9\">At a restaurant one evening the customers were asked to rate the service they received. Scores could range from 1 to 10. The following twenty-nine responses (scores) were provided:\r\n1\u00a0 \u00a01\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a02\u00a0 \u00a03\u00a0 \u00a03\u00a0 \u00a03\u00a0 \u00a04\u00a0 \u00a0 4\u00a0 \u00a0 4\u00a0 \u00a05\u00a0 \u00a05\u00a0 \u00a05\u00a0 \u00a06\u00a0 \u00a06\u00a0 6\u00a0 7\u00a0 7\u00a0 8\u00a0 8\u00a0 8\u00a0 8\u00a0 \u00a08\u00a0 9\u00a0 9\u00a0 10\u00a0 10\u00a0 10\r\n<ol type=\"a\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"a\">\r\n \t<li>Determine the mean, median and mode.<\/li>\r\n \t<li>Determine the first, second and third quartiles.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<div id=\"fs-idm10803744\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\r\n<div data-type=\"title\">\r\n<h1>Answers<\/h1>\r\n<ol>\r\n \t<li>If your score is 63% and this is in the 80th percentile this means that 80% of your classmates received scores lower than or equal to 63%\r\n<ol type=\"a\">\r\n \t<li>A score of 15 is in the10th percentile.<\/li>\r\n \t<li>A score of 25 is in the 44th percentile.<\/li>\r\n \t<li>A score of 37 is in the 84th percentile.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>We can conclude that the employee\u2019s salary is not more than $56,000\/year because the median salary is also the 50th percentile. If the employee\u2019s salary is in the 45th percentile they cannot be earning more than the median.<\/li>\r\n \t<li>You scored better than three quarters of your class mates.\r\n<ol type=\"a\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"a\">\r\n \t<li><span style=\"font-size: 14pt;\">The mean is 765\/33 = 23.18; the median is 22; and there are two modes 5\u00a0 and 31.<\/span><\/li>\r\n \t<li>A help time of 14 minutes is in the 33rd percentile;\u00a0 23 minutes is in the 52nd percentile;\u00a0 47 minutes is in the 91st perecntile<\/li>\r\n \t<li>Q<sub>1<\/sub> = 9.5 min.; Q<sub>2<\/sub> = 22 min. ;\u00a0 Q<sub>3<\/sub> =\u00a0 31 min.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Q<sub>1<\/sub> = 19;\u00a0 Q<sub>2<\/sub> = 26.5;\u00a0 Q<sub>3<\/sub> = 33<\/li>\r\n \t<li>The mode must be an actual data value\r\n<ol type=\"a\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"a\">\r\n \t<li>A score of 2 is the 10th percentile. This means that 10% of the scores were less than a score of 2. A score of 5 is the 40th percentile. This means that 40% of the scores were less than a score of 5.<\/li>\r\n \t<li>The first quartile is 3, the second quartile is 5.5, the third quartile is 8<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<ol type=\"a\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"a\">\r\n \t<li>mean\u00a0 5.6;\u00a0 \u00a0 median 6;\u00a0 mode 8<\/li>\r\n \t<li>Q<sub>1<\/sub> is 3;\u00a0 \u00a0Q<sub>2<\/sub>\u00a0 is 6;\u00a0 Q<sub>3<\/sub>\u00a0 \u00a0is 8<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<h1>Attribution<\/h1>\r\nSome content in this chapter has been adapted from \u201cMeasures of the Location of the Data\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/introductory-statistics\"><i>Introductory Statistics <\/i><\/a>(OpenStax) by Daniel Birmajer, Bryan Blount, Sheri Boyd, Matthew Einsohn, James Helmreich, Lynette Kenyon, Sheldon Lee,\u00a0 and Jeff Taub, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Kim Moshenko. See the Copyright page for more information.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"part-title\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"public domai aligncenter wp-image-5605 size-large\" title=\"https:\/\/commons.wikimedia.org\/wiki\/File:Ready_for_final_exam_at_Norwegian_University_of_Science_and_Technology.jpg\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-1024x768.jpg\" alt=\"\" width=\"1024\" height=\"768\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-1024x768.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-300x225.jpg 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-768x576.jpg 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-1536x1152.jpg 1536w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-65x49.jpg 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-225x169.jpg 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam-350x263.jpg 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-intro-image-exam.jpg 1632w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section it is expected that you will be able to:<\/p>\n<ul>\n<li>Describe the measures of location:\u00a0 percentile and quartile<\/li>\n<li>Find the percentile represented by a given data value<\/li>\n<li>Determine the first, second and third quartiles for a set of data<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h6><strong>Measures of Central Tendency<\/strong><\/h6>\n<p>The mean, median and mode, as\u00a0 <strong>measures of central tendency,<\/strong> provide us with a point of comparison. As an example, consider Company ABC where the average (mean) salary is $55,000\/year. An employee earning $38,000\/year might feel unjustly treated or at the very least the employee might explore the reasons for the substantial difference. If in the process the employee learns that the\u00a0<a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_2665_4755\">median<\/a> salary at his workplace\u00a0 is $26,000\/year the employee would learn that relative to everyone else this employee&#8217;s\u00a0 salary is in the upper half of the employee group.<\/p>\n<p>To provide additional comparison the employee could consider other measures of position or location. Two such measures are percentiles and quartiles.<\/p>\n<h6><strong>Percentiles<\/strong><\/h6>\n<\/div>\n<p>Percentiles are useful for comparing values. If a data item is in the 75th percentile then three-quarters of the values are less than this value. This is not to be confused with a score of 75%, which is something very different. A student could score 35% on an exam but be in the 75th percentile. This means that relative to the rest of the class the student had a score that was higher than 75% of the students.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Percentiles<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Percentiles divide ordered data into hundredths. A data item is said to be in the k<sup>th<\/sup> <strong>percentile<\/strong> of a data set if k% of the data items are less than the item.<\/p>\n<\/div>\n<\/div>\n<p>The notation\u00a0 P<sub>k<\/sub> can be used to represent the k<sup>th<\/sup> percentile. A data set can be divided into one hundred equal parts by ninety-nine percentiles P<sub>1<\/sub> , P<sub>2<\/sub> , P<sub>3<\/sub> , &#8230; P<sub>99<\/sub> . The 60<sup>th<\/sup> percentile would be denoted P<sub>60<\/sub> . If an item is in the 60th percentile, then 60 percent of the data items are less than this item.<\/p>\n<p>Consider a set of math exam scores. A student scoring in the 60th percentile achieved a score equal to or higher than 60 percent of the other students. This does not mean that the student scored 60% on the exam. Perhaps the student&#8217;s score was 78%, which would mean that 60 percent of the other students in the class had exam scores less than (or equal to) 78%.<\/p>\n<p>It is important to note that since percentiles divide a data set into one hundred equal parts, percentiles are best used with large data sets. Percentiles are mostly used with very large populations. For a specified percentile P<sub>k<\/sub> if you were to say that k percent of the data values are less (and not the same or less) than a specified data value, it would be acceptable because removing one particular data value is not significant.<\/p>\n<p>Refer again to the employee earning $38,000\/year at Company ABC. If the employee learns that their salary is in the 90th percentile then 90 percent of the other\u00a0 employees at Company ABC have a salary less than (or possibly equal to) this salary. In relation to the other employees this salary ranks among the upper portion of the employee group.<\/p>\n<p>Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles on entrance exams. Rather than set one value as an acceptance score, a university may set a percentile target. Perhaps all students scoring\u00a0 in the the 80th percentile or above will receive an acceptance letter. Every year there is likely to be a different acceptance score. Students will be accepted based on their score relative to all other applicants.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Determining Percentiles<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>To determine the k<sup>th<\/sup> percentile that is represented by a particular data item <strong><em>x<\/em>,<\/strong> the following formula can be used.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2744 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula.png\" alt=\"\" width=\"364\" height=\"62\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula.png 364w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula-300x51.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula-65x11.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula-225x38.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula-350x60.png 350w\" sizes=\"auto, (max-width: 364px) 100vw, 364px\" \/><\/p>\n<p>Step 1: If necessary order the data values from smallest to largest.<\/p>\n<p>Step 2: Determine the total number of data values, n. This will be the denominator in the formula.<\/p>\n<p>Step 3: Count the number of data values that are less than the value <em><strong>x.<\/strong><\/em> This will be the value in the numerator of the formula.<\/p>\n<p>Step 4: Calculate the percentile, <em>k<\/em>, that is associated <span style=\"font-size: 0.9em;\">with a score of\u00a0 <em><strong>x<\/strong><\/em> using the formula.<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"textbox__content\">\n<p>A class set of exam scores for 48 students are ranked from lowest to highest. Determine the percentiles associated with the scores of\u00a0 a) 39%\u00a0 b) 60% c) 94%.<\/p>\n<\/div>\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 100%; height: 159px;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 12.5%; height: 16px;\">39<\/td>\n<td style=\"width: 12.5%; height: 16px;\">54<\/td>\n<td style=\"width: 12.5%; height: 16px;\">59<\/td>\n<td style=\"width: 12.5%; height: 16px;\">65<\/td>\n<td style=\"width: 12.5%; height: 16px;\">75<\/td>\n<td style=\"width: 12.5%; height: 16px;\">79<\/td>\n<td style=\"width: 12.5%; height: 16px;\">84<\/td>\n<td style=\"width: 12.5%; height: 16px;\">92<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 12.5%; height: 15px;\">42<\/td>\n<td style=\"width: 12.5%; height: 15px;\">54<\/td>\n<td style=\"width: 12.5%; height: 15px;\">60<\/td>\n<td style=\"width: 12.5%; height: 15px;\">67<\/td>\n<td style=\"width: 12.5%; height: 15px;\">76<\/td>\n<td style=\"width: 12.5%; height: 15px;\">80<\/td>\n<td style=\"width: 12.5%; height: 15px;\">86<\/td>\n<td style=\"width: 12.5%; height: 15px;\">92<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 12.5%; height: 15px;\">43<\/td>\n<td style=\"width: 12.5%; height: 15px;\">55<\/td>\n<td style=\"width: 12.5%; height: 15px;\">60<\/td>\n<td style=\"width: 12.5%; height: 15px;\">69<\/td>\n<td style=\"width: 12.5%; height: 15px;\">76<\/td>\n<td style=\"width: 12.5%; height: 15px;\">80<\/td>\n<td style=\"width: 12.5%; height: 15px;\">88<\/td>\n<td style=\"width: 12.5%; height: 15px;\">94<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 12.5%; height: 15px;\">48<\/td>\n<td style=\"width: 12.5%; height: 15px;\">57<\/td>\n<td style=\"width: 12.5%; height: 15px;\">60<\/td>\n<td style=\"width: 12.5%; height: 15px;\">69<\/td>\n<td style=\"width: 12.5%; height: 15px;\">77<\/td>\n<td style=\"width: 12.5%; height: 15px;\">82<\/td>\n<td style=\"width: 12.5%; height: 15px;\">88<\/td>\n<td style=\"width: 12.5%; height: 15px;\">95<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 12.5%; height: 15px;\">51<\/td>\n<td style=\"width: 12.5%; height: 15px;\">57<\/td>\n<td style=\"width: 12.5%; height: 15px;\">63<\/td>\n<td style=\"width: 12.5%; height: 15px;\">72<\/td>\n<td style=\"width: 12.5%; height: 15px;\">77<\/td>\n<td style=\"width: 12.5%; height: 15px;\">83<\/td>\n<td style=\"width: 12.5%; height: 15px;\">89<\/td>\n<td style=\"width: 12.5%; height: 15px;\">96<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 12.5%; height: 10px;\">51<\/td>\n<td style=\"width: 12.5%; height: 10px;\">59<\/td>\n<td style=\"width: 12.5%; height: 10px;\">65<\/td>\n<td style=\"width: 12.5%; height: 10px;\">72<\/td>\n<td style=\"width: 12.5%; height: 10px;\">78<\/td>\n<td style=\"width: 12.5%; height: 10px;\">83<\/td>\n<td style=\"width: 12.5%; height: 10px;\">91<\/td>\n<td style=\"width: 12.5%; height: 10px;\">97<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Solution<\/strong><\/p>\n<p><span style=\"font-size: 0.9em;\">a) For a score of 39%:\u00a0<\/span><\/p>\n<p>Step 1: The data values are already ordered from smallest to largest.<\/p>\n<p>Step 2: Determine the number of data values.\u00a0Since there are 48 students n = 48.<\/p>\n<p>Step 3: We count\u00a0 0\u00a0 data values that are less than 39<\/p>\n<p>Step 4: Calculate the percentile, <em>k<\/em>, that is associated <span style=\"font-size: 0.9em;\">with a score of\u00a0 x using the formula<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 0.9em;\"> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"background-color: #ffffff; color: #333333; font-size: 14pt;\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Percentile-Formula-300x51.png\" width=\"229\" height=\"39\" alt=\"image\" \/><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 0.9em;\"> k = (0\/48)*100% = 0%. <\/span><\/p>\n<p><span style=\"font-size: 0.9em;\">This means that the student who scored 39% is in the 0 percentile. A score of 39% is not higher than any other score. <\/span><\/p>\n<p><span style=\"font-size: 0.9em;\">b) For a score of 60%: <\/span><\/p>\n<p><span style=\"font-size: 0.9em;\">There are 13 scores lower than 60%\u00a0 so\u00a0 \u00a0 k = (13\/48)*100% = 27%. A score of\u00a0 60% is in the 27th percentile which means that 27% (or just over one-fourth)\u00a0 of the test scores are less than 60%.\u00a0 <\/span><\/p>\n<p><span style=\"font-size: 0.9em;\">c) For a score of 94%:\u00a0 <\/span><\/p>\n<p><span style=\"font-size: 0.9em;\">There are 44 scores less than 94%\u00a0 so\u00a0 \u00a0 \u00a0k = (44\/48)*100% = 92%. A score of\u00a0 94% is in the 92nd percentile which means that 92%\u00a0 of the test scores are less than 94%.\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A set of assignment scores for a class of 32 students are provided in the table below. Determine the percentiles associated with the scores of\u00a0 a) 61%\u00a0 b) 79%\u00a0 c) 98%.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 14.2857%;\">72<\/td>\n<td style=\"width: 14.2857%;\">65<\/td>\n<td style=\"width: 14.2857%;\">85<\/td>\n<td style=\"width: 14.2857%;\">52<\/td>\n<td style=\"width: 14.2857%;\">61<\/td>\n<td style=\"width: 14.2857%;\">49<\/td>\n<td style=\"width: 7.14285%;\">65<\/td>\n<td style=\"width: 7.14285%;\">82<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;\">55<\/td>\n<td style=\"width: 14.2857%;\">99<\/td>\n<td style=\"width: 14.2857%;\">58<\/td>\n<td style=\"width: 14.2857%;\">79<\/td>\n<td style=\"width: 14.2857%;\">98<\/td>\n<td style=\"width: 14.2857%;\">79<\/td>\n<td style=\"width: 7.14285%;\">58<\/td>\n<td style=\"width: 7.14285%;\">93<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;\">88<\/td>\n<td style=\"width: 14.2857%;\">48<\/td>\n<td style=\"width: 14.2857%;\">97<\/td>\n<td style=\"width: 14.2857%;\">74<\/td>\n<td style=\"width: 14.2857%;\">65<\/td>\n<td style=\"width: 14.2857%;\">85<\/td>\n<td style=\"width: 7.14285%;\">71<\/td>\n<td style=\"width: 7.14285%;\">75<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;\">99<\/td>\n<td style=\"width: 14.2857%;\">39<\/td>\n<td style=\"width: 14.2857%;\">60<\/td>\n<td style=\"width: 14.2857%;\">96<\/td>\n<td style=\"width: 14.2857%;\">80<\/td>\n<td style=\"width: 14.2857%;\">70<\/td>\n<td style=\"width: 7.14285%;\">54<\/td>\n<td style=\"width: 7.14285%;\">77<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<details>\n<summary>Show answer<\/summary>\n<p>a) 61% is 28th percentile\u00a0 \u00a0b) 79% is 59th percentile\u00a0 \u00a0 c) 98% is 91st percentile<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p><strong style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1em;\">Quartiles<\/strong><\/p>\n<p id=\"fs-idp16986528\">Quartiles divide ordered data into quarters. Quartiles are special percentiles. The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the same as the 25<sup>th<\/sup> percentile, and the third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the same as the 75<sup>th<\/sup> percentile.\u00a0The <span data-type=\"term\">median<\/span> is a number that separates ordered data into halves. Half the values are the same as or smaller than the median, and half the values are the same as or larger than the median. The median can be called both the second quartile Q<sub>2<\/sub> and the 50<sup>th<\/sup> percentile.<\/p>\n<div class=\"part-title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Quartiles<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Quartiles divide the data set into <strong>four<\/strong> equal parts.<\/p>\n<p>The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the same as the 25<sup>th<\/sup> percentile, and the third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the same as the 75<sup>th<\/sup> percentile.\u00a0The median can be called both the second quartile, Q<sub>2 <\/sub>, and the 50<sup>th<\/sup> percentile.<\/p>\n<p>As with the median, the quartiles may or may not be part of the data set.<\/p>\n<\/div>\n<\/div>\n<p>As indicated in <a href=\"#figure1\">Figure 1<\/a> each quartile divides a data set into four equal parts so that one-fourth of the data set is located in each part.<a id=\"#figure1\"><\/a><\/p>\n<\/div>\n<figure id=\"attachment_2736\" aria-describedby=\"caption-attachment-2736\" style=\"width: 636px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2736 size-full\" title=\"created by Kim Moshenko\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles.png\" alt=\"\" width=\"636\" height=\"226\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles.png 636w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-300x107.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-65x23.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-225x80.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-350x124.png 350w\" sizes=\"auto, (max-width: 636px) 100vw, 636px\" \/><figcaption id=\"caption-attachment-2736\" class=\"wp-caption-text\">Fig. 1<\/figcaption><\/figure>\n<div class=\"part-title\">\n<h6><strong>Determining Quartiles<\/strong><\/h6>\n<p>We will consider two methods for determining quartiles. As with percentiles, the data values must first be ordered from smallest to largest. The first method involves dividing the data set into four equal parts. The second method involves the use of formulas.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Determining Quartiles: Method 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Step 1: Order the data from smallest to largest.<\/p>\n<p>Step 2: Determine the number of data values <strong>n<\/strong>.<\/p>\n<p>Step 3: Determine the median (Q<sub>2<\/sub>) of the data set. This will divide the data set into two equal parts.<\/p>\n<p>Step 4: Determine Q<sub>1<\/sub>. This will divide the first half of the data set into two equal parts.<\/p>\n<p>Step 5: Determine Q<sub>3<\/sub>. This will divide the second half of the data set into two equal parts.<\/p>\n<p><strong>Note:<\/strong> The median and the quartiles may not be actual observations from the data set.<\/p>\n<\/div>\n<\/div>\n<h6><strong>Method 1<\/strong><\/h6>\n<p>Consider the following data set:<\/p>\n<p style=\"text-align: center;\">15\u00a0 \u00a0 \u00a0 \u00a04\u00a0 \u00a0 \u00a0 \u00a020\u00a0 \u00a0 \u00a0 \u00a08\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0 \u00a012\u00a0 \u00a0 \u00a0 14\u00a0 \u00a0 \u00a0 11\u00a0 \u00a0 \u00a0 7\u00a0 \u00a0 \u00a02\u00a0 \u00a0 \u00a06\u00a0 \u00a0 \u00a023\u00a0 \u00a0 \u00a016<\/p>\n<p>Step 1: To determine the \u00a0quartiles, order the data values from smallest to largest:<\/p>\n<p style=\"text-align: center;\">2\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a04\u00a0 \u00a0 \u00a06\u00a0 \u00a0 \u00a07\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a08\u00a0\u00a0 \u00a0\u00a0\u00a011\u00a0\u00a0\u00a0\u00a0 12\u00a0\u00a0\u00a0 \u00a0\u00a014\u00a0 \u00a0\u00a0\u00a0\u00a015\u00a0 \u00a0 16\u00a0 \u00a0 \u00a020\u00a0 \u00a0 \u00a023<\/p>\n<p>Step 2: The number of data values is 13.<\/p>\n<p>Step 3: Determine the <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_2665_4755\">median<\/a><span data-type=\"term\">, which<\/span> measures the &#8220;centre&#8221; of the data. It is the number that separates ordered data into halves. Half the observations are the same number or smaller than the median, and half the observations are the same number or larger.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5562\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1.png\" alt=\"\" width=\"469\" height=\"66\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1.png 994w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-300x42.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-768x107.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-65x9.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-225x31.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-350x49.png 350w\" sizes=\"auto, (max-width: 469px) 100vw, 469px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p id=\"element-546\">Since there are 13 observations, the median will be in the seventhh position. The median, and therefore the 2nd quartile Q<sub>2<\/sub> , is eleven. The median is often referred to as\u00a0 the &#8220;middle observation,&#8221; but it is important to note that it does not actually have to be one of the observed values.<\/p>\n<p id=\"element-308\" style=\"text-align: left;\">Step 4: The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the <strong>middle value of the lower half<\/strong> of the data.<\/p>\n<p>To determine the<strong> first quartile<\/strong>, Q<sub>1<\/sub>, consider the lower half of the data observations:<\/p>\n<p style=\"text-align: center;\">2\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a04\u00a0 \u00a0 \u00a0 6\u00a0 \u00a0 \u00a0 7\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a08<\/p>\n<p>Since there are six observations, the middle observation will be the average of the third and fourth data values\u00a0 or\u00a0 (4 + 6)\/2 = 5\u00a0 therefore\u00a0 \u00a0Q<sub>1<\/sub>\u00a0 is 5<\/p>\n<p id=\"element-308\" style=\"text-align: left;\">Step 5: The third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the <strong>middle value of the upper half<\/strong> of the data.<\/p>\n<p>To determine the<strong> third quartile<\/strong>, Q<sub>3<\/sub>, consider the upper half of the data observations:<\/p>\n<p style=\"text-align: center;\">12\u00a0\u00a0\u00a0 \u00a0\u00a014\u00a0 \u00a0\u00a0\u00a0\u00a015\u00a0 \u00a0 \u00a016\u00a0 \u00a0 20\u00a0 \u00a0 23<\/p>\n<p><span data-type=\"newline\">Since there are six observations, the middle observation will be the average of 15 and 16 , or 15.5 therefore\u00a0 \u00a0 Q<sub>3<\/sub>\u00a0 is 15.5.<\/span><\/p>\n<p><a href=\"#figure2\">Figure 2<\/a> illustrates the three quartiles, which divide the data set into four equal parts.<a id=\"#figure2\"><\/a><\/p>\n<figure id=\"attachment_5558\" aria-describedby=\"caption-attachment-5558\" style=\"width: 610px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5558\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno.png\" alt=\"\" width=\"610\" height=\"149\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno.png 1010w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-300x73.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-768x188.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-225x55.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-350x86.png 350w\" sizes=\"auto, (max-width: 610px) 100vw, 610px\" \/><figcaption id=\"caption-attachment-5558\" class=\"wp-caption-text\">Fig. 2<\/figcaption><\/figure>\n<p id=\"element-227\">The number 4.5 is the <span data-type=\"term\">first quartile, Q<sub>1<\/sub><\/span>. One-fourth of the entire set of observations lie below 4.5 and\u00a0 three-fourths of the data observations lie above 4.5.<\/p>\n<p>The <span data-type=\"term\">third quartile<\/span>, <em data-effect=\"italics\">Q<sub>3<\/sub><\/em>, is 15.5. Three-fourths (75%) of the ordered data set lie below 15.5. One-fourth (25%) of the ordered data set lie above 15.5.<\/p>\n<\/div>\n<p>It is important to note that a quartile may not be a data observation. Sometimes there may be a need to average or weight the data values when determining the quartiles.<\/p>\n<div class=\"part-title\">\n<p><span style=\"text-align: initial; font-size: 14pt;\">A second method for determining quartiles is to use a formula to determine the position of each quartile. This is especially useful when there is a large number of data items.<\/span><\/p>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Determining Quartiles: Method 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Quartile Formula<\/strong><\/p>\n<p>The following formulas, where <strong>n<\/strong> is the <strong>number of data values<\/strong>,\u00a0 can be used to determine the <strong>position<\/strong> of the three quartiles.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2796 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_56_13-8.1-DA2-Images.docx-Word-300x73.png\" alt=\"\" width=\"300\" height=\"73\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_56_13-8.1-DA2-Images.docx-Word-300x73.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_56_13-8.1-DA2-Images.docx-Word-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_56_13-8.1-DA2-Images.docx-Word-225x55.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_56_13-8.1-DA2-Images.docx-Word.png 315w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>It is important to note that these results indicate the <strong>positions<\/strong> of the quartiles, not the actual data obervations. If, for example, the calculation gives Q<sub>1<\/sub>=3, this indicates that the first quartile will be the data obervation in the 3rd <strong>position.<\/strong> If\u00a0 Q<sub>3<\/sub> = 32, this indicates that the third quartile will be the data observation in the 32nd <strong>position.<\/strong><\/p>\n<p>Step 1: Order the data from smallest to largest.<\/p>\n<p>Step 2:\u00a0 Determine<strong> n.<\/strong><\/p>\n<p>Step 3: Use the formula to determine the <strong>position<\/strong> for the median (Q<sub>2<\/sub>) of the data set. Count from left to right to determine the corresponding data value. If the position is a fraction then two data values will need to be weighted to determine the median value.<\/p>\n<p>Step 4: Use the formula to determine the <strong>position<\/strong> for the first quartile Q<sub>1<\/sub> of the data set. Count from left to right to determine the corresponding data value. If the position is a fraction then two data values will need to be weighted to determine the value of Q<sub>1<\/sub>.<\/p>\n<p>Step 5: Use the formula to determine the <strong>position<\/strong> for the third quartile\u00a0 Q<sub>3<\/sub> of the data set. Count from left to right to determine the corresponding data value. If the position is a fraction then two data values will need to be weighted to determine the value of Q<sub>3<\/sub>.<\/p>\n<\/div>\n<\/div>\n<h6><strong>Method 2:<\/strong><\/h6>\n<p>Consider the following data set:<\/p>\n<p style=\"text-align: center;\">15\u00a0 \u00a0 \u00a0 \u00a04\u00a0 \u00a0 \u00a0 \u00a020\u00a0 \u00a0 \u00a0 \u00a08\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0 \u00a012\u00a0 \u00a0 \u00a0 14\u00a0 \u00a0 \u00a0 11\u00a0 \u00a0 \u00a0 7\u00a0 \u00a0 \u00a02\u00a0 \u00a0 \u00a06\u00a0 \u00a0 \u00a023\u00a0 \u00a0 \u00a016<\/p>\n<p>Step 1: To determine the \u00a0quartiles, order the data values from smallest to largest:<\/p>\n<p style=\"text-align: center;\">2\u00a0 \u00a0 \u00a0 3\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a04\u00a0 \u00a0 \u00a06\u00a0 \u00a0 \u00a07\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a08\u00a0\u00a0 \u00a0\u00a0\u00a011\u00a0\u00a0\u00a0\u00a0 12\u00a0\u00a0\u00a0 \u00a0\u00a014\u00a0 \u00a0\u00a0\u00a0\u00a015\u00a0 \u00a0 16\u00a0 \u00a0 \u00a020\u00a0 \u00a0 \u00a023<\/p>\n<p>Step 2: The number of data values is 13.<\/p>\n<div class=\"part-title\">\n<p>Step 3: Use the formula to determine the <strong>position<\/strong> for the median (Q<sub>2<\/sub>) of the data set.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5557 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno1-300x76.png\" alt=\"\" width=\"300\" height=\"76\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno1-300x76.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno1-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno1-225x57.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno1-350x89.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno1.png 536w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Count from left to right to determine the corresponding data value in the 7th position. The corresponding value is 11.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5562\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1.png\" alt=\"\" width=\"469\" height=\"66\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1.png 994w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-300x42.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-768x107.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-65x9.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-225x31.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-medianoddno-1-350x49.png 350w\" sizes=\"auto, (max-width: 469px) 100vw, 469px\" \/><\/p>\n<p style=\"text-align: left;\">Step 4:\u00a0Use the formula to determine the <strong>position<\/strong> for the first quartile (Q<sub>1<\/sub>) of the data set.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5609 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1Q3-method2-300x76.png\" alt=\"\" width=\"300\" height=\"76\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1Q3-method2-300x76.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1Q3-method2-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1Q3-method2-225x57.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1Q3-method2-350x88.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1Q3-method2.png 527w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Since 3.5 is a fraction, the first quartile will be the average of the two data values that are in the 3rd and 4th positions. Count from left to right to determine the corresponding data values. The data value 4 is in the 3rd position and the data value 6 is in the 4th position so these will be averaged (4 + 6)\/2 = 5. The first quartile will be 5.<\/p>\n<p style=\"text-align: left;\">Step 5:\u00a0Use the formula to determine the <strong>position<\/strong> for the third quartile (Q<sub>3<\/sub>) of the data set.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5560 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Q3method1oddno-300x64.png\" alt=\"\" width=\"300\" height=\"64\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Q3method1oddno-300x64.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Q3method1oddno-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Q3method1oddno-225x48.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Q3method1oddno-350x74.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Q3method1oddno.png 636w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Since 10.5 is a fraction, the third quartile will be the average of the two data values that are in the 10th and 11th positions. Count from left to right to determine the corresponding data values. The data value 15 is in the 10th position and the data value 16\u00a0 is in the 11th position so these will be averaged (15 + 16)\/2 = 15.5. The third quartile will be 15.5.<\/p>\n<p><a href=\"#figure3\">Figure 3<\/a> illustrates the three quartiles, which divide the data set into four equal parts.<a id=\"#figure3\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_5558\" aria-describedby=\"caption-attachment-5558\" style=\"width: 610px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5558\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno.png\" alt=\"\" width=\"610\" height=\"149\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno.png 1010w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-300x73.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-768x188.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-225x55.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-quartilesmethod1oddno-350x86.png 350w\" sizes=\"auto, (max-width: 610px) 100vw, 610px\" \/><figcaption id=\"caption-attachment-5558\" class=\"wp-caption-text\">Fig. 3<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"textbox__content\">\n<p>A shoe store wanted to determine the popularity of different shoe sizes for women&#8217;s tennis shoes. It planned to place its next order using this information. In\u00a0 a five day period it sold nine pairs of women&#8217;s tennis shoes in the following sizes:\u00a0 \u00a0 7,\u00a0 8, 11,\u00a0 10,\u00a0 7,\u00a0 \u00a06,\u00a0 \u00a09, 10,\u00a0 7<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>Method 1:<\/strong><\/p>\n<p>To determine the quartiles:<\/p>\n<ol>\n<li>Order the shoe sizes from smallest to largest:\u00a0 6,\u00a0 7,\u00a0 \u00a07,\u00a0 \u00a07,\u00a0 8,\u00a0 \u00a09,\u00a0 10,\u00a0 \u00a010,\u00a0 \u00a011<\/li>\n<li>Count the number of values: n = 9<\/li>\n<li>Determine Q<sub>2<\/sub>, the median, which is the middle observation. Since there are nine data observations (shoe sizes) t<span style=\"text-align: initial; font-size: 0.9em;\">he median, or second quartile, will be the 5th <\/span><span style=\"text-align: initial; font-size: 0.9em;\">data value. The 5th data value is 8.\u00a0<\/span><\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-4775 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-300x110.png\" alt=\"\" width=\"300\" height=\"110\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-300x110.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-65x24.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-225x83.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-350x129.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1.png 685w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>4. Determine the first quartile Q<sub style=\"text-align: initial;\">1. <\/sub>It <span style=\"font-size: 0.9em; text-align: initial;\">will be the middle observation of the <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">lower half<\/strong><span style=\"font-size: 0.9em; text-align: initial;\"> of data values. This will be the average of the 2nd and 3rd data values\u00a0 so (7 +7)\/2 = 7.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2804\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"301\" height=\"123\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word.png 333w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word-300x123.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word-65x27.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word-225x92.png 225w\" sizes=\"auto, (max-width: 301px) 100vw, 301px\" \/><\/p>\n<p><span style=\"font-size: 0.9em; text-align: initial;\">5. Determine the third quartile Q<\/span><sub style=\"text-align: initial;\">3<\/sub>.<span style=\"font-size: 0.9em; text-align: initial;\">\u00a0This will be the middle observation of the <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">upper half<\/strong><span style=\"font-size: 0.9em; text-align: initial;\">. This will be the average of the 7th and 8th data values\u00a0 so (10+10)\/2 = 10\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Method 2:<\/strong><\/p>\n<p>The formulas can be used to determine the quartiles.<\/p>\n<ol>\n<li>Order the shoe sizes from smallest to largest:\u00a0 6,\u00a0 7,\u00a0 \u00a07,\u00a0 \u00a07,\u00a0 8,\u00a0 \u00a09,\u00a0 10,\u00a0 \u00a010,\u00a0 \u00a011 .<\/li>\n<li>Determine the number of data values, n.\u00a0 \u00a0 \u00a0n = 9<\/li>\n<li>Use the formula to determine the median. The median , or second quartile,\u00a0 can be determined as follows:<\/li>\n<\/ol>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2794 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_54_42-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"259\" height=\"64\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_54_42-8.1-DA2-Images.docx-Word.png 259w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_54_42-8.1-DA2-Images.docx-Word-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_54_42-8.1-DA2-Images.docx-Word-225x56.png 225w\" sizes=\"auto, (max-width: 259px) 100vw, 259px\" \/><\/p>\n<p>Counting from left to right, the 5<sup>th<\/sup> data value is 8. The median, or 2nd quartile Q<sub>2<\/sub>, is 8.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-4775 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-300x110.png\" alt=\"\" width=\"300\" height=\"110\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-300x110.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-65x24.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-225x83.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1-350x129.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles-3-1.png 685w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>4 &amp; 5.\u00a0 The first and third quartiles can be determined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2795 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_55_16-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"608\" height=\"80\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_55_16-8.1-DA2-Images.docx-Word.png 608w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_55_16-8.1-DA2-Images.docx-Word-300x39.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_55_16-8.1-DA2-Images.docx-Word-65x9.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_55_16-8.1-DA2-Images.docx-Word-225x30.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_55_16-8.1-DA2-Images.docx-Word-350x46.png 350w\" sizes=\"auto, (max-width: 608px) 100vw, 608px\" \/><\/p>\n<p>The first quartile is the 2.5th data value. To determine the 2.5<sup>th<\/sup> data value we must take the average of the 2nd and 3rd data values. The 2nd data value is 7 and the 3rd data value is 7 so\u00a0 (7+7)\/2 = 7.<\/p>\n<p>The first quartile, Q<sub>1<\/sub> = 7<\/p>\n<p>The third quartile is the 7.5th data value. This will be the average of the 7th and 8th data values. The 7th data value is 10 and the 8th data value is also 10\u00a0 so\u00a0 \u00a0(10+10)\/2 = 10.<\/p>\n<p>The third quartile, Q<sub>3<\/sub> = 10<\/p>\n<p>We can see that\u00a0 Q<sub>2 <\/sub>= 8\u00a0 splits the data set into two halves. <span style=\"text-align: initial; font-size: 0.9em;\">Q<\/span><sub style=\"text-align: initial;\">1<\/sub>= 7\u00a0<span style=\"text-align: initial; font-size: 0.9em;\"> is the middle value of the lower half of the data set and Q<\/span><sub style=\"text-align: initial;\">3<\/sub> = 10<span style=\"text-align: initial; font-size: 0.9em;\"> is the middle value of the upper half of the data set.\u00a0 \u00a0<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2804 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"333\" height=\"136\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word.png 333w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word-300x123.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word-65x27.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_01-8.1-DA2-Images.docx-Word-225x92.png 225w\" sizes=\"auto, (max-width: 333px) 100vw, 333px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In Example 2 the number of data items was <strong>odd.<\/strong> When <em>n<\/em>\u00a0 is odd the median or Q<sub>2<\/sub> will be one of the data observations. When <em>n <\/em>is odd the formula for finding quartiles is straight forward.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Determine the quartiles for the following temperature data that was recorded over a 3-week period in May:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2817 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_17_54-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"665\" height=\"161\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_17_54-8.1-DA2-Images.docx-Word.png 665w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_17_54-8.1-DA2-Images.docx-Word-300x73.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_17_54-8.1-DA2-Images.docx-Word-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_17_54-8.1-DA2-Images.docx-Word-225x54.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_17_54-8.1-DA2-Images.docx-Word-350x85.png 350w\" sizes=\"auto, (max-width: 665px) 100vw, 665px\" \/><\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Show answer<\/summary>\n<p>Q<sub>2<\/sub> = 21; Q<sub>1<\/sub> = 18; Q<sub>3<\/sub> = 25<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p>It is important to note that a quartile may <strong>not<\/strong> be a data observation. When the number of data values <em>n<\/em>\u00a0 is <strong>even<\/strong> the median or Q<sub>2<\/sub> will <strong>not<\/strong> be one of the actual data observations. As a result, when <em>n <\/em>is <strong>even<\/strong> an adjustment must be made to the value of <strong>n<\/strong> that is to be used in the formula to determine the <strong>first<\/strong> and <strong>third<\/strong> quartiles.<\/p>\n<p><strong>Method 1<\/strong>:<\/p>\n<p>Consider the following data set:<\/p>\n<p style=\"text-align: center;\">1;\u00a0 11.5;\u00a0 6;\u00a0 7.2;\u00a0 4;\u00a0 8;\u00a0 9;\u00a0 10;\u00a0 6.8;\u00a0 \u00a08.3;\u00a0 \u00a02;\u00a0 \u00a02;\u00a0 \u00a010;\u00a0 1<\/p>\n<p>Step 1: To determine the \u00a0quartiles, order the data values from smallest to largest:<\/p>\n<p style=\"text-align: center;\">1\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a02\u00a0 \u00a04\u00a0 \u00a06\u00a0 \u00a06.8\u00a0 \u00a07.2\u00a0 \u00a08\u00a0 \u00a0 8.3\u00a0 \u00a09\u00a0 \u00a0 10\u00a0 \u00a010\u00a0 \u00a011.5<\/p>\n<p>Step 2:\u00a0 The number of data values is 14<\/p>\n<p>Step 3: Determine the <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_2665_4755\">median<\/a><span data-type=\"term\">, which<\/span> measures the &#8220;centre&#8221; of the data. It is the number that separates ordered data into halves. Half the observations are the same number or smaller than the median, and half the observations are the same number or larger.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2787 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_18_38-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"524\" height=\"78\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_18_38-8.1-DA2-Images.docx-Word.png 524w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_18_38-8.1-DA2-Images.docx-Word-300x45.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_18_38-8.1-DA2-Images.docx-Word-65x10.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_18_38-8.1-DA2-Images.docx-Word-225x33.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_18_38-8.1-DA2-Images.docx-Word-350x52.png 350w\" sizes=\"auto, (max-width: 524px) 100vw, 524px\" \/><\/p>\n<p id=\"element-546\">Since there are 14 observations, the median lies between the seventh observation, 6.8, and the eighth observation, 7.2. To find the median, add the two values together and divide by two.\u00a0 \u00a0Median = (6.8 + 7.2)\/2 = 7<\/p>\n<p>The median, and therefore the 2nd quartile Q<sub>2<\/sub> , is seven. It is important to note that the median is not actually one of the observed data values.<\/p>\n<p id=\"element-308\" style=\"text-align: left;\">Step 4: The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the <strong>middle value of the lower half<\/strong> of the data.<\/p>\n<p>To determine the<strong> first quartile<\/strong>, Q<sub>1<\/sub>, consider the lower half of the data observations:<\/p>\n<p style=\"text-align: center;\">1\u00a0 \u00a0 \u00a01\u00a0 \u00a0 \u00a02\u00a0 \u00a0 \u00a02\u00a0 \u00a0 4\u00a0 \u00a0 6\u00a0 \u00a0 6.8.<\/p>\n<p>Since there are seven observations, the middle observation will be the 4th item. The middle or 4<sup>th<\/sup> item of these data observations\u00a0 is 2.<\/p>\n<p id=\"element-308\" style=\"text-align: left;\">Step 5: The third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the <strong>middle value of the upper half<\/strong> of the data.<\/p>\n<p>To determine the<strong> third quartile<\/strong>, Q<sub>3<\/sub>, consider the upper half of the data observations:<\/p>\n<p style=\"text-align: center;\">7.2\u00a0 \u00a0 \u00a08\u00a0 \u00a0 \u00a08.3\u00a0 \u00a0 9\u00a0 \u00a0 10\u00a0 \u00a0 10\u00a0 \u00a0 \u00a011.5.<\/p>\n<p>Since there are seven observations, the middle observation will be the 4th item in the upper half. The middle item of these data observations is 9.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><a href=\"#figure4\">Figure 4<\/a> illustrates the three quartiles, which divide the data set into four equal parts.<a id=\"#figure4\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_2788\" aria-describedby=\"caption-attachment-2788\" style=\"width: 526px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2788 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"526\" height=\"114\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word.png 526w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word-300x65.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word-225x49.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word-350x76.png 350w\" sizes=\"auto, (max-width: 526px) 100vw, 526px\" \/><figcaption id=\"caption-attachment-2788\" class=\"wp-caption-text\">Fig. 4<\/figcaption><\/figure>\n<p id=\"element-227\">The number 2 is the <span data-type=\"term\">first quartile, Q<sub>1<\/sub><\/span>. One-fourth of the entire set of observations lie below 2 and\u00a0 three-fourths of the data observations lie above 2.<\/p>\n<p>The <span data-type=\"term\">third quartile<\/span>, <em data-effect=\"italics\">Q<sub>3<\/sub><\/em>, is 9. Three-fourths (75%) of the ordered data set lie below 9. One-fourth (25%) of the ordered data set lie above 9.<\/p>\n<p><strong>Method 2<\/strong>:<\/p>\n<p>Consider the following data set:<\/p>\n<p style=\"text-align: center;\">1;\u00a0 11.5;\u00a0 6;\u00a0 7.2;\u00a0 4;\u00a0 8;\u00a0 9;\u00a0 10;\u00a0 6.8;\u00a0 \u00a08.3;\u00a0 \u00a02;\u00a0 \u00a02;\u00a0 \u00a010;\u00a0 1<\/p>\n<p>Step 1: To determine the \u00a0quartiles, order the data values from smallest to largest:<\/p>\n<p style=\"text-align: center;\">1\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a02\u00a0 \u00a04\u00a0 \u00a06\u00a0 \u00a06.8\u00a0 \u00a07.2\u00a0 \u00a08\u00a0 \u00a0 8.3\u00a0 \u00a09\u00a0 \u00a0 10\u00a0 \u00a010\u00a0 \u00a011.5<\/p>\n<p>Step 2:\u00a0 The number of data values is 14 so <strong>n is an even number.<\/strong><\/p>\n<p>Step 3: Use the formula to determine the position of\u00a0 Q<sub>2<\/sub>, the median. The position will be (14 + 1)\/2 = 7.5. This means that the median,\u00a0 or Q<sub>2<\/sub>, will be in the 7.5th observation or halfway between the 7th and 8th position. The observation 6.8 is in the 7th position and the observation 7.2 is in the 8th position therefore the average of these (6.8 + 7.2)\/ 2 is the median or Q<sub>2<\/sub>.<\/p>\n<p>Note that the median is <strong>not<\/strong> an actual observation in the data set. If we use the formula to find Q<sub>1<\/sub> and Q<sub>3<\/sub> then we must adjust &#8220;n&#8221; to include this additional item so in effect &#8220;n&#8221; will be 15. <span style=\"text-align: initial; font-size: 0.9em;\">This is done <strong>only<\/strong> when determining the positions of Q<sub>1<\/sub> and Q<sub>3\u00a0<\/sub>(and not for determining the position of Q<sub>2<\/sub>)<\/span><\/p>\n<p>Step 4: Use the formula to determine the position of\u00a0 Q<sub>1<\/sub>, the first quartile. Remember than<strong> n<\/strong> will now be 15, not 14. The position will be\u00a0(15 + 1)\/4 = 4 th. This means that\u00a0 Q<sub>1<\/sub> will be in the 4th position.\u00a0Counting from the left, the data value 2 is in the 4th position so Q<sub>1<\/sub>= 2.<\/p>\n<p>Step 5: Use the formula to determine the position of\u00a0 Q<sub>3<\/sub>, the third quartile. Remember than<strong> n<\/strong> will now be 15, not 14. The position will be 3(15 + 1)\/4 = 12th. This means that\u00a0 Q<sub>3<\/sub> will be in the 12th position. Refer to <a href=\"#figure5\">Figure 5<\/a>. Counting from the left, we include the median value of\u00a0 7,\u00a0 to determine that the data value in the 12th position. This value is 9\u00a0 so \u00a0Q<sub>3<\/sub> will be 9.<a id=\"#figure5\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_5513\" aria-describedby=\"caption-attachment-5513\" style=\"width: 544px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5513\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4-1024x214.png\" alt=\"\" width=\"544\" height=\"114\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4-1024x214.png 1024w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4-300x63.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4-768x160.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4-225x47.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4-350x73.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/05\/8.1-Quartiles-ex4.png 1069w\" sizes=\"auto, (max-width: 544px) 100vw, 544px\" \/><figcaption id=\"caption-attachment-5513\" class=\"wp-caption-text\">Fig. 5<\/figcaption><\/figure>\n<p>It is also important to recognize that the median of 7 is not an actual data value in this set. It was included in <a href=\"#figure5\">Figure 5<\/a> to illustrate that its <strong>position<\/strong> must be counted when determing the position of the third quartile. It is not actually part of the data set. The actual data set is illustrated in <a href=\"#figure6\">Figure 6<\/a>\u00a0 (and <a href=\"#figure4\">Figure 4<\/a>).<a id=\"#figure6\"><\/a><\/p>\n<figure id=\"attachment_2788\" aria-describedby=\"caption-attachment-2788\" style=\"width: 526px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2788 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"526\" height=\"114\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word.png 526w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word-300x65.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word-225x49.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-14_26_02-8.1-DA2-Images.docx-Word-350x76.png 350w\" sizes=\"auto, (max-width: 526px) 100vw, 526px\" \/><figcaption id=\"caption-attachment-2788\" class=\"wp-caption-text\">Fig. 6<\/figcaption><\/figure>\n<p><span style=\"font-size: 14pt; text-align: initial;\">Consider <a href=\"#figure7\">Figure 7<\/a> where the data set that has an even number of data values:\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a0 4\u00a0 \u00a0 5\u00a0 <a id=\"#figure7\"><\/a><\/span><\/p>\n<figure id=\"attachment_3043\" aria-describedby=\"caption-attachment-3043\" style=\"width: 274px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3043\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-2-300x204.png\" alt=\"\" width=\"274\" height=\"186\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-2-300x204.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-2-65x44.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-2-225x153.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-2-350x238.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.1-Quartiles-2.png 461w\" sizes=\"auto, (max-width: 274px) 100vw, 274px\" \/><figcaption id=\"caption-attachment-3043\" class=\"wp-caption-text\">Fig. 7<\/figcaption><\/figure>\n<p>In this data set\u00a0 Q<sub>1<\/sub> = 1.5,\u00a0 Q<sub>2<\/sub> = 3,\u00a0 and Q<sub>3<\/sub> = 4.5\u00a0 \u00a0This illustrates that quartile values need not be actual values in the data set. The second quartile Q<sub>2<\/sub> is 3 which \u00a0is the average of the data values 2 and 4. Similarily the first quartile of 1.5 is the average of two data values 1 and 2 and the third quartile of 4.5 is the average of the two data values 4 and 5. Determining the quartile values can become complex as it may require different weightings of the data values but this is beyond the scope of this textbook.<\/p>\n<p>Example 3 illustrates two techniques for determining quartiles when the number of data observations is <strong>even<\/strong>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"textbox__content\">\n<p>Consider again the\u00a0 shoe store and a different week. Over a five day period it sold ten pairs of tennis shoes in the following sizes:<\/p>\n<p style=\"text-align: center;\">6,\u00a0 8, 11,\u00a0 10,\u00a0 7,\u00a0 \u00a06,\u00a0 \u00a09, 10,\u00a0 8,\u00a0 \u00a09<\/p>\n<p>Note that there is an <strong>even<\/strong> number of data values\u00a0 n = 10<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>Method 1:<\/strong><\/p>\n<p>To determine the quartiles:<\/p>\n<ol>\n<li>Rank the sizes from smallest to largest:\u00a0 \u00a06,\u00a0 6,\u00a0 \u00a07,\u00a0 \u00a08,\u00a0 8,\u00a0 \u00a09,\u00a0 \u00a09,\u00a0 \u00a010,\u00a0 \u00a010,\u00a0 11 and divide the data set into four equal quarters.<\/li>\n<li>n = 10<\/li>\n<li>Start with the median which is the middle observation. <span style=\"text-align: initial; font-size: 0.9em;\">The median, or second quartile, will <\/span><span style=\"text-align: initial; font-size: 0.9em;\">lie between the 5th and 6th data values. The 5th data value is 8\u00a0 and the 6th data value is 9\u00a0 so the average of 8 and 9, or 8.5, is the median.<\/span><\/li>\n<li>Determine the first quartile Q<sub style=\"text-align: initial;\">1. <\/sub>It <span style=\"font-size: 0.9em; text-align: initial;\">will be the middle observation of the <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">lower half<\/strong><span style=\"font-size: 0.9em; text-align: initial;\"> of data values. This is the 3rd data value or the observation of\u00a0 7.<\/span><\/li>\n<li><span style=\"font-size: 0.9em; text-align: initial;\">Determine the third quartile Q<\/span><sub style=\"text-align: initial;\">3<\/sub>.<span style=\"font-size: 0.9em; text-align: initial;\">\u00a0This will be the middle observation of the <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">upper half<\/strong><span style=\"font-size: 0.9em; text-align: initial;\">. This will be the data observation of 10.<\/span><\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2807 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_40_24-8.1-DA2-Images.docx-Word-300x106.png\" alt=\"\" width=\"308\" height=\"109\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_40_24-8.1-DA2-Images.docx-Word-300x106.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_40_24-8.1-DA2-Images.docx-Word-65x23.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_40_24-8.1-DA2-Images.docx-Word-225x79.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_40_24-8.1-DA2-Images.docx-Word-350x123.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_40_24-8.1-DA2-Images.docx-Word.png 352w\" sizes=\"auto, (max-width: 308px) 100vw, 308px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Note that each quartile divides the data values such that there are an equal number of data values in each of the four sections.<\/p>\n<p><strong>Method 2:<\/strong><\/p>\n<p><strong style=\"font-size: 0.9em; text-align: initial;\">An alternative is to use the formulas <\/strong><span style=\"font-size: 0.9em; text-align: initial;\">\u00a0to determine the quartiles.\u00a0<\/span><\/p>\n<p>To determine the quartiles:<\/p>\n<ol>\n<li>Rank the sizes from smallest to largest:\u00a0 \u00a06,\u00a0 6,\u00a0 \u00a07,\u00a0 \u00a08,\u00a0 8,\u00a0 \u00a09,\u00a0 \u00a09,\u00a0 \u00a010,\u00a0 \u00a010,\u00a0 11<\/li>\n<li>n = 10<\/li>\n<li>Determine the position of the median\u00a0 using the formula. <img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2805 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_35-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"287\" height=\"72\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_35-8.1-DA2-Images.docx-Word.png 287w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_35-8.1-DA2-Images.docx-Word-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_35-8.1-DA2-Images.docx-Word-225x56.png 225w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><\/li>\n<\/ol>\n<p>The 5.5th data value will be the average of the 5th and 6th data values. The 5th data value is 8\u00a0 and the 6th data value is 9\u00a0 so<\/p>\n<p style=\"text-align: center;\">(8 + 9) \/2 = 8.5\u00a0 The median or Q<sub>2<\/sub>\u00a0 is 8.5.<\/p>\n<p><span style=\"font-size: 0.9em; text-align: initial;\"><strong>Note:<\/strong><\/span><span style=\"font-size: 0.9em; text-align: initial;\">\u00a0Q<sub style=\"text-align: initial;\">2<\/sub> is <strong>not<\/strong> one of the <strong>actual<\/strong> data values. <\/span><span style=\"text-align: initial; font-size: 0.9em;\">\u00a0In this example Q<sub>2<\/sub> is the 5.5th data value or 8.5. It is the data value that lies between the 5th and 6th data values but it is not one of the original data values.\u00a0<\/span><\/p>\n<p>4 &amp; 5.\u00a0 \u00a0Determine the first quartile Q<sub style=\"text-align: initial;\">1\u00a0<\/sub>and the third quartile Q<sub>3<\/sub>.<\/p>\n<p><span style=\"font-size: 0.9em; text-align: initial;\">Since the number of data values <strong><em>n<\/em><\/strong> is <\/span><strong style=\"font-size: 0.9em; text-align: initial;\">even<\/strong><span style=\"font-size: 0.9em; text-align: initial;\"> the median or\u00a0 Q<sub style=\"text-align: initial;\">2<\/sub> is <strong>not<\/strong> one of the <strong>actual<\/strong> data values so when we use<\/span><span style=\"text-align: initial; font-size: 0.9em;\">\u00a0the formula to determine Q<\/span><sub style=\"text-align: initial;\">1<\/sub><span style=\"text-align: initial; font-size: 0.9em;\">\u00a0 and Q<\/span><sub style=\"text-align: initial;\">3<\/sub><span style=\"text-align: initial; font-size: 0.9em;\"> we must <strong>increase <\/strong><strong>the value of n by 1<\/strong>. In effect the number of data values has increased by one and therefore the value of <strong>n<\/strong> in the formula must be increased by 1. This is done only when determining the positions of Q<sub>1<\/sub> and Q<sub>3\u00a0<\/sub>(and not for determining the position of Q<sub>2<\/sub>)<\/span><\/p>\n<p><span style=\"text-align: initial; font-size: 0.9em;\">In this example, when determining Q<sub>1<\/sub> and Q<sub>3\u00a0<\/sub> the original value of\u00a0 <strong>n = 10<\/strong> will now be increased by 1. The new number for <strong>n<\/strong> to be used in the formula will be<strong> n = 11.<\/strong> Using the formula, the first and third quartile positions can be determined as follows:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"text-align: initial; font-size: 0.9em;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4787\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles3i-300x97.png\" alt=\"\" width=\"204\" height=\"66\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles3i-300x97.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles3i-65x21.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles3i-225x73.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles3i-350x114.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2021\/04\/8.1-Quartiles3i.png 545w\" sizes=\"auto, (max-width: 204px) 100vw, 204px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox__content\">\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2806 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_56-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"501\" height=\"61\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_56-8.1-DA2-Images.docx-Word.png 501w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_56-8.1-DA2-Images.docx-Word-300x37.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_56-8.1-DA2-Images.docx-Word-65x8.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_56-8.1-DA2-Images.docx-Word-225x27.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-15_39_56-8.1-DA2-Images.docx-Word-350x43.png 350w\" sizes=\"auto, (max-width: 501px) 100vw, 501px\" \/><\/p>\n<p>Using the results from the formula we count to get the 3rd and 9th data values. When determining these values\u00a0 be sure to include and count the <strong>\u00a0position occupied by the new median value<\/strong> of\u00a0 8.5. The 3rd data value is 7\u00a0 and the 9th data value is 10.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-3045 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3-300x93.png\" alt=\"\" width=\"300\" height=\"93\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3-300x93.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3-1024x316.png 1024w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3-768x237.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3-65x20.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3-225x69.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3-350x108.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/8.2-Quartiles3.png 1056w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Note that Method 1 and Method 2 yield the same results.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We have seen that either of Method 1 or Method 2 will produce the same quartile values although the formula method can be less intuitive when <strong>n<\/strong> is <strong>even.<\/strong><\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use either technique to determine the quartiles for the following temperature data that was recorded over the month of April:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2818 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_18_19-8.1-DA2-Images.docx-Word.png\" alt=\"\" width=\"656\" height=\"213\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_18_19-8.1-DA2-Images.docx-Word.png 656w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_18_19-8.1-DA2-Images.docx-Word-300x97.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_18_19-8.1-DA2-Images.docx-Word-65x21.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_18_19-8.1-DA2-Images.docx-Word-225x73.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/06\/2020-06-11-17_18_19-8.1-DA2-Images.docx-Word-350x114.png 350w\" sizes=\"auto, (max-width: 656px) 100vw, 656px\" \/><\/p>\n<details>\n<summary>Show answer<\/summary>\n<p>Q<sub>2<\/sub> = 18.5 Q<sub>1<\/sub> = 15; Q<sub>3<\/sub> = 22<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Consider the data set:\u00a0 3, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15.\u00a0 \u00a0Determine the three quartiles using either technique.<\/p>\n<p><strong>Method 1:<\/strong><\/p>\n<p>Step 1: Order the data values\u00a0 \u00a0 \u00a03, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15<\/p>\n<p>Step 2:\u00a0 n = 16<\/p>\n<p>Step 3:\u00a0 The median will be the average of 9 and 10, so 9.5. This is not one of the observed values.<\/p>\n<p>Step 4:\u00a0 Q<sub>1<\/sub> is the value that splits the lower half, which will be the average of 6 and 7, so 6.5.<\/p>\n<p>Step 5:\u00a0 Q<sub>3<\/sub> is the value that splits the upper half, which will be the average of 12 and 13, so 12.5.<\/p>\n<p><strong>Method 2:<\/strong><\/p>\n<p>Step 1: Order the data values\u00a0 \u00a0 \u00a03, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15<\/p>\n<p>Step 2:\u00a0 n = 16<\/p>\n<p>Step 3: Use the formula\u00a0 (16 + 1) \/2 = 8.5. The median will be in the 8.5th position. This is the average of the 8th value of\u00a0 9 and the 9th value of 10 so the median is 9.5<\/p>\n<p>Step 4 and 5: Since n is <strong>even,<\/strong> we will use a value of 17, not 16, in the formulas to determine Q<sub>1<\/sub> and Q<sub>3<\/sub>.<\/p>\n<p>Q<sub>1<\/sub> will be (17 + 1)\/4 = 4.5 th. This means that\u00a0 Q<sub>1<\/sub> will be in the 4.5th position or the average of the 4th and 5th data values. The 4th value is 6 and the 5th value is 7 so\u00a0 Q<sub>1<\/sub>= 6.5.<\/p>\n<p>Q<sub>3<\/sub> will be 3(17 + 1)\/4 = 13.5 th. This means that\u00a0 Q<sub>3<\/sub> will be in the 13.5th position or the average of the 13th and 14th data values. Including the median&#8217;s position when we count, the 13th value is 12 and the 14th value is 13 so\u00a0 Q<sub>3<\/sub> = (12 + 13)\/2 = 12.5.<\/p>\n<p><strong>Note<\/strong> that these identical results were obtained without using\u00a0 the formulas. It is also important to recognize that the median of 9.5 is not an actual data value in this set. It serves only to divide the data set into two equal halves and it is not actually part of the data set.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>An athlete was training for a race and logged the following distances (in km) over a 36 day period. Determine the three quartiles for the distances covered by the athlete.<\/p>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse; width: 100%; height: 60px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">22<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">34<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">38<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">42<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">14<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">22<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">0<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 11.1111%; height: 15px;\">21<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">30<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">41<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">56<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">11<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">30<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 11.1111%; height: 15px;\">24<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">52<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">11<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">16<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">28<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">36<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">25<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">25<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">11<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 11.1111%; height: 15px;\">0<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">18<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">24<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">20<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">46<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">38<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">40<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">27<\/td>\n<td style=\"width: 11.1111%; height: 15px;\">10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p>Q<sub>1<\/sub>= 17,\u00a0 Q<sub>2<\/sub> = 23,\u00a0 Q<sub>3<\/sub>= 35<\/p>\n<\/details>\n<\/div>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul>\n<li>A data set can be divided into one hundred equal parts by ninety-nine percentiles P<sub>1<\/sub> , P<sub>2<\/sub> , P<sub>3<\/sub> , &#8230; P<sub>99<\/sub> . Percentiles are best used with large sets of data.<\/li>\n<li>Quartiles divide the data set into <strong>four<\/strong> equal parts.\u00a0The first quartile, <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>, is the same as the 25<sup>th<\/sup> percentile, and the third quartile, <em data-effect=\"italics\">Q<\/em><sub>3<\/sub>, is the same as the 75<sup>th<\/sup> percentile.\u00a0The median can be called both the second quartile, Q<sub>2 <\/sub>, and the 50<sup>th<\/sup> percentile.<\/li>\n<li>Quartiles may or may not be actual observations within a set of data.<\/li>\n<\/ul>\n<h1><strong>Glossary<\/strong><\/h1>\n<div class=\"textbox shaded\">\n<p><strong>Percentiles<\/strong><\/p>\n<p>divide ordered data into hundredths.<\/p>\n<p><strong>Quartiles<\/strong><\/p>\n<p>divide ordered data into four equal parts.<\/p>\n<\/div>\n<h1>8.1 Exercise Set<\/h1>\n<ol>\n<li><span style=\"text-align: initial; font-size: 14pt;\">Your instructor announces to the class that anyone with a midterm exam score of 63% scored in the 80th percentile.\u00a0 If you received a score of 63% how did you do in relation to your classmates?<\/span><\/li>\n<li>A test consists of 40 marks. Fifty students wrote the test and their scores are in the table below. Determine the percentiles that are associated with scores of:\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li>15<\/li>\n<li>25<\/li>\n<li>37<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3330 size-large\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table-1024x266.png\" alt=\"\" width=\"1024\" height=\"266\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table-1024x266.png 1024w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table-300x78.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table-768x200.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table-65x17.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table-225x58.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table-350x91.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set2-Table.png 1170w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<div class=\"part-title\">\n<ol start=\"3\">\n<li>An employee at a large manufacturing company learns that their salary is in the 45th percentile. If the median salary at the company is $56,000\/year can we conclude that this employee\u2019s annual salary is more than $56,000?<\/li>\n<li><span style=\"text-align: initial; font-size: 14pt;\">Your instructor announces to the class that the third quartile for the midterm exam was a score of 88%.\u00a0 \u00a0If you received a score of 88% how did you do in relation to your classmates?<\/span><\/li>\n<li>A cell phone provider is trying to improve its service by reducing the amount of time that its help desk spends with each customer. It kept track of the average length of time (to the nearest minute) each of its 33\u00a0 employees spent with customers.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3331 size-large\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table-1024x178.png\" alt=\"\" width=\"1024\" height=\"178\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table-1024x178.png 1024w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table-300x52.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table-768x134.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table-65x11.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table-225x39.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table-350x61.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set4-Table.png 1162w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/>\n<ol type=\"a\">\n<li>Determine the mean, median and mode for average wait times.<\/li>\n<li>Determine the percentiles that correspond to help times of 14 minutes, 23 minutes and 47 minutes.<\/li>\n<li>Determine the 1st, 2nd and 3rd quartiles for average help times.<\/li>\n<\/ol>\n<\/li>\n<li value=\"6\">A test consists of 40 marks. Fifty students wrote the test and their scores are as recorded. Determine the 1st, 2nd and 3rd quartiles.<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3329 size-large\" src=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table-1024x277.png\" alt=\"\" width=\"1024\" height=\"277\" srcset=\"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table-1024x277.png 1024w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table-300x81.png 300w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table-768x208.png 768w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table-65x18.png 65w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table-225x61.png 225w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table-350x95.png 350w, https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-content\/uploads\/sites\/794\/2020\/07\/8.1Ex-Set5-Table.png 1159w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<ol start=\"7\">\n<li>Which of the following must be an actual data value:\u00a0 mean, median, mode, first quartile, third quartile?<\/li>\n<li>At a restaurant one evening the customers were asked to rate the service they received. Scores could range from 1 to 10. The following thirty responses (scores) were provided:1\u00a0 1\u00a0 1\u00a0 2\u00a0 2\u00a0 2\u00a0 3\u00a0 3\u00a0 3\u00a0 4\u00a0 4\u00a0 4\u00a0 5\u00a0 5\u00a0 5\u00a0 6\u00a0 6\u00a0 6\u00a0 7\u00a0 7\u00a0 7\u00a0 8\u00a0 8\u00a0 8\u00a0 9\u00a0 9\u00a0 9\u00a0 10\u00a0 10\u00a0 10\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li>Determine the percentiles that correspond to scores of\u00a0 2 and 5. Explain what this means.<\/li>\n<li>Determine the first, second and third quartiles.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<li value=\"9\">At a restaurant one evening the customers were asked to rate the service they received. Scores could range from 1 to 10. The following twenty-nine responses (scores) were provided:<br \/>\n1\u00a0 \u00a01\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a02\u00a0 \u00a03\u00a0 \u00a03\u00a0 \u00a03\u00a0 \u00a04\u00a0 \u00a0 4\u00a0 \u00a0 4\u00a0 \u00a05\u00a0 \u00a05\u00a0 \u00a05\u00a0 \u00a06\u00a0 \u00a06\u00a0 6\u00a0 7\u00a0 7\u00a0 8\u00a0 8\u00a0 8\u00a0 8\u00a0 \u00a08\u00a0 9\u00a0 9\u00a0 10\u00a0 10\u00a0 10<\/p>\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li>Determine the mean, median and mode.<\/li>\n<li>Determine the first, second and third quartiles.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-idm10803744\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\">\n<h1>Answers<\/h1>\n<ol>\n<li>If your score is 63% and this is in the 80th percentile this means that 80% of your classmates received scores lower than or equal to 63%\n<ol type=\"a\">\n<li>A score of 15 is in the10th percentile.<\/li>\n<li>A score of 25 is in the 44th percentile.<\/li>\n<li>A score of 37 is in the 84th percentile.<\/li>\n<\/ol>\n<\/li>\n<li>We can conclude that the employee\u2019s salary is not more than $56,000\/year because the median salary is also the 50th percentile. If the employee\u2019s salary is in the 45th percentile they cannot be earning more than the median.<\/li>\n<li>You scored better than three quarters of your class mates.\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li><span style=\"font-size: 14pt;\">The mean is 765\/33 = 23.18; the median is 22; and there are two modes 5\u00a0 and 31.<\/span><\/li>\n<li>A help time of 14 minutes is in the 33rd percentile;\u00a0 23 minutes is in the 52nd percentile;\u00a0 47 minutes is in the 91st perecntile<\/li>\n<li>Q<sub>1<\/sub> = 9.5 min.; Q<sub>2<\/sub> = 22 min. ;\u00a0 Q<sub>3<\/sub> =\u00a0 31 min.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<li>Q<sub>1<\/sub> = 19;\u00a0 Q<sub>2<\/sub> = 26.5;\u00a0 Q<sub>3<\/sub> = 33<\/li>\n<li>The mode must be an actual data value\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li>A score of 2 is the 10th percentile. This means that 10% of the scores were less than a score of 2. A score of 5 is the 40th percentile. This means that 40% of the scores were less than a score of 5.<\/li>\n<li>The first quartile is 3, the second quartile is 5.5, the third quartile is 8<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li>mean\u00a0 5.6;\u00a0 \u00a0 median 6;\u00a0 mode 8<\/li>\n<li>Q<sub>1<\/sub> is 3;\u00a0 \u00a0Q<sub>2<\/sub>\u00a0 is 6;\u00a0 Q<sub>3<\/sub>\u00a0 \u00a0is 8<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h1>Attribution<\/h1>\n<p>Some content in this chapter has been adapted from \u201cMeasures of the Location of the Data\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/introductory-statistics\"><i>Introductory Statistics <\/i><\/a>(OpenStax) by Daniel Birmajer, Bryan Blount, Sheri Boyd, Matthew Einsohn, James Helmreich, Lynette Kenyon, Sheldon Lee,\u00a0 and Jeff Taub, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Kim Moshenko. See the Copyright page for more information.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_2665_4755\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_2665_4755\"><div tabindex=\"-1\"><p>The median is the data item in the middle of each set of ranked, or ordered, data. The median separates the upper half and the lower half of a data set.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":781,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2665","chapter","type-chapter","status-publish","hentry"],"part":2662,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/pressbooks\/v2\/chapters\/2665","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/wp\/v2\/users\/781"}],"version-history":[{"count":26,"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/pressbooks\/v2\/chapters\/2665\/revisions"}],"predecessor-version":[{"id":7134,"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/pressbooks\/v2\/chapters\/2665\/revisions\/7134"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/pressbooks\/v2\/parts\/2662"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/pressbooks\/v2\/chapters\/2665\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/wp\/v2\/media?parent=2665"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/pressbooks\/v2\/chapter-type?post=2665"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/wp\/v2\/contributor?post=2665"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/mycopy\/wp-json\/wp\/v2\/license?post=2665"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}