{"id":35,"date":"2021-06-22T15:30:10","date_gmt":"2021-06-22T15:30:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/measures-of-the-location-of-the-data\/"},"modified":"2023-12-05T08:54:01","modified_gmt":"2023-12-05T08:54:01","slug":"measures-of-the-location-of-the-data","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/measures-of-the-location-of-the-data\/","title":{"raw":"Measures of the Location","rendered":"Measures of the Location"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list123523\">\r\n \t<li>Calculate the median and quartiles for a set of data<\/li>\r\n \t<li>Find the interquartile range and use it to identify outliers<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe common measures of location are\u00a0<strong>quartiles<\/strong> and <strong>percentiles<\/strong>.\r\n\r\nQuartiles are special percentiles. The first quartile,\u00a0[latex]Q_1[\/latex], is the same as the [latex]25[\/latex]th percentile, and the third quartile, [latex]Q_3[\/latex], is the same as the [latex]75[\/latex]th percentile. The median, [latex]M[\/latex], is called both the second quartile and the [latex]50[\/latex]th percentile.\r\n\r\nThe following video gives an introduction to median, quartiles, and interquartile range, the topics you will learn in this section.\r\n\r\n<iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7114970&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=wNamjO-JzUg&amp;video_target=tpm-plugin-rm8vgszu-wNamjO-JzUg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n\r\nTo calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. To score in the [latex]90[\/latex]th percentile of an exam does not mean, necessarily, that you received [latex]90[\/latex]% on a test. It means that [latex]90[\/latex]% of test scores are the same or less than your score and [latex]10[\/latex]% of the test scores are the same or greater than your test score.\r\n\r\nPercentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the [latex]75[\/latex]th percentile. That translates into a score of at least [latex]1220[\/latex].\r\n\r\nPercentiles are mostly used with very large populations. Therefore, if you were to say that [latex]90[\/latex]% of the test scores are less (and not the same or less) than your score, it would be acceptable because removing one particular data value is not significant.\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Ordering Decimals<\/h3>\r\nIt is helpful to use a number line to order decimals. The best way to think about decimals is a form of a fraction. You need to break the number line up with intervals equal to the denominator of the fraction.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nLocate [latex]0.4[\/latex] on a number line.\r\n\r\nSolution\r\nThe decimal [latex]0.4[\/latex] is equivalent to [latex]{\\Large\\frac{4}{10}}[\/latex], so [latex]0.4[\/latex] is located between [latex]0[\/latex] and [latex]1[\/latex]. On a number line, divide the interval between [latex]0[\/latex] and [latex]1[\/latex] into [latex]10[\/latex] equal parts and place marks to separate the parts.\r\n\r\nLabel the marks [latex]0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0[\/latex]. We write [latex]0[\/latex] as [latex]0.0[\/latex] and [latex]1[\/latex] as [latex]1.0[\/latex], so that the numbers are consistently in tenths. Finally, mark [latex]0.4[\/latex] on the number line.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221458\/CNX_BMath_Figure_05_01_010_img.png\" alt=\"A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a red dot at 0.4.\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\nThe\u00a0<strong>median<\/strong> is a number that measures the \"center\" of the data. You can think of the median as the \"middle value,\" but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data:\r\n<p style=\"margin-left: 20px;\">[latex]1[\/latex]; [latex]11.5[\/latex]; [latex]6[\/latex]; [latex]7.2[\/latex]; [latex]4[\/latex]; [latex]8[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]; [latex]6.8[\/latex]; [latex]8.3[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]10[\/latex]; [latex]1[\/latex]<\/p>\r\nOrdered from smallest to largest:\r\n<p style=\"margin-left: 20px;\">[latex]1[\/latex]; [latex]1[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]4[\/latex]; [latex]6[\/latex]; [latex]6.8[\/latex]; [latex]7.2[\/latex]; [latex]8[\/latex]; [latex]8.3[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]11.5[\/latex]<\/p>\r\nSince there are [latex]14[\/latex] observations, the median is between the seventh value, [latex]6.8[\/latex], and the eighth value, [latex]7.2[\/latex]. To find the median, add the two values together and divide by two.\r\n\r\n<center>[latex]\\displaystyle\\frac{{{6.8}+{7.2}}}{{2}}={7}[\/latex]<\/center>The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Multiply Decimal Numbers<\/h3>\r\n<ol id=\"eip-id1168469803517\" class=\"stepwise\">\r\n \t<li>Determine the sign of the product<\/li>\r\n \t<li>Write the numbers in vertical format, lining up the numbers on the right<\/li>\r\n \t<li>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points<\/li>\r\n \t<li>Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.<\/li>\r\n \t<li>Write the product with the appropriate sign<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n\r\n<strong>Quartiles<\/strong> are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median or second quartile. The first quartile,\u00a0[latex]Q_1[\/latex], is the middle value of the lower half of the data, and the third quartile, [latex]Q_3[\/latex], is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:\r\n<p style=\"margin-left: 20px;\">[latex]1[\/latex]; [latex]1[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]4[\/latex]; [latex]6[\/latex]; [latex]6.8[\/latex]; [latex]7.2[\/latex]; [latex]8[\/latex]; [latex]8.3[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]11.5[\/latex]<\/p>\r\nThe median or\u00a0<strong>second quartile<\/strong> is seven. The lower half of the data are [latex]1[\/latex], [latex]1[\/latex], [latex]2[\/latex], [latex]2[\/latex], [latex]4[\/latex], [latex]6[\/latex], [latex]6.8[\/latex]. The middle value of the lower half is two.\r\n<p style=\"margin-left: 20px;\">[latex]1[\/latex]; [latex]1[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]4[\/latex]; [latex]6[\/latex]; [latex]6.8[\/latex]<\/p>\r\nThe number two, which is part of the data, is the\u00a0<strong>first quartile<\/strong>. One-fourth of the entire sets of values are the same as or less than two and three-fourths of the values are more than two.\r\n\r\nThe upper half of the data is [latex]7.2[\/latex], [latex]8[\/latex], [latex]8.3[\/latex], [latex]9[\/latex], [latex]10[\/latex], [latex]10[\/latex], [latex]11.5[\/latex]. The middle value of the upper half is nine.\r\n\r\nThe\u00a0<strong>third quartile<\/strong>,\u00a0[latex]Q_3[\/latex], is nine. Three-fourths ([latex]75[\/latex]%) of the ordered data set are less than nine. One-fourth ([latex]25[\/latex]%) of the ordered data set are greater than nine. The third quartile is part of the data set in this example.\r\n\r\nThe\u00a0<strong>interquartile range (IQR)<\/strong>\u00a0is a number that indicates the spread of the middle half or the middle [latex]50[\/latex]% of the data. It is the difference between the third quartile ([latex]Q_3[\/latex]) and the first quartile ([latex]Q_1[\/latex]).\r\n\r\n[latex]IQR[\/latex] = [latex]Q_3[\/latex] \u2013 [latex]Q_1[\/latex]\r\n\r\nThe IQR can help to determine potential <strong>outliers<\/strong>. <strong>A value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile<\/strong>. Potential outliers always require further investigation.\r\n<div class=\"textbox shaded\">\r\n<h3>NOTE<\/h3>\r\nA potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality or they may be a key to understanding the data.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFor the following [latex]13[\/latex] real estate prices, calculate the [latex]IQR[\/latex] and determine if any prices are potential outliers. Prices are in dollars.\r\n[latex]389{,}950[\/latex]; [latex]230{,}500[\/latex]; [latex]158{,}000[\/latex]; [latex]479{,}000[\/latex]; [latex]639{,}000[\/latex]; [latex]114{,}950[\/latex]; [latex]5{,}500{,}000[\/latex]; [latex]387{,}000[\/latex]; [latex]659{,}000[\/latex]; [latex]529{,}000[\/latex]; [latex]575{,}000[\/latex]; [latex]488{,}800[\/latex]; [latex]1{,}095{,}000[\/latex]\r\n[reveal-answer q=\"283390\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283390\"]\r\nOrder the data from smallest to largest.\r\n[latex]114{,}950[\/latex]; [latex]158{,}000[\/latex]; [latex]230{,}500[\/latex]; [latex]387{,}000[\/latex]; [latex]389{,}950[\/latex]; [latex]479{,}000[\/latex]; [latex]488{,}800[\/latex]; [latex]529{,}000[\/latex]; [latex]575{,}000[\/latex]; [latex]639{,}000[\/latex]; [latex]659{,}000[\/latex]; [latex]1{,}095{,}000[\/latex]; [latex]5{,}500{,}000[\/latex]\r\n\r\n[latex]M = 488,800[\/latex]\r\n\r\n[latex]Q_1 = \\frac{230{,}500 + 387{,}000}{2} = 308{,}750[\/latex]\r\n\r\n[latex]Q_3 = \\frac{639{,}000 + 659{,}000}{2} = 649{,}000[\/latex]\r\n\r\n[latex]IQR = 649{,}000 \u2013 308{,}750 = 340{,}250[\/latex]\r\n\r\n[latex](1.5)(IQR) = (1.5)(340{,}250) = 510{,}375[\/latex]\r\n\r\n[latex]Q_1 \u2013 (1.5)(IQR) = 308{,}750 \u2013 510{,}375 = \u2013201{,}625[\/latex]\r\n\r\n[latex]Q_3 + (1.5)(IQR) = 649{,}000 + 510{,}375 = 1{,}159{,}375[\/latex]\r\n\r\nNo house price is less than [latex]\u2013201{,}625[\/latex]. However, [latex]5{,}500{,}000[\/latex] is more than [latex]1{,}159{,}375[\/latex]. Therefore, [latex]5{,}500{,}000[\/latex] is a potential <strong>outlier<\/strong>.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFor the following [latex]11[\/latex] salaries, calculate the [latex]IQR[\/latex] and determine if any salaries are outliers. The salaries are in dollars.\r\n\r\n[latex]$33{,}000[\/latex], [latex]$64{,}500[\/latex], [latex]$28{,}000[\/latex], [latex]$54{,}000[\/latex], [latex]$72{,}000[\/latex], [latex]$68{,}500[\/latex], [latex]$69{,}000[\/latex], [latex]$42{,}000[\/latex], [latex]$54{,}000[\/latex] [latex]$120{,}000[\/latex], [latex]$40{,}500[\/latex]\r\n[reveal-answer q=\"283391\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283391\"]\r\nOrder the data from smallest to largest.\r\n\r\n[latex]$28{,}000[\/latex], [latex]$33{,}000[\/latex], [latex]$40{,}500[\/latex], [latex]$42{,}000[\/latex], [latex]$54{,}000[\/latex], [latex]$54{,}000[\/latex], [latex]$64{,}500[\/latex], [latex]$68{,}500[\/latex], [latex]$69{,}000[\/latex], [latex]$72{,}000[\/latex], [latex]$120{,}000[\/latex]\r\n\r\nMedian = [latex]$54,000[\/latex]\r\n\r\n[latex]Q_1[\/latex] = [latex]$40{,}500[\/latex]\r\n\r\n[latex]Q_3[\/latex] = [latex]$69{,}000[\/latex]\r\n\r\n[latex]IQR[\/latex] = [latex]$69{,}000[\/latex] \u2013 [latex]$40{,}500[\/latex] = [latex]$28{,}500[\/latex]\r\n\r\n(1.5)([latex]IQR[\/latex]) = (1.5)($28{,}500) = [latex]$42{,}750[\/latex]\r\n\r\n[latex]Q_1[\/latex] \u2013 (1.5)([latex]IQR[\/latex]) = [latex]$40{,}500[\/latex] \u2013 [latex]$42{,}750[\/latex] = [latex]\u2013$2{,}250[\/latex]\r\n\r\n[latex]Q_3[\/latex] + (1.5)([latex]IQR[\/latex]) = [latex]$69{,}000[\/latex] + [latex]$42{,}750[\/latex] = [latex]$111{,}750[\/latex]\r\n\r\nNo salary is less than [latex]$2{,}250[\/latex]. However, [latex]$120{,}000[\/latex] is more than [latex]$111{,}750[\/latex], so [latex]$120{,}000[\/latex] is a potential outlier.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"element-880\" class=\" \">For the two data sets in the\u00a0test scores example, find the following:<\/p>\r\n\r\n<ol id=\"element-971\" type=\"a\" data-mark-suffix=\".\">\r\n \t<li data-mark-suffix=\".\">The interquartile range. Compare the two interquartile ranges.<\/li>\r\n \t<li data-mark-suffix=\".\">Any outliers in either set.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"563920\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"563920\"]\r\n\r\nThe five number summary for the day and night classes is\r\n<div id=\"fs-idp36487328\" class=\"os-table \">\r\n<table summary=\"Table 2.21 \" data-id=\"fs-idp36487328\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\"><\/th>\r\n<th scope=\"col\">Minimum<\/th>\r\n<th scope=\"col\"><em data-effect=\"italics\">Q<\/em><sub>1<\/sub><\/th>\r\n<th scope=\"col\">Median<\/th>\r\n<th scope=\"col\"><em data-effect=\"italics\">Q<\/em><sub>3<\/sub><\/th>\r\n<th scope=\"col\">Maximum<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong data-effect=\"bold\">Day<\/strong><\/td>\r\n<td>32<\/td>\r\n<td>56<\/td>\r\n<td>74.5<\/td>\r\n<td>82.5<\/td>\r\n<td>99<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">Night<\/strong><\/td>\r\n<td>25.5<\/td>\r\n<td>78<\/td>\r\n<td>81<\/td>\r\n<td>89<\/td>\r\n<td>98<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol id=\"fs-idm23962720\" type=\"a\">\r\n \t<li>The IQR for the day group is\u00a0<em data-effect=\"italics\">Q<\/em><sub>3<\/sub>\u00a0\u2013\u00a0<em data-effect=\"italics\">Q<\/em><sub>1<\/sub>\u00a0= 82.5 \u2013 56 = 26.5\r\n<p id=\"fs-idm7044352\" class=\" \">The IQR for the night group is\u00a0<em data-effect=\"italics\">Q<\/em><sub>3<\/sub>\u00a0\u2013\u00a0<em data-effect=\"italics\">Q<\/em><sub>1<\/sub>\u00a0= 89 \u2013 78 = 11<\/p>\r\n<p id=\"fs-idp42547504\" class=\" \">The interquartile range (the spread or variability) for the day class is larger than the night class\u00a0<em data-effect=\"italics\">IQR<\/em>. This suggests more variation will be found in the day class\u2019s class test scores.<\/p>\r\n<\/li>\r\n \t<li>Day class outliers are found using the IQR times 1.5 rule. So,\r\n<ul id=\"fs-idm52257968\" data-labeled-item=\"true\">\r\n \t<li><em data-effect=\"italics\">Q<\/em><sub>1<\/sub>\u00a0-\u00a0<em data-effect=\"italics\">IQR<\/em>(1.5) = 56 \u2013 26.5(1.5) = 16.25<\/li>\r\n \t<li><em data-effect=\"italics\">Q<\/em><sub>3<\/sub>\u00a0+\u00a0<em data-effect=\"italics\">IQR<\/em>(1.5) = 82.5 + 26.5(1.5) = 122.25<\/li>\r\n<\/ul>\r\n<p id=\"fs-idp38341744\" class=\" \">Since the minimum and maximum values for the day class are greater than 16.25 and less than 122.25, there are no outliers.<\/p>\r\n<p id=\"fs-idm23940160\" class=\" \">Night class outliers are calculated as:<\/p>\r\n\r\n<ul id=\"fs-idp29569184\" data-labeled-item=\"true\">\r\n \t<li><em data-effect=\"italics\">Q<\/em><sub>1<\/sub>\u00a0\u2013\u00a0<em data-effect=\"italics\">IQR<\/em>\u00a0(1.5) = 78 \u2013 11(1.5) = 61.5<\/li>\r\n \t<li><em data-effect=\"italics\">Q<\/em><sub>3<\/sub>\u00a0+ IQR(1.5) = 89 + 11(1.5) = 105.5<\/li>\r\n<\/ul>\r\n<p id=\"fs-idp5005056\" class=\" \">For this class, any test score less than 61.5 is an outlier. Therefore, the scores of 45 and 25.5 are outliers. Since no test score is greater than 105.5, there is no upper end outlier.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the interquartile range for the following two data sets and compare them.\r\n\r\nTest Scores for Class\u00a0<em>A<\/em>\r\n<p style=\"margin-left: 20px;\">[latex]69[\/latex]; [latex]96[\/latex]; [latex]81[\/latex]; [latex]79[\/latex]; [latex]65[\/latex]; [latex]76[\/latex]; [latex]83[\/latex]; [latex]99[\/latex]; [latex]89[\/latex]; [latex]67[\/latex]; [latex]90[\/latex]; [latex]77[\/latex]; [latex]85[\/latex]; [latex]98[\/latex]; [latex]66[\/latex]; [latex]91[\/latex]; [latex]77[\/latex]; [latex]69[\/latex]; [latex]80[\/latex]; [latex]94[\/latex]<\/p>\r\nTest Scores for Class\u00a0<em>B<\/em>\r\n<p style=\"margin-left: 20px;\">[latex]90[\/latex]; [latex]72[\/latex]; [latex]80[\/latex]; [latex]92[\/latex]; [latex]90[\/latex]; [latex]97[\/latex]; [latex]92[\/latex]; [latex]75[\/latex]; [latex]79[\/latex]; [latex]68[\/latex]; [latex]70[\/latex]; [latex]80[\/latex]; [latex]99[\/latex]; [latex]95[\/latex]; [latex]78[\/latex]; [latex]73[\/latex]; [latex]71[\/latex]; [latex]68[\/latex]; [latex]95[\/latex]; [latex]100[\/latex]<\/p>\r\n[reveal-answer q=\"283392\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283392\"]\r\n<h4>Class\u00a0A<\/h4>\r\nOrder the data from smallest to largest.\r\n\r\n[latex]65[\/latex]; [latex]66[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]; [latex]77[\/latex]; [latex]79[\/latex]; [latex]80[\/latex]; [latex]81[\/latex]; [latex]83[\/latex]; [latex]85[\/latex]; [latex]89[\/latex]; [latex]90[\/latex]; [latex]91[\/latex]; [latex]94[\/latex]; [latex]96[\/latex]; [latex]98[\/latex]; [latex]99[\/latex]\r\n\r\n[latex]\\displaystyle {Median}=\\frac{{{80}+{81}}}{{2}}={80.5}[\/latex]\r\n\r\n[latex]{Q}_{{1}}=\\frac{{{69}+{76}}}{{2}}={72.5}[\/latex]\r\n\r\n[latex]{Q}_{{3}}=\\frac{{{90}+{91}}}{{2}}={90.5}[\/latex]\r\n\r\n[latex]IQR[\/latex] = [latex]90.5[\/latex] \u2013 [latex]72.5[\/latex] = [latex]18[\/latex]\r\n<h4>Class\u00a0B<\/h4>\r\nOrder the data from smallest to largest.\r\n\r\n[latex]68[\/latex]; [latex]68[\/latex]; [latex]70[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]75[\/latex]; [latex]78[\/latex]; [latex]79[\/latex]; [latex]80[\/latex]; [latex]80[\/latex]; [latex]90[\/latex]; [latex]90[\/latex]; [latex]92[\/latex]; [latex]92[\/latex]; [latex]95[\/latex]; [latex]95[\/latex]; [latex]97[\/latex]; [latex]99[\/latex]; [latex]100[\/latex]\r\n\r\n[latex]\\displaystyle{Median}=\\frac{{{80}+{80}}}{{2}}={80}[\/latex]\r\n\r\n[latex]{Q}_{{1}}=\\frac{{{72}+{73}}}{{2}}={72.5}[\/latex]\r\n\r\n[latex]{Q}_{{3}}=\\frac{{{92}+{95}}}{{2}}={93.5}[\/latex]\r\n\r\n[latex]IQR[\/latex] = [latex]93.5[\/latex] \u2013 [latex]72.5[\/latex] = [latex]21[\/latex]\r\n\r\nThe data for Class\u00a0<em>B<\/em> has a larger [latex]IQR[\/latex], so the scores between [latex]Q_3[\/latex] and [latex]Q_1[\/latex] (middle [latex]50[\/latex]%) for the data for Class <em>B<\/em> are more spread out and not clustered about the median.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Amount of Sleep per School Night (Hours)<\/th>\r\n<th>Frequency<\/th>\r\n<th>Relative Frequency<\/th>\r\n<th>Cumulative Relative Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]0.04[\/latex]<\/td>\r\n<td>[latex]0.04[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]0.10[\/latex]<\/td>\r\n<td>[latex]0.14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]0.14[\/latex]<\/td>\r\n<td>[latex]0.28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]0.24[\/latex]<\/td>\r\n<td>[latex]0.52[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<td>[latex]0.28[\/latex]<\/td>\r\n<td>[latex]0.80[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]0.14[\/latex]<\/td>\r\n<td>[latex]0.94[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]0.06[\/latex]<\/td>\r\n<td>[latex]1.00[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Find the [latex]28[\/latex]th percentile.<\/strong> Notice the [latex]0.28[\/latex] in the \"cumulative relative frequency\" column. Twenty-eight percent of [latex]50[\/latex] data values is [latex]14[\/latex] values. There are [latex]14[\/latex] values less than the [latex]28[\/latex]th percentile. They include the two [latex]4[\/latex]s, the five [latex]5[\/latex]s, and the seven [latex]6[\/latex]s. The [latex]28[\/latex]th percentile is between the last six and the first seven. <strong>The [latex]28[\/latex]th percentile is [latex]6.5[\/latex].<\/strong>\r\n\r\n<strong>Find the median.<\/strong> Look again at the \"cumulative relative frequency\" column and find [latex]0.52[\/latex]. The median is the [latex]50[\/latex]th percentile or the second quartile. [latex]50[\/latex]% of [latex]50[\/latex] is [latex]25[\/latex]. There are [latex]25[\/latex] values less than the median. They include the two [latex]4[\/latex]s, the five [latex]5[\/latex]s, the seven [latex]6[\/latex]s, and eleven of the [latex]7[\/latex]s. The median or [latex]50[\/latex]th percentile is between the [latex]25[\/latex]th, or seven, and [latex]26[\/latex]th, or seven, values. <strong>The median is seven.<\/strong>\r\n\r\n<strong>Find the third quartile. <\/strong>The third quartile is the same as the [latex]75[\/latex]th percentile. You can \"eyeball\" this answer. If you look at the \"cumulative relative frequency\" column, you find [latex]0.52[\/latex] and [latex]0.80[\/latex]. When you have all the fours, fives, sixes and sevens, you have [latex]52[\/latex]% of the data. When you include all the [latex]8[\/latex]s, you have [latex]80[\/latex]% of the data. <strong>The [latex]75[\/latex]th percentile, then, must be an eight.<\/strong> Another way to look at the problem is to find [latex]75[\/latex]% of [latex]50[\/latex], which is [latex]37.5[\/latex],and round up to [latex]38[\/latex]. The third quartile, [latex]Q_3[\/latex], is the 38th value, which is an 8. You can check this answer by counting the values. (There are [latex]37[\/latex] values below the third quartile and 12 values above.)\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nForty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the [latex]65[\/latex]th percentile.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Amount of time spent on route (hours)<\/th>\r\n<th>Frequency<\/th>\r\n<th>Relative Frequency<\/th>\r\n<th>Cumulative Relative Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]0.30[\/latex]<\/td>\r\n<td>[latex]0.30[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<td>[latex]0.35[\/latex]<\/td>\r\n<td>[latex]0.65[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]0.25[\/latex]<\/td>\r\n<td>[latex]0.90[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]0.10[\/latex]<\/td>\r\n<td>[latex]1.00[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"283393\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283393\"]\r\nThe [latex]65[\/latex]th percentile is between the last three and the first four.\r\n\r\nThe [latex]65[\/latex]th percentile is [latex]3.5[\/latex].\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Amount of Sleep per School Night (Hours)<\/th>\r\n<th>Frequency<\/th>\r\n<th>Relative Frequency<\/th>\r\n<th>Cumulative Relative Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]0.04[\/latex]<\/td>\r\n<td>[latex]0.04[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]0.10[\/latex]<\/td>\r\n<td>[latex]0.14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]0.14[\/latex]<\/td>\r\n<td>[latex]0.28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]0.24[\/latex]<\/td>\r\n<td>[latex]0.52[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<td>[latex]0.28[\/latex]<\/td>\r\n<td>[latex]0.80[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]0.14[\/latex]<\/td>\r\n<td>[latex]0.94[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]0.06[\/latex]<\/td>\r\n<td>[latex]1.00[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\r\n \t<li>Find the [latex]80[\/latex]th percentile.<\/li>\r\n \t<li>Find the [latex]90[\/latex]th percentile.<\/li>\r\n \t<li>Find the first quartile. What is another name for the first quartile?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"283394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283394\"]\r\n\r\nUsing the data from the frequency table, we have:\r\n<ol>\r\n \t<li>The [latex]80[\/latex]th percentile is between the last eight and the first nine in the table (between the [latex]40[\/latex]th and [latex]41[\/latex]st values). Therefore, we need to take the mean of the [latex]40[\/latex]th an [latex]41[\/latex]st values. The 80th percentile [latex]\\displaystyle\\frac{{{8}+{9}}}{{2}}={8.5}[\/latex]<\/li>\r\n \t<li>The [latex]90[\/latex]th percentile will be the [latex]45[\/latex]th data value (location is [latex]0.90(50) = 45[\/latex]) and the [latex]45[\/latex]th data value is nine.<\/li>\r\n \t<li>[latex]Q_1[\/latex] is also the [latex]25[\/latex]th percentile. The [latex]25[\/latex]th percentile location calculation: [latex]P_{25}[\/latex] = [latex]0.25(50) = 12.5 \u2248 13[\/latex] the [latex]13[\/latex]th data value. Thus, the [latex]25[\/latex]th percentile is six.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Amount of time spent on route (hours)<\/th>\r\n<th>Frequency<\/th>\r\n<th>Relative Frequency<\/th>\r\n<th>Cumulative Relative Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]0.30[\/latex]<\/td>\r\n<td>[latex]0.30[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<td>[latex]0.35[\/latex]<\/td>\r\n<td>[latex]0.65[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]0.25[\/latex]<\/td>\r\n<td>[latex]0.90[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]0.10[\/latex]<\/td>\r\n<td>[latex]1.00[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFind the third quartile. What is another name for the third quartile?\r\n[reveal-answer q=\"283395\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283395\"]\r\nThe third quartile is the [latex]75[\/latex]th percentile, which is four. The [latex]65[\/latex]th percentile is between three and four, and the [latex]90[\/latex]th percentile is between four and [latex]5.75[\/latex]. The third quartile is between [latex]65[\/latex] and [latex]90[\/latex], so it must be four.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\"><header>\r\n<h3 class=\"title\" data-type=\"title\">Collaborative Exercise<\/h3>\r\nYour instructor or a member of the class will ask everyone in class how many sweaters they own. Answer the following questions:\r\n<ol>\r\n \t<li>How many students were surveyed?<\/li>\r\n \t<li>What kind of sampling did you do?<\/li>\r\n \t<li>Construct two different histograms. For each, starting value = _____ ending value = ____.<\/li>\r\n \t<li>Find the median, first quartile, and third quartile.<\/li>\r\n \t<li>Construct a table of the data to find the following:<\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>the 10th percentile<\/li>\r\n \t<li>the 70th percentile<\/li>\r\n \t<li>the percent of students who own less than four sweaters<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/header><\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list123523\">\n<li>Calculate the median and quartiles for a set of data<\/li>\n<li>Find the interquartile range and use it to identify outliers<\/li>\n<\/ul>\n<\/div>\n<p>The common measures of location are\u00a0<strong>quartiles<\/strong> and <strong>percentiles<\/strong>.<\/p>\n<p>Quartiles are special percentiles. The first quartile,\u00a0[latex]Q_1[\/latex], is the same as the [latex]25[\/latex]th percentile, and the third quartile, [latex]Q_3[\/latex], is the same as the [latex]75[\/latex]th percentile. The median, [latex]M[\/latex], is called both the second quartile and the [latex]50[\/latex]th percentile.<\/p>\n<p>The following video gives an introduction to median, quartiles, and interquartile range, the topics you will learn in this section.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7114970&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=wNamjO-JzUg&amp;video_target=tpm-plugin-rm8vgszu-wNamjO-JzUg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. To score in the [latex]90[\/latex]th percentile of an exam does not mean, necessarily, that you received [latex]90[\/latex]% on a test. It means that [latex]90[\/latex]% of test scores are the same or less than your score and [latex]10[\/latex]% of the test scores are the same or greater than your test score.<\/p>\n<p>Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the [latex]75[\/latex]th percentile. That translates into a score of at least [latex]1220[\/latex].<\/p>\n<p>Percentiles are mostly used with very large populations. Therefore, if you were to say that [latex]90[\/latex]% of the test scores are less (and not the same or less) than your score, it would be acceptable because removing one particular data value is not significant.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Ordering Decimals<\/h3>\n<p>It is helpful to use a number line to order decimals. The best way to think about decimals is a form of a fraction. You need to break the number line up with intervals equal to the denominator of the fraction.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Locate [latex]0.4[\/latex] on a number line.<\/p>\n<p>Solution<br \/>\nThe decimal [latex]0.4[\/latex] is equivalent to [latex]{\\Large\\frac{4}{10}}[\/latex], so [latex]0.4[\/latex] is located between [latex]0[\/latex] and [latex]1[\/latex]. On a number line, divide the interval between [latex]0[\/latex] and [latex]1[\/latex] into [latex]10[\/latex] equal parts and place marks to separate the parts.<\/p>\n<p>Label the marks [latex]0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0[\/latex]. We write [latex]0[\/latex] as [latex]0.0[\/latex] and [latex]1[\/latex] as [latex]1.0[\/latex], so that the numbers are consistently in tenths. Finally, mark [latex]0.4[\/latex] on the number line.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221458\/CNX_BMath_Figure_05_01_010_img.png\" alt=\"A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a red dot at 0.4.\" \/><\/p>\n<\/div>\n<\/div>\n<p>The\u00a0<strong>median<\/strong> is a number that measures the &#8220;center&#8221; of the data. You can think of the median as the &#8220;middle value,&#8221; but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data:<\/p>\n<p style=\"margin-left: 20px;\">[latex]1[\/latex]; [latex]11.5[\/latex]; [latex]6[\/latex]; [latex]7.2[\/latex]; [latex]4[\/latex]; [latex]8[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]; [latex]6.8[\/latex]; [latex]8.3[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]10[\/latex]; [latex]1[\/latex]<\/p>\n<p>Ordered from smallest to largest:<\/p>\n<p style=\"margin-left: 20px;\">[latex]1[\/latex]; [latex]1[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]4[\/latex]; [latex]6[\/latex]; [latex]6.8[\/latex]; [latex]7.2[\/latex]; [latex]8[\/latex]; [latex]8.3[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]11.5[\/latex]<\/p>\n<p>Since there are [latex]14[\/latex] observations, the median is between the seventh value, [latex]6.8[\/latex], and the eighth value, [latex]7.2[\/latex]. To find the median, add the two values together and divide by two.<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\frac{{{6.8}+{7.2}}}{{2}}={7}[\/latex]<\/div>\n<p>The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Multiply Decimal Numbers<\/h3>\n<ol id=\"eip-id1168469803517\" class=\"stepwise\">\n<li>Determine the sign of the product<\/li>\n<li>Write the numbers in vertical format, lining up the numbers on the right<\/li>\n<li>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points<\/li>\n<li>Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.<\/li>\n<li>Write the product with the appropriate sign<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<p><strong>Quartiles<\/strong> are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median or second quartile. The first quartile,\u00a0[latex]Q_1[\/latex], is the middle value of the lower half of the data, and the third quartile, [latex]Q_3[\/latex], is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:<\/p>\n<p style=\"margin-left: 20px;\">[latex]1[\/latex]; [latex]1[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]4[\/latex]; [latex]6[\/latex]; [latex]6.8[\/latex]; [latex]7.2[\/latex]; [latex]8[\/latex]; [latex]8.3[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]11.5[\/latex]<\/p>\n<p>The median or\u00a0<strong>second quartile<\/strong> is seven. The lower half of the data are [latex]1[\/latex], [latex]1[\/latex], [latex]2[\/latex], [latex]2[\/latex], [latex]4[\/latex], [latex]6[\/latex], [latex]6.8[\/latex]. The middle value of the lower half is two.<\/p>\n<p style=\"margin-left: 20px;\">[latex]1[\/latex]; [latex]1[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]4[\/latex]; [latex]6[\/latex]; [latex]6.8[\/latex]<\/p>\n<p>The number two, which is part of the data, is the\u00a0<strong>first quartile<\/strong>. One-fourth of the entire sets of values are the same as or less than two and three-fourths of the values are more than two.<\/p>\n<p>The upper half of the data is [latex]7.2[\/latex], [latex]8[\/latex], [latex]8.3[\/latex], [latex]9[\/latex], [latex]10[\/latex], [latex]10[\/latex], [latex]11.5[\/latex]. The middle value of the upper half is nine.<\/p>\n<p>The\u00a0<strong>third quartile<\/strong>,\u00a0[latex]Q_3[\/latex], is nine. Three-fourths ([latex]75[\/latex]%) of the ordered data set are less than nine. One-fourth ([latex]25[\/latex]%) of the ordered data set are greater than nine. The third quartile is part of the data set in this example.<\/p>\n<p>The\u00a0<strong>interquartile range (IQR)<\/strong>\u00a0is a number that indicates the spread of the middle half or the middle [latex]50[\/latex]% of the data. It is the difference between the third quartile ([latex]Q_3[\/latex]) and the first quartile ([latex]Q_1[\/latex]).<\/p>\n<p>[latex]IQR[\/latex] = [latex]Q_3[\/latex] \u2013 [latex]Q_1[\/latex]<\/p>\n<p>The IQR can help to determine potential <strong>outliers<\/strong>. <strong>A value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile<\/strong>. Potential outliers always require further investigation.<\/p>\n<div class=\"textbox shaded\">\n<h3>NOTE<\/h3>\n<p>A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality or they may be a key to understanding the data.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>For the following [latex]13[\/latex] real estate prices, calculate the [latex]IQR[\/latex] and determine if any prices are potential outliers. Prices are in dollars.<br \/>\n[latex]389{,}950[\/latex]; [latex]230{,}500[\/latex]; [latex]158{,}000[\/latex]; [latex]479{,}000[\/latex]; [latex]639{,}000[\/latex]; [latex]114{,}950[\/latex]; [latex]5{,}500{,}000[\/latex]; [latex]387{,}000[\/latex]; [latex]659{,}000[\/latex]; [latex]529{,}000[\/latex]; [latex]575{,}000[\/latex]; [latex]488{,}800[\/latex]; [latex]1{,}095{,}000[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283390\">Show Solution<\/span><\/p>\n<div id=\"q283390\" class=\"hidden-answer\" style=\"display: none\">\nOrder the data from smallest to largest.<br \/>\n[latex]114{,}950[\/latex]; [latex]158{,}000[\/latex]; [latex]230{,}500[\/latex]; [latex]387{,}000[\/latex]; [latex]389{,}950[\/latex]; [latex]479{,}000[\/latex]; [latex]488{,}800[\/latex]; [latex]529{,}000[\/latex]; [latex]575{,}000[\/latex]; [latex]639{,}000[\/latex]; [latex]659{,}000[\/latex]; [latex]1{,}095{,}000[\/latex]; [latex]5{,}500{,}000[\/latex]<\/p>\n<p>[latex]M = 488,800[\/latex]<\/p>\n<p>[latex]Q_1 = \\frac{230{,}500 + 387{,}000}{2} = 308{,}750[\/latex]<\/p>\n<p>[latex]Q_3 = \\frac{639{,}000 + 659{,}000}{2} = 649{,}000[\/latex]<\/p>\n<p>[latex]IQR = 649{,}000 \u2013 308{,}750 = 340{,}250[\/latex]<\/p>\n<p>[latex](1.5)(IQR) = (1.5)(340{,}250) = 510{,}375[\/latex]<\/p>\n<p>[latex]Q_1 \u2013 (1.5)(IQR) = 308{,}750 \u2013 510{,}375 = \u2013201{,}625[\/latex]<\/p>\n<p>[latex]Q_3 + (1.5)(IQR) = 649{,}000 + 510{,}375 = 1{,}159{,}375[\/latex]<\/p>\n<p>No house price is less than [latex]\u2013201{,}625[\/latex]. However, [latex]5{,}500{,}000[\/latex] is more than [latex]1{,}159{,}375[\/latex]. Therefore, [latex]5{,}500{,}000[\/latex] is a potential <strong>outlier<\/strong>.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>For the following [latex]11[\/latex] salaries, calculate the [latex]IQR[\/latex] and determine if any salaries are outliers. The salaries are in dollars.<\/p>\n<p>[latex]$33{,}000[\/latex], [latex]$64{,}500[\/latex], [latex]$28{,}000[\/latex], [latex]$54{,}000[\/latex], [latex]$72{,}000[\/latex], [latex]$68{,}500[\/latex], [latex]$69{,}000[\/latex], [latex]$42{,}000[\/latex], [latex]$54{,}000[\/latex] [latex]$120{,}000[\/latex], [latex]$40{,}500[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283391\">Show Solution<\/span><\/p>\n<div id=\"q283391\" class=\"hidden-answer\" style=\"display: none\">\nOrder the data from smallest to largest.<\/p>\n<p>[latex]$28{,}000[\/latex], [latex]$33{,}000[\/latex], [latex]$40{,}500[\/latex], [latex]$42{,}000[\/latex], [latex]$54{,}000[\/latex], [latex]$54{,}000[\/latex], [latex]$64{,}500[\/latex], [latex]$68{,}500[\/latex], [latex]$69{,}000[\/latex], [latex]$72{,}000[\/latex], [latex]$120{,}000[\/latex]<\/p>\n<p>Median = [latex]$54,000[\/latex]<\/p>\n<p>[latex]Q_1[\/latex] = [latex]$40{,}500[\/latex]<\/p>\n<p>[latex]Q_3[\/latex] = [latex]$69{,}000[\/latex]<\/p>\n<p>[latex]IQR[\/latex] = [latex]$69{,}000[\/latex] \u2013 [latex]$40{,}500[\/latex] = [latex]$28{,}500[\/latex]<\/p>\n<p>(1.5)([latex]IQR[\/latex]) = (1.5)($28{,}500) = [latex]$42{,}750[\/latex]<\/p>\n<p>[latex]Q_1[\/latex] \u2013 (1.5)([latex]IQR[\/latex]) = [latex]$40{,}500[\/latex] \u2013 [latex]$42{,}750[\/latex] = [latex]\u2013$2{,}250[\/latex]<\/p>\n<p>[latex]Q_3[\/latex] + (1.5)([latex]IQR[\/latex]) = [latex]$69{,}000[\/latex] + [latex]$42{,}750[\/latex] = [latex]$111{,}750[\/latex]<\/p>\n<p>No salary is less than [latex]$2{,}250[\/latex]. However, [latex]$120{,}000[\/latex] is more than [latex]$111{,}750[\/latex], so [latex]$120{,}000[\/latex] is a potential outlier.\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"element-880\" class=\"\">For the two data sets in the\u00a0test scores example, find the following:<\/p>\n<ol id=\"element-971\" type=\"a\" data-mark-suffix=\".\">\n<li data-mark-suffix=\".\">The interquartile range. Compare the two interquartile ranges.<\/li>\n<li data-mark-suffix=\".\">Any outliers in either set.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q563920\">Show Answer<\/span><\/p>\n<div id=\"q563920\" class=\"hidden-answer\" style=\"display: none\">\n<p>The five number summary for the day and night classes is<\/p>\n<div id=\"fs-idp36487328\" class=\"os-table\">\n<table summary=\"Table 2.21\" data-id=\"fs-idp36487328\">\n<thead>\n<tr>\n<th scope=\"col\"><\/th>\n<th scope=\"col\">Minimum<\/th>\n<th scope=\"col\"><em data-effect=\"italics\">Q<\/em><sub>1<\/sub><\/th>\n<th scope=\"col\">Median<\/th>\n<th scope=\"col\"><em data-effect=\"italics\">Q<\/em><sub>3<\/sub><\/th>\n<th scope=\"col\">Maximum<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong data-effect=\"bold\">Day<\/strong><\/td>\n<td>32<\/td>\n<td>56<\/td>\n<td>74.5<\/td>\n<td>82.5<\/td>\n<td>99<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">Night<\/strong><\/td>\n<td>25.5<\/td>\n<td>78<\/td>\n<td>81<\/td>\n<td>89<\/td>\n<td>98<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-idm23962720\" type=\"a\">\n<li>The IQR for the day group is\u00a0<em data-effect=\"italics\">Q<\/em><sub>3<\/sub>\u00a0\u2013\u00a0<em data-effect=\"italics\">Q<\/em><sub>1<\/sub>\u00a0= 82.5 \u2013 56 = 26.5\n<p id=\"fs-idm7044352\" class=\"\">The IQR for the night group is\u00a0<em data-effect=\"italics\">Q<\/em><sub>3<\/sub>\u00a0\u2013\u00a0<em data-effect=\"italics\">Q<\/em><sub>1<\/sub>\u00a0= 89 \u2013 78 = 11<\/p>\n<p id=\"fs-idp42547504\" class=\"\">The interquartile range (the spread or variability) for the day class is larger than the night class\u00a0<em data-effect=\"italics\">IQR<\/em>. This suggests more variation will be found in the day class\u2019s class test scores.<\/p>\n<\/li>\n<li>Day class outliers are found using the IQR times 1.5 rule. So,\n<ul id=\"fs-idm52257968\" data-labeled-item=\"true\">\n<li><em data-effect=\"italics\">Q<\/em><sub>1<\/sub>\u00a0&#8211;\u00a0<em data-effect=\"italics\">IQR<\/em>(1.5) = 56 \u2013 26.5(1.5) = 16.25<\/li>\n<li><em data-effect=\"italics\">Q<\/em><sub>3<\/sub>\u00a0+\u00a0<em data-effect=\"italics\">IQR<\/em>(1.5) = 82.5 + 26.5(1.5) = 122.25<\/li>\n<\/ul>\n<p id=\"fs-idp38341744\" class=\"\">Since the minimum and maximum values for the day class are greater than 16.25 and less than 122.25, there are no outliers.<\/p>\n<p id=\"fs-idm23940160\" class=\"\">Night class outliers are calculated as:<\/p>\n<ul id=\"fs-idp29569184\" data-labeled-item=\"true\">\n<li><em data-effect=\"italics\">Q<\/em><sub>1<\/sub>\u00a0\u2013\u00a0<em data-effect=\"italics\">IQR<\/em>\u00a0(1.5) = 78 \u2013 11(1.5) = 61.5<\/li>\n<li><em data-effect=\"italics\">Q<\/em><sub>3<\/sub>\u00a0+ IQR(1.5) = 89 + 11(1.5) = 105.5<\/li>\n<\/ul>\n<p id=\"fs-idp5005056\" class=\"\">For this class, any test score less than 61.5 is an outlier. Therefore, the scores of 45 and 25.5 are outliers. Since no test score is greater than 105.5, there is no upper end outlier.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the interquartile range for the following two data sets and compare them.<\/p>\n<p>Test Scores for Class\u00a0<em>A<\/em><\/p>\n<p style=\"margin-left: 20px;\">[latex]69[\/latex]; [latex]96[\/latex]; [latex]81[\/latex]; [latex]79[\/latex]; [latex]65[\/latex]; [latex]76[\/latex]; [latex]83[\/latex]; [latex]99[\/latex]; [latex]89[\/latex]; [latex]67[\/latex]; [latex]90[\/latex]; [latex]77[\/latex]; [latex]85[\/latex]; [latex]98[\/latex]; [latex]66[\/latex]; [latex]91[\/latex]; [latex]77[\/latex]; [latex]69[\/latex]; [latex]80[\/latex]; [latex]94[\/latex]<\/p>\n<p>Test Scores for Class\u00a0<em>B<\/em><\/p>\n<p style=\"margin-left: 20px;\">[latex]90[\/latex]; [latex]72[\/latex]; [latex]80[\/latex]; [latex]92[\/latex]; [latex]90[\/latex]; [latex]97[\/latex]; [latex]92[\/latex]; [latex]75[\/latex]; [latex]79[\/latex]; [latex]68[\/latex]; [latex]70[\/latex]; [latex]80[\/latex]; [latex]99[\/latex]; [latex]95[\/latex]; [latex]78[\/latex]; [latex]73[\/latex]; [latex]71[\/latex]; [latex]68[\/latex]; [latex]95[\/latex]; [latex]100[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283392\">Show Solution<\/span><\/p>\n<div id=\"q283392\" class=\"hidden-answer\" style=\"display: none\">\n<h4>Class\u00a0A<\/h4>\n<p>Order the data from smallest to largest.<\/p>\n<p>[latex]65[\/latex]; [latex]66[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]; [latex]77[\/latex]; [latex]79[\/latex]; [latex]80[\/latex]; [latex]81[\/latex]; [latex]83[\/latex]; [latex]85[\/latex]; [latex]89[\/latex]; [latex]90[\/latex]; [latex]91[\/latex]; [latex]94[\/latex]; [latex]96[\/latex]; [latex]98[\/latex]; [latex]99[\/latex]<\/p>\n<p>[latex]\\displaystyle {Median}=\\frac{{{80}+{81}}}{{2}}={80.5}[\/latex]<\/p>\n<p>[latex]{Q}_{{1}}=\\frac{{{69}+{76}}}{{2}}={72.5}[\/latex]<\/p>\n<p>[latex]{Q}_{{3}}=\\frac{{{90}+{91}}}{{2}}={90.5}[\/latex]<\/p>\n<p>[latex]IQR[\/latex] = [latex]90.5[\/latex] \u2013 [latex]72.5[\/latex] = [latex]18[\/latex]<\/p>\n<h4>Class\u00a0B<\/h4>\n<p>Order the data from smallest to largest.<\/p>\n<p>[latex]68[\/latex]; [latex]68[\/latex]; [latex]70[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]75[\/latex]; [latex]78[\/latex]; [latex]79[\/latex]; [latex]80[\/latex]; [latex]80[\/latex]; [latex]90[\/latex]; [latex]90[\/latex]; [latex]92[\/latex]; [latex]92[\/latex]; [latex]95[\/latex]; [latex]95[\/latex]; [latex]97[\/latex]; [latex]99[\/latex]; [latex]100[\/latex]<\/p>\n<p>[latex]\\displaystyle{Median}=\\frac{{{80}+{80}}}{{2}}={80}[\/latex]<\/p>\n<p>[latex]{Q}_{{1}}=\\frac{{{72}+{73}}}{{2}}={72.5}[\/latex]<\/p>\n<p>[latex]{Q}_{{3}}=\\frac{{{92}+{95}}}{{2}}={93.5}[\/latex]<\/p>\n<p>[latex]IQR[\/latex] = [latex]93.5[\/latex] \u2013 [latex]72.5[\/latex] = [latex]21[\/latex]<\/p>\n<p>The data for Class\u00a0<em>B<\/em> has a larger [latex]IQR[\/latex], so the scores between [latex]Q_3[\/latex] and [latex]Q_1[\/latex] (middle [latex]50[\/latex]%) for the data for Class <em>B<\/em> are more spread out and not clustered about the median.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were:<\/p>\n<table>\n<thead>\n<tr>\n<th>Amount of Sleep per School Night (Hours)<\/th>\n<th>Frequency<\/th>\n<th>Relative Frequency<\/th>\n<th>Cumulative Relative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]0.04[\/latex]<\/td>\n<td>[latex]0.04[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]0.10[\/latex]<\/td>\n<td>[latex]0.14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]0.14[\/latex]<\/td>\n<td>[latex]0.28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]0.24[\/latex]<\/td>\n<td>[latex]0.52[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]0.28[\/latex]<\/td>\n<td>[latex]0.80[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]0.14[\/latex]<\/td>\n<td>[latex]0.94[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0.06[\/latex]<\/td>\n<td>[latex]1.00[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Find the [latex]28[\/latex]th percentile.<\/strong> Notice the [latex]0.28[\/latex] in the &#8220;cumulative relative frequency&#8221; column. Twenty-eight percent of [latex]50[\/latex] data values is [latex]14[\/latex] values. There are [latex]14[\/latex] values less than the [latex]28[\/latex]th percentile. They include the two [latex]4[\/latex]s, the five [latex]5[\/latex]s, and the seven [latex]6[\/latex]s. The [latex]28[\/latex]th percentile is between the last six and the first seven. <strong>The [latex]28[\/latex]th percentile is [latex]6.5[\/latex].<\/strong><\/p>\n<p><strong>Find the median.<\/strong> Look again at the &#8220;cumulative relative frequency&#8221; column and find [latex]0.52[\/latex]. The median is the [latex]50[\/latex]th percentile or the second quartile. [latex]50[\/latex]% of [latex]50[\/latex] is [latex]25[\/latex]. There are [latex]25[\/latex] values less than the median. They include the two [latex]4[\/latex]s, the five [latex]5[\/latex]s, the seven [latex]6[\/latex]s, and eleven of the [latex]7[\/latex]s. The median or [latex]50[\/latex]th percentile is between the [latex]25[\/latex]th, or seven, and [latex]26[\/latex]th, or seven, values. <strong>The median is seven.<\/strong><\/p>\n<p><strong>Find the third quartile. <\/strong>The third quartile is the same as the [latex]75[\/latex]th percentile. You can &#8220;eyeball&#8221; this answer. If you look at the &#8220;cumulative relative frequency&#8221; column, you find [latex]0.52[\/latex] and [latex]0.80[\/latex]. When you have all the fours, fives, sixes and sevens, you have [latex]52[\/latex]% of the data. When you include all the [latex]8[\/latex]s, you have [latex]80[\/latex]% of the data. <strong>The [latex]75[\/latex]th percentile, then, must be an eight.<\/strong> Another way to look at the problem is to find [latex]75[\/latex]% of [latex]50[\/latex], which is [latex]37.5[\/latex],and round up to [latex]38[\/latex]. The third quartile, [latex]Q_3[\/latex], is the 38th value, which is an 8. You can check this answer by counting the values. (There are [latex]37[\/latex] values below the third quartile and 12 values above.)<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the [latex]65[\/latex]th percentile.<\/p>\n<table>\n<thead>\n<tr>\n<th>Amount of time spent on route (hours)<\/th>\n<th>Frequency<\/th>\n<th>Relative Frequency<\/th>\n<th>Cumulative Relative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]0.30[\/latex]<\/td>\n<td>[latex]0.30[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]0.35[\/latex]<\/td>\n<td>[latex]0.65[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td>[latex]0.90[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]0.10[\/latex]<\/td>\n<td>[latex]1.00[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283393\">Show Solution<\/span><\/p>\n<div id=\"q283393\" class=\"hidden-answer\" style=\"display: none\">\nThe [latex]65[\/latex]th percentile is between the last three and the first four.<\/p>\n<p>The [latex]65[\/latex]th percentile is [latex]3.5[\/latex].\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<thead>\n<tr>\n<th>Amount of Sleep per School Night (Hours)<\/th>\n<th>Frequency<\/th>\n<th>Relative Frequency<\/th>\n<th>Cumulative Relative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]0.04[\/latex]<\/td>\n<td>[latex]0.04[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]0.10[\/latex]<\/td>\n<td>[latex]0.14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]0.14[\/latex]<\/td>\n<td>[latex]0.28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]0.24[\/latex]<\/td>\n<td>[latex]0.52[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]0.28[\/latex]<\/td>\n<td>[latex]0.80[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]0.14[\/latex]<\/td>\n<td>[latex]0.94[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0.06[\/latex]<\/td>\n<td>[latex]1.00[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Find the [latex]80[\/latex]th percentile.<\/li>\n<li>Find the [latex]90[\/latex]th percentile.<\/li>\n<li>Find the first quartile. What is another name for the first quartile?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283394\">Show Solution<\/span><\/p>\n<div id=\"q283394\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the data from the frequency table, we have:<\/p>\n<ol>\n<li>The [latex]80[\/latex]th percentile is between the last eight and the first nine in the table (between the [latex]40[\/latex]th and [latex]41[\/latex]st values). Therefore, we need to take the mean of the [latex]40[\/latex]th an [latex]41[\/latex]st values. The 80th percentile [latex]\\displaystyle\\frac{{{8}+{9}}}{{2}}={8.5}[\/latex]<\/li>\n<li>The [latex]90[\/latex]th percentile will be the [latex]45[\/latex]th data value (location is [latex]0.90(50) = 45[\/latex]) and the [latex]45[\/latex]th data value is nine.<\/li>\n<li>[latex]Q_1[\/latex] is also the [latex]25[\/latex]th percentile. The [latex]25[\/latex]th percentile location calculation: [latex]P_{25}[\/latex] = [latex]0.25(50) = 12.5 \u2248 13[\/latex] the [latex]13[\/latex]th data value. Thus, the [latex]25[\/latex]th percentile is six.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<table>\n<thead>\n<tr>\n<th>Amount of time spent on route (hours)<\/th>\n<th>Frequency<\/th>\n<th>Relative Frequency<\/th>\n<th>Cumulative Relative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]0.30[\/latex]<\/td>\n<td>[latex]0.30[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]0.35[\/latex]<\/td>\n<td>[latex]0.65[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td>[latex]0.90[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]0.10[\/latex]<\/td>\n<td>[latex]1.00[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Find the third quartile. What is another name for the third quartile?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283395\">Show Solution<\/span><\/p>\n<div id=\"q283395\" class=\"hidden-answer\" style=\"display: none\">\nThe third quartile is the [latex]75[\/latex]th percentile, which is four. The [latex]65[\/latex]th percentile is between three and four, and the [latex]90[\/latex]th percentile is between four and [latex]5.75[\/latex]. The third quartile is between [latex]65[\/latex] and [latex]90[\/latex], so it must be four.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<header>\n<h3 class=\"title\" data-type=\"title\">Collaborative Exercise<\/h3>\n<p>Your instructor or a member of the class will ask everyone in class how many sweaters they own. Answer the following questions:<\/p>\n<ol>\n<li>How many students were surveyed?<\/li>\n<li>What kind of sampling did you do?<\/li>\n<li>Construct two different histograms. For each, starting value = _____ ending value = ____.<\/li>\n<li>Find the median, first quartile, and third quartile.<\/li>\n<li>Construct a table of the data to find the following:<\/li>\n<\/ol>\n<ol style=\"list-style-type: lower-alpha;\">\n<li style=\"list-style-type: none;\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>the 10th percentile<\/li>\n<li>the 70th percentile<\/li>\n<li>the percent of students who own less than four sweaters<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/header>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-35\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>OpenStax, Statistics, Measures of the Location of Data. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\">https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Median, Quartiles and Interquartile Range: ExamSolutions. <strong>Authored by<\/strong>: ExamSolutions. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/wNamjO-JzUg\">https:\/\/youtu.be\/wNamjO-JzUg<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Measures of the Location of Data\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"copyrighted_video\",\"description\":\"Median, Quartiles and Interquartile Range: ExamSolutions\",\"author\":\"ExamSolutions\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/wNamjO-JzUg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at 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