{"id":76,"date":"2016-04-21T22:43:45","date_gmt":"2016-04-21T22:43:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&#038;p=76"},"modified":"2021-07-04T17:02:17","modified_gmt":"2021-07-04T17:02:17","slug":"measures-of-the-location-of-the-data","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/chapter\/measures-of-the-location-of-the-data\/","title":{"raw":"Measures of the Location of the Data","rendered":"Measures of the Location of the Data"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list123523\">\r\n \t<li>Recognize, describe, and calculate the measures of location of data: quartiles and percentiles.<\/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 the Median, Quartiles and the Interquartile Range, topics you will learn in this section.\r\n\r\nhttps:\/\/youtu.be\/wNamjO-JzUg\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\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\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<\/strong> is 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 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\n\r\nOrder the data from smallest to largest.\r\n\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(1.5)([latex]IQR[\/latex]) = (1.5)(340,250) = [latex]510,375[\/latex]\r\n\r\n[latex]Q_1[\/latex] \u2013 (1.5)([latex]IQR[\/latex]) = [latex]308,750[\/latex] \u2013 [latex]510,375[\/latex] = [latex]\u2013201625[\/latex]\r\n\r\n[latex]Q_3[\/latex] + (1.5)([latex]IQR[\/latex]) = [latex]649,000[\/latex] + [latex]510,375[\/latex] = [latex]1,159,375[\/latex]\r\n\r\nNo house price is less than [latex]-201,625[\/latex]. However, [latex]5,500,00[\/latex] is more than [latex]1,159,375[\/latex], so [latex]5,500,000[\/latex] is a potential outlier.\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 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 eight. 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>\r\n<h3>A Formula for Finding the [latex]k[\/latex]th Percentile<\/h3>\r\nIf you were to do a little research, you would find several formulas for calculating the [latex]k[\/latex]th percentile. Here is one of them.\r\n\r\n[latex]k[\/latex] = the [latex]k[\/latex]th percentile. It may or may not be part of the data.\r\n\r\n[latex]i[\/latex] = the index (ranking or position of a data value)\r\n\r\n[latex]n[\/latex] = the total number of data\r\n<ul>\r\n \t<li>Order the data from smallest to largest.<\/li>\r\n \t<li>Calculate [latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}[\/latex]<\/li>\r\n \t<li>If [latex]i[\/latex] is an integer, then the [latex]k[\/latex]th percentile is the data value in the [latex]i[\/latex]th position in the ordered set of data.<\/li>\r\n \t<li>If [latex]i[\/latex] is not an integer, then round [latex]i[\/latex] up and round [latex]i[\/latex] down to the nearest integers. Average the two data values in these two positions in the ordered data set. This is easier to understand in an example.<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nListed are twenty-nine ages for trees found in the Saint Louis Botanical Garden\u00a0<em>in order from smallest to largest.<\/em>\r\n\r\n[latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]30[\/latex]; [latex]31[\/latex]; [latex]33[\/latex]; [latex]36[\/latex]; [latex]37[\/latex]; [latex]41[\/latex]; [latex]42[\/latex]; [latex]47[\/latex]; [latex]52[\/latex]; [latex]55[\/latex]; [latex]57[\/latex]; [latex]58[\/latex]; [latex]62[\/latex]; [latex]64[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]74[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]\r\n<ol>\r\n \t<li>Find the [latex]70[\/latex]th percentile.<\/li>\r\n \t<li>Find the [latex]83[\/latex]rd percentile.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"283396\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283396\"]\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>\r\n<ul>\r\n \t<li>[latex]k[\/latex] = [latex]70[\/latex]<\/li>\r\n \t<li>[latex]i[\/latex] = the index<\/li>\r\n \t<li>[latex]n[\/latex] = [latex]29[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}={(\\frac{{70}}{{100}})}{({29}+{1})}={21}[\/latex]. Twenty-one is an integer, and the data value in the [latex]21[\/latex]st position in the ordered data set is [latex]64[\/latex]. The [latex]70[\/latex]th percentile is 64 years.\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>\r\n<ul>\r\n \t<li>[latex]k[\/latex] = [latex]83[\/latex]rd percentile<\/li>\r\n \t<li>[latex]i[\/latex] = the index<\/li>\r\n \t<li>[latex]n[\/latex] = [latex]29[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}={(\\frac{{83}}{{100}})}{({29}+{1})}={24.9}[\/latex], which is NOT an integer. Round it down to [latex]24[\/latex] and up to [latex]25[\/latex]. The age in the [latex]24[\/latex]th position is [latex]71[\/latex] and the age in the [latex]25[\/latex]th position is [latex]72[\/latex]. Average [latex]71[\/latex] and [latex]72[\/latex]. The [latex]83[\/latex]rd percentile is [latex]71.5[\/latex] years.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nListed are [latex]29[\/latex] ages for Academy Award winning best actors <em>in order from smallest to largest.<\/em>\r\n\r\n[latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]30[\/latex]; [latex]31[\/latex]; [latex]33[\/latex]; [latex]36[\/latex]; [latex]37[\/latex]; [latex]41[\/latex]; [latex]42[\/latex]; [latex]47[\/latex]; [latex]52[\/latex]; [latex]55[\/latex]; [latex]57[\/latex]; [latex]58[\/latex]; [latex]62[\/latex]; [latex]64[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]74[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]\r\n\r\nCalculate the [latex]20[\/latex]th percentile and the [latex]55[\/latex]th percentile.\r\n[reveal-answer q=\"283397\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283397\"]\r\n\r\n[latex]k[\/latex] = [latex]20[\/latex]. Index = [latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}=\\frac{{20}}{{100}}{({29}+{1})}={6}[\/latex] The age in the sixth position is [latex]27[\/latex]. The [latex]20[\/latex]th percentile is [latex]27[\/latex] years.\r\n\r\n[latex]k[\/latex] = [latex]55[\/latex]. Index = [latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}=\\frac{{55}}{{100}}{({29}+{1})}={16.5}[\/latex]<i>.<\/i> Round down to [latex]16[\/latex] and up to [latex]17[\/latex]. The age in the [latex]16[\/latex]th position is [latex]52[\/latex] and the age in the [latex]17[\/latex]th position is [latex]55[\/latex]. The average of [latex]52[\/latex] and [latex]55[\/latex] is [latex]53.5[\/latex]. The [latex]55[\/latex]th percentile is [latex]53.5[\/latex] years.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>NOTE<\/h3>\r\nYou can calculate percentiles using calculators and computers. There are a variety of online calculators.\r\n\r\n<\/div>\r\n<h2>A Formula for Finding the Percentile of a Value in a Data Set<\/h2>\r\n<ul>\r\n \t<li>Order the data from smallest to largest.<\/li>\r\n \t<li>[latex]x[\/latex] = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile.<\/li>\r\n \t<li>[latex]y[\/latex] = the number of data values equal to the data value for which you want to find the percentile.<\/li>\r\n \t<li>[latex]n[\/latex] = the total number of data.<\/li>\r\n \t<li>Calculate [latex]\\displaystyle\\frac{{{x}+{0.5}{y}}}{{n}}{({100})}[\/latex]. Then round to the nearest integer.<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nListed are [latex]29[\/latex] ages for Academy Award winning best actors <em>in order from smallest to largest.<\/em>\r\n\r\n[latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]30[\/latex]; [latex]31[\/latex]; [latex]33[\/latex]; [latex]36[\/latex]; [latex]37[\/latex]; [latex]41[\/latex]; [latex]42[\/latex]; [latex]47[\/latex]; [latex]52[\/latex]; [latex]55[\/latex]; [latex]57[\/latex]; [latex]58[\/latex]; [latex]62[\/latex]; [latex]64[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]74[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]\r\n<ol>\r\n \t<li>Find the percentile for [latex]58[\/latex].<\/li>\r\n \t<li>Find the percentile for [latex]25[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"283398\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283398\"]\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>Counting from the bottom of the list, there are [latex]18[\/latex] data values less than [latex]58[\/latex]. There is one value of [latex]58[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[latex]x=18\\quad\\text{and}\\quad{y=1}[\/latex]\r\n\r\n[latex]\\dfrac{x+0.5y}{n}(100)=\\dfrac{18+0.5(1)}{29}(100)=63.80[\/latex]\r\n\r\n[latex]58[\/latex] is the [latex]64[\/latex]th percentile.\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>Counting from the bottom of the list, there are three data values less than [latex]25[\/latex]. There is one value of [latex]25[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[latex]x=3\\quad\\text{and}\\quad{y=1}[\/latex]\r\n\r\n[latex]\\dfrac{x+0.5y}{n}(100)=\\dfrac{3+0.5(1)}{29}(100)=12.07[\/latex]\r\n\r\n[latex]25[\/latex] is the [latex]12[\/latex]th percentile.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nListed are [latex]30[\/latex] ages for New York Times published columnists <em>in order from smallest to largest.<\/em>\r\n\r\n[latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]30[\/latex]; [latex]31[\/latex], [latex]31[\/latex]; [latex]33[\/latex]; [latex]36[\/latex]; [latex]37[\/latex]; [latex]41[\/latex]; [latex]42[\/latex]; [latex]47[\/latex]; [latex]52[\/latex]; [latex]55[\/latex]; [latex]57[\/latex]; [latex]58[\/latex]; [latex]62[\/latex]; [latex]64[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]74[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]\r\n\r\nFind the percentiles for [latex]47[\/latex] and [latex]31[\/latex].\r\n\r\n[reveal-answer q=\"283399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283399\"]\r\n\r\nPercentile for [latex]47[\/latex]: Counting from the bottom of the list, there are [latex]15[\/latex] data values less than [latex]47[\/latex]. There is one value of [latex]47[\/latex].\r\n\r\n[latex]x=15\\quad\\text{and}\\quad{y=1}[\/latex]\r\n\r\n[latex]\\dfrac{x+0.5y}{n}(100)=\\dfrac{15+0.5(1)}{30}(100)=51.67[\/latex]\r\n\r\n[latex]47[\/latex] is the [latex]52[\/latex]nd percentile.\r\n\r\nPercentile for [latex]31[\/latex]: Counting from the bottom of the list, there are eight data values less than [latex]31[\/latex]. There are [latex]two[\/latex] values of [latex]31[\/latex].\r\n\r\n[latex]x=8\\quad\\text{and}\\quad{y=2}[\/latex]\r\n\r\n[latex]\\dfrac{x+0.5y}{n}(100)=\\dfrac{8+0.5(2)}{30}(100)=30[\/latex]\r\n\r\n[latex]31[\/latex] is the [latex]30[\/latex]th percentile.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Interpreting Percentiles, Quartiles, and Median<\/h2>\r\nA percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the [latex]p[\/latex]th percentile. For example, [latex]15[\/latex]% of data values are less than or equal to the [latex]15[\/latex]th percentile.\r\n<ul>\r\n \t<li>Low percentiles always correspond to lower data values.<\/li>\r\n \t<li>High percentiles always correspond to higher data values.<\/li>\r\n<\/ul>\r\nA percentile may or may not correspond to a value judgment about whether it is \"good\" or \"bad.\" The interpretation of whether a certain percentile is \"good\" or \"bad\" depends on the context of the situation to which the data applies. In some situations, a low percentile would be considered \"good;\" in other contexts a high percentile might be considered \"good\". In many situations, there is no value judgment that applies.\r\n\r\nUnderstanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text.\r\n<h3>Guideline<\/h3>\r\nWhen writing the interpretation of a percentile in the context of the given data, the sentence should contain the following information.\r\n<ul>\r\n \t<li>information about the context of the situation being considered<\/li>\r\n \t<li>the data value (value of the variable) that represents the percentile<\/li>\r\n \t<li>the percent of individuals or items with data values below the percentile<\/li>\r\n \t<li>the percent of individuals or items with data values above the percentile.<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nOn a timed math test, the first quartile for time it took to finish the exam was [latex]35[\/latex] minutes. Interpret the first quartile in the context of this situation.\r\n\r\n[reveal-answer q=\"283400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283400\"]\r\n<ul>\r\n \t<li>Twenty-five percent of students finished the exam in [latex]35[\/latex] minutes or less.<\/li>\r\n \t<li>Seventy-five percent of students finished the exam in [latex]35[\/latex] minutes or more.<\/li>\r\n \t<li>A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. (If you take too long, you might not be able to finish.)<\/li>\r\n<\/ul>\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 [latex]100[\/latex]-meter dash, the third quartile for times for finishing the race was [latex]11.5[\/latex] seconds. Interpret the third quartile in the context of the situation.\r\n[reveal-answer q=\"283401\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283401\"]\r\nTwenty-five percent of runners finished the race in [latex]11.5[\/latex] seconds or more. Seventy-five percent of runners finished the race in [latex]11.5[\/latex] seconds or less. A lower percentile is good because finishing a race more quickly is desirable.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nOn a [latex]20[\/latex] question math test, the [latex]70[\/latex]th percentile for number of correct answers was [latex]16[\/latex]. Interpret the [latex]70[\/latex]th percentile in the context of this situation.\r\n\r\n[reveal-answer q=\"283402\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283402\"]\r\n<ul>\r\n \t<li>Seventy percent of students answered [latex]16[\/latex] or fewer questions correctly.<\/li>\r\n \t<li>Thirty percent of students answered [latex]16[\/latex] or more questions correctly.<\/li>\r\n \t<li>A higher percentile could be considered good, as answering more questions correctly is desirable.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nOn a [latex]60[\/latex] point written assignment, the [latex]80[\/latex]th percentile for the number of points earned was [latex]49[\/latex]. Interpret the [latex]80[\/latex]th percentile in the context of this situation.\r\n[reveal-answer q=\"283403\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283403\"]\r\nEighty percent of students earned [latex]49[\/latex] points or fewer. Twenty percent of students earned 49 or more points. A higher percentile is good because getting more points on an assignment is desirable.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAt a community college, it was found that the [latex]30[\/latex]th percentile of credit units that students are enrolled for is seven units. Interpret the [latex]30[\/latex]th percentile in the context of this situation.\r\n\r\n[reveal-answer q=\"283404\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283404\"]\r\n<ul>\r\n \t<li>Thirty percent of students are enrolled in seven or fewer credit units.<\/li>\r\n \t<li>Seventy percent of students are enrolled in seven or more credit units.<\/li>\r\n \t<li>In this example, there is no \"good\" or \"bad\" value judgment associated with a higher or lower percentile. Students attend community college for varied reasons and needs, and their course load varies according to their needs.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDuring a season, the [latex]40[\/latex]th percentile for points scored per player in a game is eight. Interpret the [latex]40[\/latex]th percentile in the context of this situation.\r\n[reveal-answer q=\"283405\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283405\"]\r\nForty percent of players scored eight points or fewer. Sixty percent of players scored eight points or more. A higher percentile is good because getting more points in a basketball game is desirable.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSharpe Middle School is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed [latex]15[\/latex] anonymous students to determine how many minutes a day the students spend exercising. The results from the [latex]15[\/latex] anonymous students are shown.\r\n\r\n[latex]0[\/latex] minutes; [latex]40[\/latex] minutes; [latex]60[\/latex] minutes; [latex]30[\/latex] minutes; [latex]60[\/latex] minutes\r\n\r\n[latex]10[\/latex] minutes; [latex]45[\/latex] minutes; [latex]30[\/latex] minutes; [latex]300[\/latex] minutes; [latex]90[\/latex] minutes;\r\n\r\n[latex]30[\/latex] minutes; [latex]120[\/latex] minutes; [latex]60[\/latex] minutes; [latex]0[\/latex] minutes; [latex]20[\/latex] minutes\r\n\r\nDetermine the following five values.\r\n<ul>\r\n \t<li>Min = [latex]0[\/latex]<\/li>\r\n \t<li>[latex]Q_1[\/latex] = [latex]20[\/latex]<\/li>\r\n \t<li>Med = [latex]40[\/latex]<\/li>\r\n \t<li>[latex]Q_3[\/latex] = [latex]60[\/latex]<\/li>\r\n \t<li>Max = [latex]300[\/latex]<\/li>\r\n<\/ul>\r\nIf you were the principal, would you be justified in purchasing new fitness equipment?\r\n\r\n[reveal-answer q=\"283406\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283406\"]\r\n\r\nSince [latex]75[\/latex]% of the students exercise for [latex]60[\/latex] minutes or less daily, and since the [latex]IQR[\/latex] is [latex]40[\/latex] minutes [latex](60 \u2013 20 = 40)[\/latex], we know that half of the students surveyed exercise between [latex]20[\/latex] minutes and [latex]60[\/latex] minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment.\r\n\r\nHowever, the principal needs to be careful. The value [latex]300[\/latex] appears to be a potential outlier.\r\n\r\n[latex]Q_3[\/latex] + [latex]1.5[\/latex]([latex]IQR[\/latex]) = [latex]60 + (1.5)(40) = 120[\/latex].\r\n\r\nThe value [latex]300[\/latex] is greater than [latex]120[\/latex] so it is a potential outlier. If we delete it and calculate the five values, we get the following values:\r\n<ul>\r\n \t<li>Min = [latex]0[\/latex]<\/li>\r\n \t<li>[latex]Q_1[\/latex] = [latex]20[\/latex]<\/li>\r\n \t<li>[latex]Q_3[\/latex] = [latex]60[\/latex]<\/li>\r\n \t<li>Max = [latex]120[\/latex]<\/li>\r\n<\/ul>\r\nWe still have [latex]75[\/latex]% of the students exercising for [latex]60[\/latex] minutes or less daily and half of the students exercising between [latex]20[\/latex] and [latex]60[\/latex] minutes a day. However, [latex]15[\/latex] students is a small sample and the principal should survey more students to be sure of his survey results.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Concept Review<\/h2>\r\nThe values that divide a rank-ordered set of data into [latex]100[\/latex] equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the [latex]50[\/latex]th percentile would be greater than [latex]50[\/latex] percent of the other observations in the set. Quartiles divide data into quarters. The first quartile ([latex]Q_1[\/latex]) is the 25th percentile, the second quartile ([latex]Q_2[\/latex] or median) is [latex]50[\/latex]th percentile, and the third quartile ([latex]Q_3[\/latex]) is the the [latex]75[\/latex]th percentile. The interquartile range, or [latex]IQR[\/latex], is the range of the middle [latex]50[\/latex] percent of the data values. The [latex]IQR[\/latex] is found by subtracting [latex]Q_1[\/latex] from [latex]Q_3[\/latex], and can help determine outliers by using the following two expressions.\r\n<ul>\r\n \t<li>[latex]Q_3[\/latex] + [latex]IQR[\/latex]([latex]1.5[\/latex])<\/li>\r\n \t<li>[latex]Q_1[\/latex] \u2013 [latex]IQR[\/latex]([latex]1.5[\/latex])<\/li>\r\n<\/ul>\r\n<h2>Formula Review<\/h2>\r\n[latex]\\displaystyle{i}={(\\frac{{k}}{{100}})}{({n}+{1})}[\/latex]where\r\n[latex]i[\/latex] = the ranking or position of a data value,\r\n\r\n[latex]k[\/latex] = the kth percentile,\r\n\r\n[latex]n[\/latex] = total number of data.\r\n\r\nExpression for finding the percentile of a data value:\r\n\r\n[latex]\\displaystyle{(\\frac{{{x}+{0.5}{y}}}{{n}})}{({100})}[\/latex]\r\n\r\nwhere\r\n\r\n[latex]x[\/latex] = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile,\r\n\r\n[latex]y[\/latex]= the number of data values equal to the data value for which you want to find the percentile,\r\n\r\n[latex]n[\/latex] = total number of data\r\n<h3>References<\/h3>\r\nCauchon, Dennis, Paul Overberg. \"Census data shows minorities now a majority of U.S. births.\" USA Today, 2012. Available online at http:\/\/usatoday30.usatoday.com\/news\/nation\/story\/2012-05-17\/minority-birthscensus\/55029100\/1 (accessed April 3, 2013).\r\n\r\nData from the United States Department of Commerce: United States Census Bureau. Available online at http:\/\/www.census.gov\/ (accessed April 3, 2013).\r\n\r\n\"1990 Census.\" United States Department of Commerce: United States Census Bureau. Available online at http:\/\/www.census.gov\/main\/www\/cen1990.html (accessed April 3, 2013).\r\n\r\nData from\r\n<em>San Jose Mercury News<\/em>.\r\n\r\nData from\r\n<em>Time Magazine<\/em>; survey by Yankelovich Partners, Inc.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list123523\">\n<li>Recognize, describe, and calculate the measures of location of data: quartiles and percentiles.<\/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 the Median, Quartiles and the Interquartile Range, topics you will learn in this section.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Median, Quartiles and Interquartile Range : ExamSolutions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/wNamjO-JzUg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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<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<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<\/strong> is 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 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\">\n<p>Order the data from smallest to largest.<\/p>\n<p>[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>(1.5)([latex]IQR[\/latex]) = (1.5)(340,250) = [latex]510,375[\/latex]<\/p>\n<p>[latex]Q_1[\/latex] \u2013 (1.5)([latex]IQR[\/latex]) = [latex]308,750[\/latex] \u2013 [latex]510,375[\/latex] = [latex]\u2013201625[\/latex]<\/p>\n<p>[latex]Q_3[\/latex] + (1.5)([latex]IQR[\/latex]) = [latex]649,000[\/latex] + [latex]510,375[\/latex] = [latex]1,159,375[\/latex]<\/p>\n<p>No house price is less than [latex]-201,625[\/latex]. However, [latex]5,500,00[\/latex] is more than [latex]1,159,375[\/latex], so [latex]5,500,000[\/latex] is a potential outlier.\n<\/p><\/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 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 eight. 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<h3>A Formula for Finding the [latex]k[\/latex]th Percentile<\/h3>\n<p>If you were to do a little research, you would find several formulas for calculating the [latex]k[\/latex]th percentile. Here is one of them.<\/p>\n<p>[latex]k[\/latex] = the [latex]k[\/latex]th percentile. It may or may not be part of the data.<\/p>\n<p>[latex]i[\/latex] = the index (ranking or position of a data value)<\/p>\n<p>[latex]n[\/latex] = the total number of data<\/p>\n<ul>\n<li>Order the data from smallest to largest.<\/li>\n<li>Calculate [latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}[\/latex]<\/li>\n<li>If [latex]i[\/latex] is an integer, then the [latex]k[\/latex]th percentile is the data value in the [latex]i[\/latex]th position in the ordered set of data.<\/li>\n<li>If [latex]i[\/latex] is not an integer, then round [latex]i[\/latex] up and round [latex]i[\/latex] down to the nearest integers. Average the two data values in these two positions in the ordered data set. This is easier to understand in an example.<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Listed are twenty-nine ages for trees found in the Saint Louis Botanical Garden\u00a0<em>in order from smallest to largest.<\/em><\/p>\n<p>[latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]30[\/latex]; [latex]31[\/latex]; [latex]33[\/latex]; [latex]36[\/latex]; [latex]37[\/latex]; [latex]41[\/latex]; [latex]42[\/latex]; [latex]47[\/latex]; [latex]52[\/latex]; [latex]55[\/latex]; [latex]57[\/latex]; [latex]58[\/latex]; [latex]62[\/latex]; [latex]64[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]74[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]<\/p>\n<ol>\n<li>Find the [latex]70[\/latex]th percentile.<\/li>\n<li>Find the [latex]83[\/latex]rd percentile.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283396\">Show Solution<\/span><\/p>\n<div id=\"q283396\" class=\"hidden-answer\" style=\"display: none\">\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>\n<ul>\n<li>[latex]k[\/latex] = [latex]70[\/latex]<\/li>\n<li>[latex]i[\/latex] = the index<\/li>\n<li>[latex]n[\/latex] = [latex]29[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>[latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}={(\\frac{{70}}{{100}})}{({29}+{1})}={21}[\/latex]. Twenty-one is an integer, and the data value in the [latex]21[\/latex]st position in the ordered data set is [latex]64[\/latex]. The [latex]70[\/latex]th percentile is 64 years.<\/p>\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>\n<ul>\n<li>[latex]k[\/latex] = [latex]83[\/latex]rd percentile<\/li>\n<li>[latex]i[\/latex] = the index<\/li>\n<li>[latex]n[\/latex] = [latex]29[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>[latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}={(\\frac{{83}}{{100}})}{({29}+{1})}={24.9}[\/latex], which is NOT an integer. Round it down to [latex]24[\/latex] and up to [latex]25[\/latex]. The age in the [latex]24[\/latex]th position is [latex]71[\/latex] and the age in the [latex]25[\/latex]th position is [latex]72[\/latex]. Average [latex]71[\/latex] and [latex]72[\/latex]. The [latex]83[\/latex]rd percentile is [latex]71.5[\/latex] years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Listed are [latex]29[\/latex] ages for Academy Award winning best actors <em>in order from smallest to largest.<\/em><\/p>\n<p>[latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]30[\/latex]; [latex]31[\/latex]; [latex]33[\/latex]; [latex]36[\/latex]; [latex]37[\/latex]; [latex]41[\/latex]; [latex]42[\/latex]; [latex]47[\/latex]; [latex]52[\/latex]; [latex]55[\/latex]; [latex]57[\/latex]; [latex]58[\/latex]; [latex]62[\/latex]; [latex]64[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]74[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]<\/p>\n<p>Calculate the [latex]20[\/latex]th percentile and the [latex]55[\/latex]th percentile.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283397\">Show Solution<\/span><\/p>\n<div id=\"q283397\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]k[\/latex] = [latex]20[\/latex]. Index = [latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}=\\frac{{20}}{{100}}{({29}+{1})}={6}[\/latex] The age in the sixth position is [latex]27[\/latex]. The [latex]20[\/latex]th percentile is [latex]27[\/latex] years.<\/p>\n<p>[latex]k[\/latex] = [latex]55[\/latex]. Index = [latex]\\displaystyle{i}=\\frac{{k}}{{100}}{({n}+{1})}=\\frac{{55}}{{100}}{({29}+{1})}={16.5}[\/latex]<i>.<\/i> Round down to [latex]16[\/latex] and up to [latex]17[\/latex]. The age in the [latex]16[\/latex]th position is [latex]52[\/latex] and the age in the [latex]17[\/latex]th position is [latex]55[\/latex]. The average of [latex]52[\/latex] and [latex]55[\/latex] is [latex]53.5[\/latex]. The [latex]55[\/latex]th percentile is [latex]53.5[\/latex] years.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>NOTE<\/h3>\n<p>You can calculate percentiles using calculators and computers. There are a variety of online calculators.<\/p>\n<\/div>\n<h2>A Formula for Finding the Percentile of a Value in a Data Set<\/h2>\n<ul>\n<li>Order the data from smallest to largest.<\/li>\n<li>[latex]x[\/latex] = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile.<\/li>\n<li>[latex]y[\/latex] = the number of data values equal to the data value for which you want to find the percentile.<\/li>\n<li>[latex]n[\/latex] = the total number of data.<\/li>\n<li>Calculate [latex]\\displaystyle\\frac{{{x}+{0.5}{y}}}{{n}}{({100})}[\/latex]. Then round to the nearest integer.<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Listed are [latex]29[\/latex] ages for Academy Award winning best actors <em>in order from smallest to largest.<\/em><\/p>\n<p>[latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]30[\/latex]; [latex]31[\/latex]; [latex]33[\/latex]; [latex]36[\/latex]; [latex]37[\/latex]; [latex]41[\/latex]; [latex]42[\/latex]; [latex]47[\/latex]; [latex]52[\/latex]; [latex]55[\/latex]; [latex]57[\/latex]; [latex]58[\/latex]; [latex]62[\/latex]; [latex]64[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]74[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]<\/p>\n<ol>\n<li>Find the percentile for [latex]58[\/latex].<\/li>\n<li>Find the percentile for [latex]25[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283398\">Show Solution<\/span><\/p>\n<div id=\"q283398\" class=\"hidden-answer\" style=\"display: none\">\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>Counting from the bottom of the list, there are [latex]18[\/latex] data values less than [latex]58[\/latex]. There is one value of [latex]58[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>[latex]x=18\\quad\\text{and}\\quad{y=1}[\/latex]<\/p>\n<p>[latex]\\dfrac{x+0.5y}{n}(100)=\\dfrac{18+0.5(1)}{29}(100)=63.80[\/latex]<\/p>\n<p>[latex]58[\/latex] is the [latex]64[\/latex]th percentile.<\/p>\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>Counting from the bottom of the list, there are three data values less than [latex]25[\/latex]. There is one value of [latex]25[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>[latex]x=3\\quad\\text{and}\\quad{y=1}[\/latex]<\/p>\n<p>[latex]\\dfrac{x+0.5y}{n}(100)=\\dfrac{3+0.5(1)}{29}(100)=12.07[\/latex]<\/p>\n<p>[latex]25[\/latex] is the [latex]12[\/latex]th percentile.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Listed are [latex]30[\/latex] ages for New York Times published columnists <em>in order from smallest to largest.<\/em><\/p>\n<p>[latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]30[\/latex]; [latex]31[\/latex], [latex]31[\/latex]; [latex]33[\/latex]; [latex]36[\/latex]; [latex]37[\/latex]; [latex]41[\/latex]; [latex]42[\/latex]; [latex]47[\/latex]; [latex]52[\/latex]; [latex]55[\/latex]; [latex]57[\/latex]; [latex]58[\/latex]; [latex]62[\/latex]; [latex]64[\/latex]; [latex]67[\/latex]; [latex]69[\/latex]; [latex]71[\/latex]; [latex]72[\/latex]; [latex]73[\/latex]; [latex]74[\/latex]; [latex]76[\/latex]; [latex]77[\/latex]<\/p>\n<p>Find the percentiles for [latex]47[\/latex] and [latex]31[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283399\">Show Solution<\/span><\/p>\n<div id=\"q283399\" class=\"hidden-answer\" style=\"display: none\">\n<p>Percentile for [latex]47[\/latex]: Counting from the bottom of the list, there are [latex]15[\/latex] data values less than [latex]47[\/latex]. There is one value of [latex]47[\/latex].<\/p>\n<p>[latex]x=15\\quad\\text{and}\\quad{y=1}[\/latex]<\/p>\n<p>[latex]\\dfrac{x+0.5y}{n}(100)=\\dfrac{15+0.5(1)}{30}(100)=51.67[\/latex]<\/p>\n<p>[latex]47[\/latex] is the [latex]52[\/latex]nd percentile.<\/p>\n<p>Percentile for [latex]31[\/latex]: Counting from the bottom of the list, there are eight data values less than [latex]31[\/latex]. There are [latex]two[\/latex] values of [latex]31[\/latex].<\/p>\n<p>[latex]x=8\\quad\\text{and}\\quad{y=2}[\/latex]<\/p>\n<p>[latex]\\dfrac{x+0.5y}{n}(100)=\\dfrac{8+0.5(2)}{30}(100)=30[\/latex]<\/p>\n<p>[latex]31[\/latex] is the [latex]30[\/latex]th percentile.\n<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Interpreting Percentiles, Quartiles, and Median<\/h2>\n<p>A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the [latex]p[\/latex]th percentile. For example, [latex]15[\/latex]% of data values are less than or equal to the [latex]15[\/latex]th percentile.<\/p>\n<ul>\n<li>Low percentiles always correspond to lower data values.<\/li>\n<li>High percentiles always correspond to higher data values.<\/li>\n<\/ul>\n<p>A percentile may or may not correspond to a value judgment about whether it is &#8220;good&#8221; or &#8220;bad.&#8221; The interpretation of whether a certain percentile is &#8220;good&#8221; or &#8220;bad&#8221; depends on the context of the situation to which the data applies. In some situations, a low percentile would be considered &#8220;good;&#8221; in other contexts a high percentile might be considered &#8220;good&#8221;. In many situations, there is no value judgment that applies.<\/p>\n<p>Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text.<\/p>\n<h3>Guideline<\/h3>\n<p>When writing the interpretation of a percentile in the context of the given data, the sentence should contain the following information.<\/p>\n<ul>\n<li>information about the context of the situation being considered<\/li>\n<li>the data value (value of the variable) that represents the percentile<\/li>\n<li>the percent of individuals or items with data values below the percentile<\/li>\n<li>the percent of individuals or items with data values above the percentile.<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>On a timed math test, the first quartile for time it took to finish the exam was [latex]35[\/latex] minutes. Interpret the first quartile in the context of this situation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283400\">Show Solution<\/span><\/p>\n<div id=\"q283400\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Twenty-five percent of students finished the exam in [latex]35[\/latex] minutes or less.<\/li>\n<li>Seventy-five percent of students finished the exam in [latex]35[\/latex] minutes or more.<\/li>\n<li>A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. (If you take too long, you might not be able to finish.)<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>For the [latex]100[\/latex]-meter dash, the third quartile for times for finishing the race was [latex]11.5[\/latex] seconds. Interpret the third quartile in the context of the situation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283401\">Show Solution<\/span><\/p>\n<div id=\"q283401\" class=\"hidden-answer\" style=\"display: none\">\nTwenty-five percent of runners finished the race in [latex]11.5[\/latex] seconds or more. Seventy-five percent of runners finished the race in [latex]11.5[\/latex] seconds or less. A lower percentile is good because finishing a race more quickly is desirable.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>On a [latex]20[\/latex] question math test, the [latex]70[\/latex]th percentile for number of correct answers was [latex]16[\/latex]. Interpret the [latex]70[\/latex]th percentile in the context of this situation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283402\">Show Solution<\/span><\/p>\n<div id=\"q283402\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Seventy percent of students answered [latex]16[\/latex] or fewer questions correctly.<\/li>\n<li>Thirty percent of students answered [latex]16[\/latex] or more questions correctly.<\/li>\n<li>A higher percentile could be considered good, as answering more questions correctly is desirable.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>On a [latex]60[\/latex] point written assignment, the [latex]80[\/latex]th percentile for the number of points earned was [latex]49[\/latex]. Interpret the [latex]80[\/latex]th percentile in the context of this situation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283403\">Show Solution<\/span><\/p>\n<div id=\"q283403\" class=\"hidden-answer\" style=\"display: none\">\nEighty percent of students earned [latex]49[\/latex] points or fewer. Twenty percent of students earned 49 or more points. A higher percentile is good because getting more points on an assignment is desirable.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>At a community college, it was found that the [latex]30[\/latex]th percentile of credit units that students are enrolled for is seven units. Interpret the [latex]30[\/latex]th percentile in the context of this situation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283404\">Show Solution<\/span><\/p>\n<div id=\"q283404\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Thirty percent of students are enrolled in seven or fewer credit units.<\/li>\n<li>Seventy percent of students are enrolled in seven or more credit units.<\/li>\n<li>In this example, there is no &#8220;good&#8221; or &#8220;bad&#8221; value judgment associated with a higher or lower percentile. Students attend community college for varied reasons and needs, and their course load varies according to their needs.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>During a season, the [latex]40[\/latex]th percentile for points scored per player in a game is eight. Interpret the [latex]40[\/latex]th percentile in the context of this situation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283405\">Show Solution<\/span><\/p>\n<div id=\"q283405\" class=\"hidden-answer\" style=\"display: none\">\nForty percent of players scored eight points or fewer. Sixty percent of players scored eight points or more. A higher percentile is good because getting more points in a basketball game is desirable.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Sharpe Middle School is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed [latex]15[\/latex] anonymous students to determine how many minutes a day the students spend exercising. The results from the [latex]15[\/latex] anonymous students are shown.<\/p>\n<p>[latex]0[\/latex] minutes; [latex]40[\/latex] minutes; [latex]60[\/latex] minutes; [latex]30[\/latex] minutes; [latex]60[\/latex] minutes<\/p>\n<p>[latex]10[\/latex] minutes; [latex]45[\/latex] minutes; [latex]30[\/latex] minutes; [latex]300[\/latex] minutes; [latex]90[\/latex] minutes;<\/p>\n<p>[latex]30[\/latex] minutes; [latex]120[\/latex] minutes; [latex]60[\/latex] minutes; [latex]0[\/latex] minutes; [latex]20[\/latex] minutes<\/p>\n<p>Determine the following five values.<\/p>\n<ul>\n<li>Min = [latex]0[\/latex]<\/li>\n<li>[latex]Q_1[\/latex] = [latex]20[\/latex]<\/li>\n<li>Med = [latex]40[\/latex]<\/li>\n<li>[latex]Q_3[\/latex] = [latex]60[\/latex]<\/li>\n<li>Max = [latex]300[\/latex]<\/li>\n<\/ul>\n<p>If you were the principal, would you be justified in purchasing new fitness equipment?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283406\">Show Solution<\/span><\/p>\n<div id=\"q283406\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since [latex]75[\/latex]% of the students exercise for [latex]60[\/latex] minutes or less daily, and since the [latex]IQR[\/latex] is [latex]40[\/latex] minutes [latex](60 \u2013 20 = 40)[\/latex], we know that half of the students surveyed exercise between [latex]20[\/latex] minutes and [latex]60[\/latex] minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment.<\/p>\n<p>However, the principal needs to be careful. The value [latex]300[\/latex] appears to be a potential outlier.<\/p>\n<p>[latex]Q_3[\/latex] + [latex]1.5[\/latex]([latex]IQR[\/latex]) = [latex]60 + (1.5)(40) = 120[\/latex].<\/p>\n<p>The value [latex]300[\/latex] is greater than [latex]120[\/latex] so it is a potential outlier. If we delete it and calculate the five values, we get the following values:<\/p>\n<ul>\n<li>Min = [latex]0[\/latex]<\/li>\n<li>[latex]Q_1[\/latex] = [latex]20[\/latex]<\/li>\n<li>[latex]Q_3[\/latex] = [latex]60[\/latex]<\/li>\n<li>Max = [latex]120[\/latex]<\/li>\n<\/ul>\n<p>We still have [latex]75[\/latex]% of the students exercising for [latex]60[\/latex] minutes or less daily and half of the students exercising between [latex]20[\/latex] and [latex]60[\/latex] minutes a day. However, [latex]15[\/latex] students is a small sample and the principal should survey more students to be sure of his survey results.\n<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Concept Review<\/h2>\n<p>The values that divide a rank-ordered set of data into [latex]100[\/latex] equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the [latex]50[\/latex]th percentile would be greater than [latex]50[\/latex] percent of the other observations in the set. Quartiles divide data into quarters. The first quartile ([latex]Q_1[\/latex]) is the 25th percentile, the second quartile ([latex]Q_2[\/latex] or median) is [latex]50[\/latex]th percentile, and the third quartile ([latex]Q_3[\/latex]) is the the [latex]75[\/latex]th percentile. The interquartile range, or [latex]IQR[\/latex], is the range of the middle [latex]50[\/latex] percent of the data values. The [latex]IQR[\/latex] is found by subtracting [latex]Q_1[\/latex] from [latex]Q_3[\/latex], and can help determine outliers by using the following two expressions.<\/p>\n<ul>\n<li>[latex]Q_3[\/latex] + [latex]IQR[\/latex]([latex]1.5[\/latex])<\/li>\n<li>[latex]Q_1[\/latex] \u2013 [latex]IQR[\/latex]([latex]1.5[\/latex])<\/li>\n<\/ul>\n<h2>Formula Review<\/h2>\n<p>[latex]\\displaystyle{i}={(\\frac{{k}}{{100}})}{({n}+{1})}[\/latex]where<br \/>\n[latex]i[\/latex] = the ranking or position of a data value,<\/p>\n<p>[latex]k[\/latex] = the kth percentile,<\/p>\n<p>[latex]n[\/latex] = total number of data.<\/p>\n<p>Expression for finding the percentile of a data value:<\/p>\n<p>[latex]\\displaystyle{(\\frac{{{x}+{0.5}{y}}}{{n}})}{({100})}[\/latex]<\/p>\n<p>where<\/p>\n<p>[latex]x[\/latex] = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile,<\/p>\n<p>[latex]y[\/latex]= the number of data values equal to the data value for which you want to find the percentile,<\/p>\n<p>[latex]n[\/latex] = total number of data<\/p>\n<h3>References<\/h3>\n<p>Cauchon, Dennis, Paul Overberg. &#8220;Census data shows minorities now a majority of U.S. births.&#8221; USA Today, 2012. Available online at http:\/\/usatoday30.usatoday.com\/news\/nation\/story\/2012-05-17\/minority-birthscensus\/55029100\/1 (accessed April 3, 2013).<\/p>\n<p>Data from the United States Department of Commerce: United States Census Bureau. Available online at http:\/\/www.census.gov\/ (accessed April 3, 2013).<\/p>\n<p>&#8220;1990 Census.&#8221; United States Department of Commerce: United States Census Bureau. Available online at http:\/\/www.census.gov\/main\/www\/cen1990.html (accessed April 3, 2013).<\/p>\n<p>Data from<br \/>\n<em>San Jose Mercury News<\/em>.<\/p>\n<p>Data from<br \/>\n<em>Time Magazine<\/em>; survey by Yankelovich Partners, Inc.<\/p>\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-76\">\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><\/li><li>Introductory Statistics . <strong>Authored by<\/strong>: Barbara Illowski, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/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>: Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/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":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Measures of the Location of Data\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"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 Illowski, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-76","chapter","type-chapter","status-publish","hentry"],"part":69,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters\/76","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":23,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters\/76\/revisions"}],"predecessor-version":[{"id":2269,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters\/76\/revisions\/2269"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/parts\/69"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters\/76\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/media?parent=76"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapter-type?post=76"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/contributor?post=76"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/license?post=76"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}