{"id":660,"date":"2021-08-13T15:37:32","date_gmt":"2021-08-13T15:37:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=660"},"modified":"2023-12-05T08:54:26","modified_gmt":"2023-12-05T08:54:26","slug":"measures-of-the-location-of-the-data-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/measures-of-the-location-of-the-data-2\/","title":{"raw":"A Formula for Finding the kth Percentile","rendered":"A Formula for Finding the kth Percentile"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list123523\">\r\n \t<li>Given a data set, find the [latex]k^{th}[\/latex] percentile<\/li>\r\n<\/ul>\r\n<\/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 29 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 style=\"list-style-type: none;\">\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 style=\"list-style-type: none;\">\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\r\na. 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].\r\n\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\r\nb. 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].\r\n\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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list123523\">\n<li>Given a data set, find the [latex]k^{th}[\/latex] percentile<\/li>\n<\/ul>\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 29 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 style=\"list-style-type: none;\">\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 style=\"list-style-type: none;\">\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<p>a. 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].<\/p>\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<p>b. 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].<\/p>\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\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-660\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>OpenStax, Statistics, Measures of the Location of Data. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Measures of the Location of Data\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-660","chapter","type-chapter","status-publish","hentry"],"part":31,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/660","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/660\/revisions"}],"predecessor-version":[{"id":3294,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/660\/revisions\/3294"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/31"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/660\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=660"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=660"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=660"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}