{"id":10383,"date":"2017-05-26T18:52:51","date_gmt":"2017-05-26T18:52:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10383"},"modified":"2017-09-25T20:43:23","modified_gmt":"2017-09-25T20:43:23","slug":"finding-the-median-of-a-set-of-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/finding-the-median-of-a-set-of-numbers\/","title":{"raw":"Finding the Median of a Set of Numbers","rendered":"Finding the Median of a Set of Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the median of a set of numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p data-type=\"title\">When Ann, Bianca, Dora, Eve, and Francine sing together on stage, they line up in order of their heights. Their heights, in inches, are shown in the table below.<\/p>\r\n\r\n<table id=\"fs-id1832924\" summary=\"A table is shown with 5 columns and 2 rows. The first column says \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Ann<\/th>\r\n<th data-align=\"center\">Bianca<\/th>\r\n<th data-align=\"center\">Dora<\/th>\r\n<th data-align=\"center\">Eve<\/th>\r\n<th data-align=\"center\">Francine<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]59[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]60[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]65[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]68[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]70[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDora is in the middle of the group. Her height, [latex]65\"[\/latex], is the <em data-effect=\"italics\">median<\/em> of the girls\u2019 heights. Half of the heights are less than or equal to Dora\u2019s height, and half are greater than or equal. The median is the middle value.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221810\/CNX_BMath_Figure_05_05_001_img.png\" alt=\"The numbers 59, 60, 65, 68, and 70 are listed. 59 and 60 have a brace beneath them and in red are labeled \" data-media-type=\"image\/png\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Median<\/h3>\r\nThe median of a set of data values is the middle value.\r\n<ul id=\"fs-id2455647\" data-bullet-style=\"bullet\">\r\n \t<li>Half the data values are less than or equal to the median.<\/li>\r\n \t<li>Half the data values are greater than or equal to the median.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhat if Carmen, the pianist, joins the singing group on stage? Carmen is [latex]62[\/latex] inches tall, so she fits in the height order between Bianca and Dora. Now the data set looks like this:\r\n<p style=\"text-align: center;\">[latex]59,60,62,65,68,70[\/latex]<\/p>\r\nThere is no single middle value. The heights of the six girls can be divided into two equal parts.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221811\/CNX_BMath_Figure_05_05_006_img.png\" alt=\"The numbers 59, 60, and 62 are listed, followed by a blank space, then 65, 68, and 70.\" data-media-type=\"image\/png\" \/>\r\nStatisticians have agreed that in cases like this the median is the mean of the two values closest to the middle. So the median is the mean of [latex]62\\text{ and }65,{\\Large\\frac{62+65}{2}}[\/latex]. The median height is [latex]63.5[\/latex] inches.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221812\/CNX_BMath_Figure_05_05_008_img.png\" alt=\"The numbers 9, 11, 12, 13, 15, 18, and 19 are listed. 9, 11, and 12 have a brace beneath them and are labeled \" data-media-type=\"image\/png\" \/>\r\nNotice that when the number of girls was [latex]5[\/latex], the median was the third height, but when the number of girls was [latex]6[\/latex], the median was the mean of the third and fourth heights. In general, when the number of values is odd, the median will be the one value in the middle, but when the number is even, the median is the mean of the two middle values.\r\n<div class=\"textbox shaded\">\r\n<h3>Find the median of a set of numbers.<\/h3>\r\n<ol id=\"eip-id1168466010714\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>List the numbers from smallest to largest.<\/li>\r\n \t<li>Count how many numbers are in the set. Call this [latex]n[\/latex].<\/li>\r\n \t<li>Is [latex]n[\/latex] odd or even?\r\n<ul id=\"fs-id1733078\" data-bullet-style=\"bullet\">\r\n \t<li>If [latex]n[\/latex] is an odd number, the median is the middle value.<\/li>\r\n \t<li>If [latex]n[\/latex] is an even number, the median is the mean of the two middle values.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the median of [latex]12,13,19,9,11,15,\\text{and }18[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168468416053\" class=\"unnumbered unstyled\" summary=\"The figure shows the numbers 59, 60, and 62 separated by a small space from the numbers 65, 68, and 70. Each set of three has a bracket underneath grouping the numbers together.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>List the numbers in order from smallest to largest.<\/td>\r\n<td data-align=\"center\">[latex]9, 11, 12, 13, 15, 18, 19[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count how many numbers are in the set. Call this [latex]n[\/latex] .<\/td>\r\n<td data-align=\"center\">[latex]n=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is [latex]n[\/latex] odd or even?<\/td>\r\n<td data-align=\"center\">odd<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The median is the middle value.<\/td>\r\n<td data-align=\"center\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221814\/CNX_BMath_Figure_05_05_009_img.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The middle is the number in the [latex]4[\/latex]th position.<\/td>\r\n<td data-align=\"center\">So the median of the data is [latex]13[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146418[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nKristen received the following scores on her weekly math quizzes:\r\n[latex]83,79,85,86,92,100,76,90,88,\\text{and }64[\/latex]. Find her median score.\r\n[reveal-answer q=\"982119\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982119\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467170036\" class=\"unnumbered unstyled\" summary=\"The problem says, 'Find the median of 83, 79, 85, 86, 92, 100, 76, 90, 88, and 64.' The first step says, 'List the numbers in order from smallest to largest,' and shows 64, 76, 79, 83, 85, 86, 88, 90, 92, 100. The next step says, 'Count how many numbers are in the set. Call this n. n equals 10.' The next step asks, 'Is n odd or even? Even.' The next line says, 'The median is the two middle values, the 5th and 6th numbers.' The ordered list of numbers is shown again with the first five numbers grouped together and labeled 5 numbers and the second five numbers are grouped together and labeled 5 numbers. The next step says 'Find the mean of 85 and 86.' The mean equals the sum of 85 plus 86 divided by 2, which equals 85.5. The last line shows 'Kristen's median score is 85.5'.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Find the median of [latex]83, 79, 85, 86, 92, 100, 76, 90, 88,\\text{ and }64[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>List the numbers in order from smallest to largest.<\/td>\r\n<td data-align=\"center\">[latex]64, 76, 79, 83, 85, 86, 88, 90, 92, 100[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of data values in the set. Call this [latex]\\mathrm{n.}[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]n=10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is [latex]n[\/latex] odd or even?<\/td>\r\n<td data-align=\"center\">even<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The median is the mean of the two middle values, the 5th and 6th numbers.<\/td>\r\n<td data-align=\"center\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221816\/CNX_BMath_Figure_05_05_010_img-01.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the mean of [latex]85[\/latex] and [latex]86[\/latex].<\/td>\r\n<td data-align=\"center\">[latex]\\text{mean}={\\Large\\frac{85+86}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">[latex]\\text{mean}=85.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">Kristen's median score is [latex]85.5[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146419[\/ohm_question]\r\n\r\n<\/div>\r\nThe next video shows another example of how to find hte median of a set of numbers.\r\n\r\nhttps:\/\/youtu.be\/CbKqFc-EPDs","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the median of a set of numbers<\/li>\n<\/ul>\n<\/div>\n<p data-type=\"title\">When Ann, Bianca, Dora, Eve, and Francine sing together on stage, they line up in order of their heights. Their heights, in inches, are shown in the table below.<\/p>\n<table id=\"fs-id1832924\" summary=\"A table is shown with 5 columns and 2 rows. The first column says\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Ann<\/th>\n<th data-align=\"center\">Bianca<\/th>\n<th data-align=\"center\">Dora<\/th>\n<th data-align=\"center\">Eve<\/th>\n<th data-align=\"center\">Francine<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]59[\/latex]<\/td>\n<td data-align=\"center\">[latex]60[\/latex]<\/td>\n<td data-align=\"center\">[latex]65[\/latex]<\/td>\n<td data-align=\"center\">[latex]68[\/latex]<\/td>\n<td data-align=\"center\">[latex]70[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Dora is in the middle of the group. Her height, [latex]65\"[\/latex], is the <em data-effect=\"italics\">median<\/em> of the girls\u2019 heights. Half of the heights are less than or equal to Dora\u2019s height, and half are greater than or equal. The median is the middle value.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221810\/CNX_BMath_Figure_05_05_001_img.png\" alt=\"The numbers 59, 60, 65, 68, and 70 are listed. 59 and 60 have a brace beneath them and in red are labeled\" data-media-type=\"image\/png\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Median<\/h3>\n<p>The median of a set of data values is the middle value.<\/p>\n<ul id=\"fs-id2455647\" data-bullet-style=\"bullet\">\n<li>Half the data values are less than or equal to the median.<\/li>\n<li>Half the data values are greater than or equal to the median.<\/li>\n<\/ul>\n<\/div>\n<p>What if Carmen, the pianist, joins the singing group on stage? Carmen is [latex]62[\/latex] inches tall, so she fits in the height order between Bianca and Dora. Now the data set looks like this:<\/p>\n<p style=\"text-align: center;\">[latex]59,60,62,65,68,70[\/latex]<\/p>\n<p>There is no single middle value. The heights of the six girls can be divided into two equal parts.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221811\/CNX_BMath_Figure_05_05_006_img.png\" alt=\"The numbers 59, 60, and 62 are listed, followed by a blank space, then 65, 68, and 70.\" data-media-type=\"image\/png\" \/><br \/>\nStatisticians have agreed that in cases like this the median is the mean of the two values closest to the middle. So the median is the mean of [latex]62\\text{ and }65,{\\Large\\frac{62+65}{2}}[\/latex]. The median height is [latex]63.5[\/latex] inches.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221812\/CNX_BMath_Figure_05_05_008_img.png\" alt=\"The numbers 9, 11, 12, 13, 15, 18, and 19 are listed. 9, 11, and 12 have a brace beneath them and are labeled\" data-media-type=\"image\/png\" \/><br \/>\nNotice that when the number of girls was [latex]5[\/latex], the median was the third height, but when the number of girls was [latex]6[\/latex], the median was the mean of the third and fourth heights. In general, when the number of values is odd, the median will be the one value in the middle, but when the number is even, the median is the mean of the two middle values.<\/p>\n<div class=\"textbox shaded\">\n<h3>Find the median of a set of numbers.<\/h3>\n<ol id=\"eip-id1168466010714\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>List the numbers from smallest to largest.<\/li>\n<li>Count how many numbers are in the set. Call this [latex]n[\/latex].<\/li>\n<li>Is [latex]n[\/latex] odd or even?\n<ul id=\"fs-id1733078\" data-bullet-style=\"bullet\">\n<li>If [latex]n[\/latex] is an odd number, the median is the middle value.<\/li>\n<li>If [latex]n[\/latex] is an even number, the median is the mean of the two middle values.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the median of [latex]12,13,19,9,11,15,\\text{and }18[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468416053\" class=\"unnumbered unstyled\" summary=\"The figure shows the numbers 59, 60, and 62 separated by a small space from the numbers 65, 68, and 70. Each set of three has a bracket underneath grouping the numbers together.\" data-label=\"\">\n<tbody>\n<tr>\n<td>List the numbers in order from smallest to largest.<\/td>\n<td data-align=\"center\">[latex]9, 11, 12, 13, 15, 18, 19[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Count how many numbers are in the set. Call this [latex]n[\/latex] .<\/td>\n<td data-align=\"center\">[latex]n=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is [latex]n[\/latex] odd or even?<\/td>\n<td data-align=\"center\">odd<\/td>\n<\/tr>\n<tr>\n<td>The median is the middle value.<\/td>\n<td data-align=\"center\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221814\/CNX_BMath_Figure_05_05_009_img.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>The middle is the number in the [latex]4[\/latex]th position.<\/td>\n<td data-align=\"center\">So the median of the data is [latex]13[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146418\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146418&theme=oea&iframe_resize_id=ohm146418&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Kristen received the following scores on her weekly math quizzes:<br \/>\n[latex]83,79,85,86,92,100,76,90,88,\\text{and }64[\/latex]. Find her median score.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982119\">Show Solution<\/span><\/p>\n<div id=\"q982119\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467170036\" class=\"unnumbered unstyled\" summary=\"The problem says, 'Find the median of 83, 79, 85, 86, 92, 100, 76, 90, 88, and 64.' The first step says, 'List the numbers in order from smallest to largest,' and shows 64, 76, 79, 83, 85, 86, 88, 90, 92, 100. The next step says, 'Count how many numbers are in the set. Call this n. n equals 10.' The next step asks, 'Is n odd or even? Even.' The next line says, 'The median is the two middle values, the 5th and 6th numbers.' The ordered list of numbers is shown again with the first five numbers grouped together and labeled 5 numbers and the second five numbers are grouped together and labeled 5 numbers. The next step says 'Find the mean of 85 and 86.' The mean equals the sum of 85 plus 86 divided by 2, which equals 85.5. The last line shows 'Kristen's median score is 85.5'.\" data-label=\"\">\n<tbody>\n<tr>\n<td>Find the median of [latex]83, 79, 85, 86, 92, 100, 76, 90, 88,\\text{ and }64[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>List the numbers in order from smallest to largest.<\/td>\n<td data-align=\"center\">[latex]64, 76, 79, 83, 85, 86, 88, 90, 92, 100[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Count the number of data values in the set. Call this [latex]\\mathrm{n.}[\/latex]<\/td>\n<td data-align=\"center\">[latex]n=10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is [latex]n[\/latex] odd or even?<\/td>\n<td data-align=\"center\">even<\/td>\n<\/tr>\n<tr>\n<td>The median is the mean of the two middle values, the 5th and 6th numbers.<\/td>\n<td data-align=\"center\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221816\/CNX_BMath_Figure_05_05_010_img-01.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the mean of [latex]85[\/latex] and [latex]86[\/latex].<\/td>\n<td data-align=\"center\">[latex]\\text{mean}={\\Large\\frac{85+86}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">[latex]\\text{mean}=85.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">Kristen&#8217;s median score is [latex]85.5[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146419\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146419&theme=oea&iframe_resize_id=ohm146419&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The next video shows another example of how to find hte median of a set of numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Find the Median of a Data Set\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/CbKqFc-EPDs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-10383\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146419, 146418. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Find the Median of a Data Set. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/CbKqFc-EPDs\">https:\/\/youtu.be\/CbKqFc-EPDs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul 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