{"id":850,"date":"2023-03-20T19:17:28","date_gmt":"2023-03-20T19:17:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/interpreting-the-mean-and-median-of-a-data-set-background-youll-need-1\/"},"modified":"2025-05-11T22:30:11","modified_gmt":"2025-05-11T22:30:11","slug":"interpreting-the-mean-and-median-of-a-data-set-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/interpreting-the-mean-and-median-of-a-data-set-background-youll-need-1\/","title":{"raw":"Describing Data Numerically: Background You'll Need 3","rendered":"Describing Data Numerically: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the median of a data set by hand&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4608,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate the median of a data set by hand<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>This support activity will give you more practice calculating mean and median, including how to interpret comparisons of mean and median.<\/p>\r\n<section class=\"textbox recall\">Before you begin, recall the definitions of mean and median.[reveal-answer q=\"702003\"]Recall the definition of the median of a set of values.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"702003\"]The <strong>median<\/strong> is the middle-most value after placing all the values in numerical order. For an even-numbered set of values, the median will be the average of the two middle values.[\/hidden-answer]<\/section>\r\n<p>In this activity, we'll be using the two data sets listed below.<\/p>\r\n<p>Suppose that the first data set lists the monthly salaries (in thousands of dollars) for all six employees at a company during the month of January. For example, Employee [latex]1[\/latex] made [latex]\\$4,000[\/latex] in salary in January. Employee [latex]2[\/latex] made [latex]\\$6,000[\/latex], and so on. We'll consider this amount the regular salary per month for each of these employees.<\/p>\r\n<p>The second data set lists the monthly salaries (in thousands of dollars) for the same six employees during the month of February.<\/p>\r\n<p>Can you locate which employee got the raise?<\/p>\r\n<div style=\"text-align: left;\" align=\"center\">\r\n<table style=\"height: 149px; width: 567px;\">\r\n<tbody>\r\n<tr style=\"height: 65px;\">\r\n<td style=\"text-align: center; height: 65px; width: 104.289px;\"><strong>Employee<\/strong><\/td>\r\n<td style=\"height: 65px; width: 131.336px;\">\r\n<p style=\"text-align: center;\"><strong>Monthly Salary in January<\/strong><\/p>\r\n<p style=\"text-align: center;\"><strong>(in thousands of dollars)<\/strong><\/p>\r\n<\/td>\r\n<td style=\"height: 131px; width: 142.375px;\">\r\n<p style=\"text-align: center;\"><strong>Monthly Salary in February<\/strong><\/p>\r\n<p style=\"text-align: center;\"><strong>(in thousands of dollars)<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 1<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 2<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 3<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 4<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 5<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 6<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2058[\/ohm2_question]<\/section>\r\n<\/div>\r\n<h2 id=\"CalcMedian\">Calculating Median<\/h2>\r\n<p>Now, let's look at the first data set only: The monthly salaries (in thousands of dollars) for all six employees at a company during the month of <strong>January<\/strong>.<\/p>\r\n<section class=\"textbox recall\">To answer the question below, you'll need to calculate the median of a data set containing an even number of values. You can refresh yourself on that information here if needed.[reveal-answer q=\"575947\"]Calculate the median of an even-numbered set of values.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"575947\"]When there are an even number of values, the median is the mean of the middle two values. Example: Consider the set [latex]1, 2, 3, 4[\/latex].To find the median, we want to find the \"middle-most\" number. If you imagine these numbers placed on a number line, where would the \"middle-most\" location of the set be?\r\n\r\n<p style=\"text-align: center;\">[latex]1 \\qquad2 \\qquad3 \\qquad4 \\qquad[\/latex]<\/p>\r\n<p>Certainly, it must fall evenly between the\u00a0[latex]2[\/latex] and the\u00a0[latex]3[\/latex].<\/p>\r\n<p>What number is halfway between\u00a0[latex]2[\/latex] and\u00a0[latex]3[\/latex]? It would be either [latex]2+\\frac{1}{2}[\/latex] or [latex]3-\\frac{1}{2}[\/latex]. Either way, that's\u00a0[latex]2.5[\/latex].<\/p>\r\n<p>Let's verify the process to find the median of a data set with an even number of values.<\/p>\r\n<p>The middle two numbers are [latex]2[\/latex] and [latex]3[\/latex]. Let's find their mean.<\/p>\r\n<p>[latex]\\dfrac{2+3}{2}=\\dfrac{5}{2}=2.5[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2059[\/ohm2_question]<\/section>\r\n<p>Let's use technology to verify the result you obtained for the median above.<\/p>\r\n<section class=\"textbox interact\">Go to the Describing and Exploring Quantitative Variables tool below and confirm your answer using the statistical tool.\r\n\r\n<p style=\"padding-left: 30px;\"><strong>Step 1:<\/strong>\u00a0Select the <strong>Single Group<\/strong> tab.<\/p>\r\n<p style=\"padding-left: 30px;\"><strong>Step 2:<\/strong>\u00a0Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>Your Own<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\"><strong>Step 3:<\/strong>\u00a0Under\u00a0<strong>Do you have,\u00a0<\/strong>select\u00a0<strong>Individual Observations<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\"><strong>Step 4:<\/strong>\u00a0Under <strong>Name of Variable<\/strong>, type \"January Salaries (in thousands $).\"<\/p>\r\n<p style=\"padding-left: 30px;\">Under <strong>Enter observations<\/strong>, enter the data list, separated by spaces: \u201c4 6 3 5 6 3.\u201d The median will be among the Descriptive Statistics listed in the tool.<\/p>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<p>How did you do? Did your calculation match the one in the tool? Now, consider what the median implies about the data. Remember that we think of the median as the [latex]50^{th}[\/latex] percentile.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2060[\/ohm2_question]\u200b<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the median of a data set by hand&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4608,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate the median of a data set by hand<\/span><\/li>\n<\/ul>\n<\/section>\n<p>This support activity will give you more practice calculating mean and median, including how to interpret comparisons of mean and median.<\/p>\n<section class=\"textbox recall\">Before you begin, recall the definitions of mean and median.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q702003\">Recall the definition of the median of a set of values.<\/button><\/p>\n<div id=\"q702003\" class=\"hidden-answer\" style=\"display: none\">The <strong>median<\/strong> is the middle-most value after placing all the values in numerical order. For an even-numbered set of values, the median will be the average of the two middle values.<\/div>\n<\/div>\n<\/section>\n<p>In this activity, we&#8217;ll be using the two data sets listed below.<\/p>\n<p>Suppose that the first data set lists the monthly salaries (in thousands of dollars) for all six employees at a company during the month of January. For example, Employee [latex]1[\/latex] made [latex]\\$4,000[\/latex] in salary in January. Employee [latex]2[\/latex] made [latex]\\$6,000[\/latex], and so on. We&#8217;ll consider this amount the regular salary per month for each of these employees.<\/p>\n<p>The second data set lists the monthly salaries (in thousands of dollars) for the same six employees during the month of February.<\/p>\n<p>Can you locate which employee got the raise?<\/p>\n<div style=\"text-align: left; margin: auto;\">\n<table style=\"height: 149px; width: 567px;\">\n<tbody>\n<tr style=\"height: 65px;\">\n<td style=\"text-align: center; height: 65px; width: 104.289px;\"><strong>Employee<\/strong><\/td>\n<td style=\"height: 65px; width: 131.336px;\">\n<p style=\"text-align: center;\"><strong>Monthly Salary in January<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>(in thousands of dollars)<\/strong><\/p>\n<\/td>\n<td style=\"height: 131px; width: 142.375px;\">\n<p style=\"text-align: center;\"><strong>Monthly Salary in February<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>(in thousands of dollars)<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 1<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 2<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 3<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 4<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]5[\/latex]<\/td>\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 5<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 104.289px;\"><strong>Employee 6<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 131.336px;\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center; height: 14px; width: 142.375px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2058\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2058&theme=lumen&iframe_resize_id=ohm2058&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<h2 id=\"CalcMedian\">Calculating Median<\/h2>\n<p>Now, let&#8217;s look at the first data set only: The monthly salaries (in thousands of dollars) for all six employees at a company during the month of <strong>January<\/strong>.<\/p>\n<section class=\"textbox recall\">To answer the question below, you&#8217;ll need to calculate the median of a data set containing an even number of values. You can refresh yourself on that information here if needed.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q575947\">Calculate the median of an even-numbered set of values.<\/button><\/p>\n<div id=\"q575947\" class=\"hidden-answer\" style=\"display: none\">When there are an even number of values, the median is the mean of the middle two values. Example: Consider the set [latex]1, 2, 3, 4[\/latex].To find the median, we want to find the &#8220;middle-most&#8221; number. If you imagine these numbers placed on a number line, where would the &#8220;middle-most&#8221; location of the set be?<\/p>\n<p style=\"text-align: center;\">[latex]1 \\qquad2 \\qquad3 \\qquad4 \\qquad[\/latex]<\/p>\n<p>Certainly, it must fall evenly between the\u00a0[latex]2[\/latex] and the\u00a0[latex]3[\/latex].<\/p>\n<p>What number is halfway between\u00a0[latex]2[\/latex] and\u00a0[latex]3[\/latex]? It would be either [latex]2+\\frac{1}{2}[\/latex] or [latex]3-\\frac{1}{2}[\/latex]. Either way, that&#8217;s\u00a0[latex]2.5[\/latex].<\/p>\n<p>Let&#8217;s verify the process to find the median of a data set with an even number of values.<\/p>\n<p>The middle two numbers are [latex]2[\/latex] and [latex]3[\/latex]. Let&#8217;s find their mean.<\/p>\n<p>[latex]\\dfrac{2+3}{2}=\\dfrac{5}{2}=2.5[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2059\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2059&theme=lumen&iframe_resize_id=ohm2059&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Let&#8217;s use technology to verify the result you obtained for the median above.<\/p>\n<section class=\"textbox interact\">Go to the Describing and Exploring Quantitative Variables tool below and confirm your answer using the statistical tool.<\/p>\n<p style=\"padding-left: 30px;\"><strong>Step 1:<\/strong>\u00a0Select the <strong>Single Group<\/strong> tab.<\/p>\n<p style=\"padding-left: 30px;\"><strong>Step 2:<\/strong>\u00a0Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>Your Own<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\"><strong>Step 3:<\/strong>\u00a0Under\u00a0<strong>Do you have,\u00a0<\/strong>select\u00a0<strong>Individual Observations<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\"><strong>Step 4:<\/strong>\u00a0Under <strong>Name of Variable<\/strong>, type &#8220;January Salaries (in thousands $).&#8221;<\/p>\n<p style=\"padding-left: 30px;\">Under <strong>Enter observations<\/strong>, enter the data list, separated by spaces: \u201c4 6 3 5 6 3.\u201d The median will be among the Descriptive Statistics listed in the tool.<\/p>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<p>How did you do? Did your calculation match the one in the tool? Now, consider what the median implies about the data. Remember that we think of the median as the [latex]50^{th}[\/latex] percentile.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2060\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2060&theme=lumen&iframe_resize_id=ohm2060&source=tnh\" width=\"100%\" height=\"150\"><\/iframe>\u200b<\/section>\n","protected":false},"author":13,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":834,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/850"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/850\/revisions"}],"predecessor-version":[{"id":6622,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/850\/revisions\/6622"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/834"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/850\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=850"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=850"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=850"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}