{"id":488,"date":"2021-12-20T14:49:29","date_gmt":"2021-12-20T14:49:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=488"},"modified":"2022-02-17T20:10:26","modified_gmt":"2022-02-17T20:10:26","slug":"corequisite-support-activity-4c","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/corequisite-support-activity-4c\/","title":{"raw":"Corequisite Support Activity for Interpreting the Mean and Median of a Dataset: 4C - 22","rendered":"Corequisite Support Activity for Interpreting the Mean and Median of a Dataset: 4C &#8211; 22"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>What you'll need to know<\/h3>\r\nIn this support activity you'll become familiar with the following:\r\n<ul>\r\n \t<li><a href=\"#CompareMeanMedian\">Compare and interpret the mean and median of different datasets.<\/a><\/li>\r\n<\/ul>\r\nYou will also have an opportunity to refresh the following skills:\r\n<ul>\r\n \t<li><a href=\"#name\">Recall the definitions of mean and median.<\/a><\/li>\r\n \t<li><a href=\"#CalcMedian\">Calculate the median of a dataset by hand.<\/a><\/li>\r\n \t<li><a href=\"#CalcMean\">Calculate the mean of a dataset by hand.<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the next section of the course material and in the following activity, you will need to compare the mean and median of a quantitative variable and calculate mean and median by hand and using technology. This support activity will give you more practice calculating mean and median and will set the stage for interpreting the comparisons of mean and median in the upcoming section.\r\n\r\nIn\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/what-to-know-about-4a\/\"><em>What to Know About Calculating Mean and Median of a Dataset: 4A<\/em><\/a>, you had a chance to calculate the mean and median of a small dataset by hand. As you work through this activity, return to that section as needed to refresh the formula and the process.\r\n<div class=\"textbox examples\">\r\n\r\nBefore you begin, recall the definitions of mean and median.\r\n\r\nCore skill:\u00a0[reveal-answer q=\"574698\"]Recall the definition of the mean of a set of values.[\/reveal-answer]\r\n[hidden-answer a=\"574698\"]\r\n\r\nThe <strong>mean<\/strong> is what we think of as the \"average\" of a set of numbers. It is the sum of the entire set divided by the number of values present.[\/hidden-answer]\r\n\r\nCore skill:\r\n[reveal-answer q=\"702003\"]Recall the definition of the median of a set of values.[\/reveal-answer]\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]\r\n\r\n<\/div>\r\n<h2>Salaries<\/h2>\r\nIn this activity, we'll be using the two datasets listed below. Suppose that the first dataset lists the monthly salaries (in thousands of dollars) for all six employees at a company during the month of January.\r\n<div align=\"center\">\r\n<table style=\"height: 149px;\">\r\n<tbody>\r\n<tr style=\"height: 65px;\">\r\n<td style=\"text-align: center; height: 65px; width: 101.766px;\"><strong>Employee<\/strong><\/td>\r\n<td style=\"height: 65px; width: 288.234px;\">\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<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 1<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 2<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 3<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 4<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 5<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 6<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nThen, suppose that one of the employees gets a large raise in February. The second dataset lists the monthly salaries (in thousands of dollars) for the same six employees during the month of February.\r\n<div align=\"center\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Employee<\/strong><\/td>\r\n<td>\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>\r\n<td style=\"text-align: center;\"><strong>Employee 1<\/strong><\/td>\r\n<td style=\"text-align: center;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Employee 2<\/strong><\/td>\r\n<td style=\"text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Employee 3<\/strong><\/td>\r\n<td style=\"text-align: center;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Employee 4<\/strong><\/td>\r\n<td style=\"text-align: center;\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Employee 5<\/strong><\/td>\r\n<td style=\"text-align: center;\">6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Employee 6<\/strong><\/td>\r\n<td style=\"text-align: center;\">3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\nCompare and contrast the two datasets: how are they similar and how are they different?\r\n\r\n[reveal-answer q=\"602799\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"602799\"]Which employee received the raise in February?[\/hidden-answer]\r\n\r\n<\/div>\r\nNow consider just the dataset of employee salaries from January. We'd like to know the median salary for the six employees for the month of January.\r\n<h3 id=\"CalcMedian\">Calculate the median of a dataset by hand<\/h3>\r\n<div class=\"textbox examples\">\r\n<h3>Recall<\/h3>\r\nTo answer the question below, you'll need to calculate the median of a dataset containing an even number of values. You can refresh that information here if needed.\r\n\r\nCore skill:\u00a0[reveal-answer q=\"575947\"]Calculate the median of an even-numbered set of values.[\/reveal-answer]\r\n[hidden-answer a=\"575947\"]\r\n\r\nWhen there are an even number of values, the median is the mean of the middle two values.\r\n\r\nEx. Consider the set 1, 2, 3 4.\r\n\r\nTo 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<p style=\"text-align: center;\">[latex]1 \\qquad2 \\qquad3 \\qquad4 \\qquad[\/latex]<\/p>\r\nCertainly, it must fall evenly between the 2 and the 3.\r\n\r\nWhat number is halfway between 2 and 3? It would be either [latex]2+\\frac{1}{2}[\/latex] or [latex]3-\\frac{1}{2}[\/latex]. Either way, that's 2.5.\r\n\r\nLet's verify that using the process to find the median of a dataset with an even number of values.\r\n\r\nThe middle two numbers are 2 and 3. Let's take their mean.\r\n\r\n[latex]\\dfrac{2+3}{2}=\\dfrac{5}{2}=2.5[\/latex]\r\n\r\nNow you try it by answering Question 2 below.\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\nBy hand or with a regular calculator (without using the median function), calculate the median employee salary in January (in dollars).\r\n\r\n[reveal-answer q=\"347645\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"347645\"]See the Recall box above to refresh the process. Don't forget to interpret the value in context, using units.[\/hidden-answer]\r\n\r\n<\/div>\r\nLet's use technology to verify the result you obtained for the median in Question 2.\r\n<div class=\"textbox\">\r\n\r\nGo to the Describing and Exploring Quantitative Variables tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a> and confirm your answer using the online tool.\r\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Single Group<\/strong> tab.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 2) Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>Your Own<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 3) Under\u00a0<strong>Do you have,\u00a0<\/strong>select\u00a0<strong>Individual Observations<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 4) Under <strong>Name of Variable<\/strong>, type \"January Salaries (in thousands $)\".<\/p>\r\n<p style=\"padding-left: 30px;\">Step 5) 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\r\n<\/div>\r\nHow did you do? Did your calculation match the one in the tool? Consider now, what the median implies about the data.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\nWhat does the median tell you about the salaries of the employees in January?\r\n\r\n[reveal-answer q=\"854456\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"854456\"]Remember that the median is the 50th percentile, and splits the data in half.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"CalcMean\">\u00a0Calculate the mean of a dataset by hand<\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\nBy hand or with a regular calculator (without using the mean function), calculate the mean employee salary in January (in dollars).\r\n\r\n[reveal-answer q=\"50836\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"50836\"]Recall the process for finding the mean. See the Recall box at the top of this page or refer to <em>What to Know About Calculating Mean and Median: 4A<\/em>.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\nConfirm your answer using the online tool.\r\n\r\n[reveal-answer q=\"180400\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"180400\"]The mean will be among the Descriptive Statistics listed in the tool[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\nWhat does the mean tell you about the salaries of the employees in January?\r\n\r\n[reveal-answer q=\"129961\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"129961\"]Consider what it means for a value to be an \"average\" value for the dataset.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\nThe median and mean employee salaries for January are the same. Why is this?\r\n\r\n[reveal-answer q=\"68470\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"68470\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n<\/div>\r\n[reveal-answer q=\"695453\"]What do the mean and median tells us about the data? What can we understand about the median and mean employee salaries for January being the same?[\/reveal-answer]\r\n[hidden-answer a=\"695453\"]The median tells us that half the employees made more than $4,500 in January and half made less.\r\n\r\nThe mean tells us that if the January salaries had been added up and evenly distributed across all six employees, each person would have received $4,500.\r\n\r\nIt was interesting that the mean and the median were identical values. This tells us that the the salaries are evenly distributed among high and low values; the distribution is symmetrical, without skew.\u00a0 But what happens if we change one of the values in the dataset? Let's move on to questions 8 - 10 to find out.[\/hidden-answer]\r\n<h3 id=\"CompareMeanMedian\">Compare the mean and median of different datasets<\/h3>\r\nNow consider the dataset of employee salaries from February. First we'll calculate the median of this set, then consider how we might expect the mean of the February salaries compares to the mean of the January salaries.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\nWhat is the median employee salary in February (in dollars)? You may use either the online tool or calculate by hand.\r\n\r\n[reveal-answer q=\"288591\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"288591\"]Remember that one of the salaries changed when an employee received a big raise in February.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\nWithout calculating the mean employee salary in February, what do you predict will be true about the February mean compared to the January mean and why?\r\n\r\n[reveal-answer q=\"100886\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"100886\"]What do <em>you<\/em> think? Consider what changed in the set of salaries from January to February. Is the total higher or lower than it was in January?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\nNow, calculate the mean employee salary in February (in dollars). You may use either the online tool or calculate by hand.\r\n\r\n[reveal-answer q=\"670906\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"670906\"]See the processes listed above.[\/hidden-answer]\r\n\r\n<\/div>\r\n[reveal-answer q=\"476661\"]Was the mean you calculated for February salaries higher, lower, or similar? Here's a way to think about that.[\/reveal-answer]\r\n[hidden-answer a=\"476661\"]The mean is now higher than the median. What caused that? Would that always happen if a data value increases?\r\n\r\nWhy did the median stay the same? Would the median always be roughly the same if a data value changes?\r\n\r\n<hr \/>\r\n\r\n[\/hidden-answer]\r\n\r\nNow consider a slightly different question.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\nSuppose that in March, Employee 6 quits and the company takes on an unpaid intern. How will the median and mean change from February to March?\r\n\r\n[reveal-answer q=\"284377\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"284377\"]Would there still be six data values (employee salary) in the set?[\/hidden-answer]\r\n\r\n<\/div>\r\nIt may take some time before you really feel comfortable interpreting means and medians and understanding what they imply about a dataset. A key idea to take from this activity is that, while the median stays relatively fixed in a dataset if one value changes by a large amount, the mean does not. This tells us that the mean is sensitive to the presence of extreme values in the dataset.\r\n\r\nIt's okay if you need more practice to process the sensitivity of the mean. But if you feel comfortable calculating the means and medians in this activity by hand and using technology, please move on to the next section and activity.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>What you&#8217;ll need to know<\/h3>\n<p>In this support activity you&#8217;ll become familiar with the following:<\/p>\n<ul>\n<li><a href=\"#CompareMeanMedian\">Compare and interpret the mean and median of different datasets.<\/a><\/li>\n<\/ul>\n<p>You will also have an opportunity to refresh the following skills:<\/p>\n<ul>\n<li><a href=\"#name\">Recall the definitions of mean and median.<\/a><\/li>\n<li><a href=\"#CalcMedian\">Calculate the median of a dataset by hand.<\/a><\/li>\n<li><a href=\"#CalcMean\">Calculate the mean of a dataset by hand.<\/a><\/li>\n<\/ul>\n<\/div>\n<p>In the next section of the course material and in the following activity, you will need to compare the mean and median of a quantitative variable and calculate mean and median by hand and using technology. This support activity will give you more practice calculating mean and median and will set the stage for interpreting the comparisons of mean and median in the upcoming section.<\/p>\n<p>In\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/what-to-know-about-4a\/\"><em>What to Know About Calculating Mean and Median of a Dataset: 4A<\/em><\/a>, you had a chance to calculate the mean and median of a small dataset by hand. As you work through this activity, return to that section as needed to refresh the formula and the process.<\/p>\n<div class=\"textbox examples\">\n<p>Before you begin, recall the definitions of mean and median.<\/p>\n<p>Core skill:\u00a0<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q574698\">Recall the definition of the mean of a set of values.<\/span><\/p>\n<div id=\"q574698\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <strong>mean<\/strong> is what we think of as the &#8220;average&#8221; of a set of numbers. It is the sum of the entire set divided by the number of values present.<\/div>\n<\/div>\n<p>Core skill:<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q702003\">Recall the definition of the median of a set of values.<\/span><\/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<\/div>\n<h2>Salaries<\/h2>\n<p>In this activity, we&#8217;ll be using the two datasets listed below. Suppose that the first dataset lists the monthly salaries (in thousands of dollars) for all six employees at a company during the month of January.<\/p>\n<div style=\"margin: auto;\">\n<table style=\"height: 149px;\">\n<tbody>\n<tr style=\"height: 65px;\">\n<td style=\"text-align: center; height: 65px; width: 101.766px;\"><strong>Employee<\/strong><\/td>\n<td style=\"height: 65px; width: 288.234px;\">\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<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 1<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 2<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 3<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">3<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 4<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">5<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 5<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 101.766px;\"><strong>Employee 6<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 288.234px;\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Then, suppose that one of the employees gets a large raise in February. The second dataset lists the monthly salaries (in thousands of dollars) for the same six employees during the month of February.<\/p>\n<div style=\"margin: auto;\">\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Employee<\/strong><\/td>\n<td>\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>\n<td style=\"text-align: center;\"><strong>Employee 1<\/strong><\/td>\n<td style=\"text-align: center;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Employee 2<\/strong><\/td>\n<td style=\"text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Employee 3<\/strong><\/td>\n<td style=\"text-align: center;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Employee 4<\/strong><\/td>\n<td style=\"text-align: center;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Employee 5<\/strong><\/td>\n<td style=\"text-align: center;\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Employee 6<\/strong><\/td>\n<td style=\"text-align: center;\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p>Compare and contrast the two datasets: how are they similar and how are they different?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q602799\">Hint<\/span><\/p>\n<div id=\"q602799\" class=\"hidden-answer\" style=\"display: none\">Which employee received the raise in February?<\/div>\n<\/div>\n<\/div>\n<p>Now consider just the dataset of employee salaries from January. We&#8217;d like to know the median salary for the six employees for the month of January.<\/p>\n<h3 id=\"CalcMedian\">Calculate the median of a dataset by hand<\/h3>\n<div class=\"textbox examples\">\n<h3>Recall<\/h3>\n<p>To answer the question below, you&#8217;ll need to calculate the median of a dataset containing an even number of values. You can refresh that information here if needed.<\/p>\n<p>Core skill:\u00a0<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q575947\">Calculate the median of an even-numbered set of values.<\/span><\/p>\n<div id=\"q575947\" class=\"hidden-answer\" style=\"display: none\">\n<p>When there are an even number of values, the median is the mean of the middle two values.<\/p>\n<p>Ex. Consider the set 1, 2, 3 4.<\/p>\n<p>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 2 and the 3.<\/p>\n<p>What number is halfway between 2 and 3? It would be either [latex]2+\\frac{1}{2}[\/latex] or [latex]3-\\frac{1}{2}[\/latex]. Either way, that&#8217;s 2.5.<\/p>\n<p>Let&#8217;s verify that using the process to find the median of a dataset with an even number of values.<\/p>\n<p>The middle two numbers are 2 and 3. Let&#8217;s take their mean.<\/p>\n<p>[latex]\\dfrac{2+3}{2}=\\dfrac{5}{2}=2.5[\/latex]<\/p>\n<p>Now you try it by answering Question 2 below.<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p>By hand or with a regular calculator (without using the median function), calculate the median employee salary in January (in dollars).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q347645\">Hint<\/span><\/p>\n<div id=\"q347645\" class=\"hidden-answer\" style=\"display: none\">See the Recall box above to refresh the process. Don&#8217;t forget to interpret the value in context, using units.<\/div>\n<\/div>\n<\/div>\n<p>Let&#8217;s use technology to verify the result you obtained for the median in Question 2.<\/p>\n<div class=\"textbox\">\n<p>Go to the Describing and Exploring Quantitative Variables tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a> and confirm your answer using the online tool.<\/p>\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Single Group<\/strong> tab.<\/p>\n<p style=\"padding-left: 30px;\">Step 2) Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>Your Own<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 3) Under\u00a0<strong>Do you have,\u00a0<\/strong>select\u00a0<strong>Individual Observations<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 4) Under <strong>Name of Variable<\/strong>, type &#8220;January Salaries (in thousands $)&#8221;.<\/p>\n<p style=\"padding-left: 30px;\">Step 5) 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<\/div>\n<p>How did you do? Did your calculation match the one in the tool? Consider now, what the median implies about the data.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p>What does the median tell you about the salaries of the employees in January?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q854456\">Hint<\/span><\/p>\n<div id=\"q854456\" class=\"hidden-answer\" style=\"display: none\">Remember that the median is the 50th percentile, and splits the data in half.<\/div>\n<\/div>\n<\/div>\n<h3 id=\"CalcMean\">\u00a0Calculate the mean of a dataset by hand<\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p>By hand or with a regular calculator (without using the mean function), calculate the mean employee salary in January (in dollars).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q50836\">Hint<\/span><\/p>\n<div id=\"q50836\" class=\"hidden-answer\" style=\"display: none\">Recall the process for finding the mean. See the Recall box at the top of this page or refer to <em>What to Know About Calculating Mean and Median: 4A<\/em>.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p>Confirm your answer using the online tool.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q180400\">Hint<\/span><\/p>\n<div id=\"q180400\" class=\"hidden-answer\" style=\"display: none\">The mean will be among the Descriptive Statistics listed in the tool<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p>What does the mean tell you about the salaries of the employees in January?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q129961\">Hint<\/span><\/p>\n<div id=\"q129961\" class=\"hidden-answer\" style=\"display: none\">Consider what it means for a value to be an &#8220;average&#8221; value for the dataset.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p>The median and mean employee salaries for January are the same. Why is this?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q68470\">Hint<\/span><\/p>\n<div id=\"q68470\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q695453\">What do the mean and median tells us about the data? What can we understand about the median and mean employee salaries for January being the same?<\/span><\/p>\n<div id=\"q695453\" class=\"hidden-answer\" style=\"display: none\">The median tells us that half the employees made more than $4,500 in January and half made less.<\/p>\n<p>The mean tells us that if the January salaries had been added up and evenly distributed across all six employees, each person would have received $4,500.<\/p>\n<p>It was interesting that the mean and the median were identical values. This tells us that the the salaries are evenly distributed among high and low values; the distribution is symmetrical, without skew.\u00a0 But what happens if we change one of the values in the dataset? Let&#8217;s move on to questions 8 &#8211; 10 to find out.<\/p><\/div>\n<\/div>\n<h3 id=\"CompareMeanMedian\">Compare the mean and median of different datasets<\/h3>\n<p>Now consider the dataset of employee salaries from February. First we&#8217;ll calculate the median of this set, then consider how we might expect the mean of the February salaries compares to the mean of the January salaries.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p>What is the median employee salary in February (in dollars)? You may use either the online tool or calculate by hand.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q288591\">Hint<\/span><\/p>\n<div id=\"q288591\" class=\"hidden-answer\" style=\"display: none\">Remember that one of the salaries changed when an employee received a big raise in February.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p>Without calculating the mean employee salary in February, what do you predict will be true about the February mean compared to the January mean and why?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q100886\">Hint<\/span><\/p>\n<div id=\"q100886\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think? Consider what changed in the set of salaries from January to February. Is the total higher or lower than it was in January?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p>Now, calculate the mean employee salary in February (in dollars). You may use either the online tool or calculate by hand.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q670906\">Hint<\/span><\/p>\n<div id=\"q670906\" class=\"hidden-answer\" style=\"display: none\">See the processes listed above.<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q476661\">Was the mean you calculated for February salaries higher, lower, or similar? Here&#8217;s a way to think about that.<\/span><\/p>\n<div id=\"q476661\" class=\"hidden-answer\" style=\"display: none\">The mean is now higher than the median. What caused that? Would that always happen if a data value increases?<\/p>\n<p>Why did the median stay the same? Would the median always be roughly the same if a data value changes?<\/p>\n<hr \/>\n<\/div>\n<\/div>\n<p>Now consider a slightly different question.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p>Suppose that in March, Employee 6 quits and the company takes on an unpaid intern. How will the median and mean change from February to March?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q284377\">Hint<\/span><\/p>\n<div id=\"q284377\" class=\"hidden-answer\" style=\"display: none\">Would there still be six data values (employee salary) in the set?<\/div>\n<\/div>\n<\/div>\n<p>It may take some time before you really feel comfortable interpreting means and medians and understanding what they imply about a dataset. A key idea to take from this activity is that, while the median stays relatively fixed in a dataset if one value changes by a large amount, the mean does not. This tells us that the mean is sensitive to the presence of extreme values in the dataset.<\/p>\n<p>It&#8217;s okay if you need more practice to process the sensitivity of the mean. But if you feel comfortable calculating the means and medians in this activity by hand and using technology, please move on to the next section and activity.<\/p>\n","protected":false},"author":25777,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-488","chapter","type-chapter","status-publish","hentry"],"part":621,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/488","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":35,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/488\/revisions"}],"predecessor-version":[{"id":3313,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/488\/revisions\/3313"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/621"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/488\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=488"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=488"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=488"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=488"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}