{"id":339,"date":"2022-02-21T17:52:51","date_gmt":"2022-02-21T17:52:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&#038;p=339"},"modified":"2022-05-20T16:46:35","modified_gmt":"2022-05-20T16:46:35","slug":"interpreting-the-mean-and-median-of-a-dataset-corequisite-support-activity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/interpreting-the-mean-and-median-of-a-dataset-corequisite-support-activity\/","title":{"raw":"Interpreting the Mean and Median of a Data Set: Corequisite Support Activity","rendered":"Interpreting the Mean and Median of a Data Set: Corequisite Support Activity"},"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 data sets.<\/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 data set by hand.<\/a><\/li>\r\n \t<li><a href=\"#CalcMean\">Calculate the mean of a data set 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>Calculating Mean and Median of a Data Set: What to Know<\/em><\/a>, you had a chance to calculate the mean and median of a small data set 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<h3>Recall<\/h3>\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 data sets listed below. 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.\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;\">[latex]4[\/latex]<\/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;\">[latex]6[\/latex]<\/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;\">[latex]3[\/latex]<\/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;\">[latex]5[\/latex]<\/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;\">[latex]6[\/latex]<\/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;\">[latex]3[\/latex]<\/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 data set lists the monthly salaries (in thousands of dollars) for the same six employees during the month of February. Can you locate which employee got the raise?\r\n<div align=\"center\">\r\n<table style=\"height: 215px;\">\r\n<tbody>\r\n<tr style=\"height: 131px;\">\r\n<td style=\"text-align: center; height: 131px; width: 102.625px;\"><strong>Employee<\/strong><\/td>\r\n<td style=\"height: 131px; width: 287.375px;\">\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: 102.625px;\"><strong>Employee 1<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 2<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 3<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 4<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 5<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 6<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]3[\/latex]<\/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\n[ohm_question hide_question_numbers=1]241070[\/ohm_question]\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 data set 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\">Calculating Median<\/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 data set 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\u00a0[latex]1, 2, 3 4[\/latex].\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\u00a0[latex]2[\/latex] and the\u00a0[latex]3[\/latex].\r\n\r\nWhat 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].\r\n\r\nLet's verify that using the process to find the median of a data set with an even number of values.\r\n\r\nThe middle two numbers are\u00a0[latex]2[\/latex] and\u00a0[latex]3[\/latex]. 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\n[ohm_question hide_question_numbers=1]241071[\/ohm_question]\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? Now consider what the median implies about the data. Remember that we think of the median as the 50th percentile.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\n[ohm_question hide_question_numbers=1]241072[\/ohm_question]\r\n\r\n[reveal-answer q=\"854456\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"854456\"]Remember that the median is the\u00a0[latex]50[\/latex]<sup>th<\/sup> percentile, and splits the data in half.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"CalcMean\">\u00a0Calculating Mean<\/h3>\r\n<div class=\"textbox examples\">\r\n<h3>Recall<\/h3>\r\nDo you recall how to calculate the mean of a data set?\r\n\r\n[reveal-answer q=\"330489\"]Core Skill: Calculating a Mean[\/reveal-answer]\r\n[hidden-answer a=\"330489\"]\r\n\r\nThe formula to calculate the mean is [latex]\\bar{x}=\\dfrac{\\sum{x}}{n}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\n[ohm_question hide_question_numbers=1]241073[\/ohm_question]\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>Calculating Mean and Median: What to Know<\/em>.[\/hidden-answer]\r\n\r\n<\/div>\r\nRepeat the steps above to locate the mean in the tool under Descriptive Statistics then compare your answer to the technology calculation. Did they match?\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\n[ohm_question hide_question_numbers=1]241076[\/ohm_question]\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\nNow, consider what the mean implies about the data. See the recall box at the top of the page for a hint.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]241078[\/ohm_question]\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 data set.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]241084[\/ohm_question]\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\nWe saw that the median and the mean employee salaries for January were the same. What understanding can we take from that information?\r\n<div class=\"textbox exercises\">\r\n<h3>Interactive Example<\/h3>\r\nWhat can we understand about the median and mean employee salaries for January being the same? Fill in the blanks to answer the following questions.\r\n<ol>\r\n \t<li>The median of the data set implies that ____________ made more than\u00a0[latex]\\$4,500[\/latex] in January and _________ made less.<\/li>\r\n \t<li>The mean of the data set implies that if the January salaries had been added up and evenly distributed across all six employees, each person would have received ________________.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"968607\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"968607\"]\r\n<ol>\r\n \t<li>Half the employees made more than\u00a0[latex]\\$4,500[\/latex] and half made less.<\/li>\r\n \t<li>Each person would have received\u00a0[latex]\\$4,500[\/latex]. That is, the average salary was\u00a0[latex]\\$4,500[\/latex] for January.<\/li>\r\n<\/ol>\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 data set? Let's move on to questions 8 - 10 to find out.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 id=\"CompareMeanMedian\">Comparing Mean and Median<\/h3>\r\nWhat happens to the mean and median if we change one of the values in the data set?\r\n\r\nRecall that the data set of employee salaries from February includes a big raise from one employee. First calculate the median of this set to answer Question 8 below, then consider how we might expect the mean of the February salaries compares to the mean of the January salaries.\r\n\r\n[reveal-answer q=\"845599\"]Here is the February salary table again for convenience.[\/reveal-answer]\r\n[hidden-answer a=\"845599\"]\r\n<table style=\"height: 215px;\">\r\n<tbody>\r\n<tr style=\"height: 131px;\">\r\n<td style=\"text-align: center; height: 131px; width: 102.625px;\"><strong>Employee<\/strong><\/td>\r\n<td style=\"height: 131px; width: 287.375px;\">\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: 102.625px;\"><strong>Employee 1<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 2<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 3<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 4<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 5<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 6<\/strong><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nBefore you get the median and the mean from the technology, or before you calculate the mean by hand, first think about what you think will be true about the February mean compared to the January mean and why.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\n[ohm_question hide_question_numbers=1]241079[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241080[\/ohm_question]\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\n[ohm_question hide_question_numbers=1]241082[\/ohm_question]\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<div class=\"textbox exercises\">\r\n<h3>Interactive example<\/h3>\r\nWas the mean you calculated for February salaries higher, lower, or similar? What do you think caused that to be true? Click below for a discussion\u00a0<strong>after<\/strong> you enter your answers to Questions 8 - 10.\r\n\r\n[reveal-answer q=\"476661\"]Here are a couple of good questions to ask about it.[\/reveal-answer]\r\n[hidden-answer a=\"476661\"]The mean is now higher than the median. They were identical in January.\r\n<ul>\r\n \t<li>Did the increase in one salary cause the mean to rise?<\/li>\r\n \t<li>Would that always happen if a data value increases?<\/li>\r\n \t<li>How could we predict mathematically how much the mean would increase under the increase of a single value?\r\n<ul>\r\n \t<li>We could predict the increase in mean mathematically by taking the difference in the January salary and the February salary then distributing that difference out evenly among the employees.<\/li>\r\n \t<li>Ex. One salary increased by [latex]$2,000[\/latex]. If we divide the\u00a0[latex]$2,000[\/latex] across all six employees, we'll have the amount by which the new mean is higher.\r\n<ul>\r\n \t<li>[latex]\\dfrac{$2,000}{6}=\\$333.33[\/latex]<\/li>\r\n \t<li>For January, [latex]\\bar{x}=$4,500[\/latex] and for February,\u00a0[latex]\\bar{x}=$4,833.33[\/latex]. The mean increased by $333.33.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nBut why did the median stay the same? Would the median always be roughly the same if a data value changes?\r\n<ul>\r\n \t<li>If the middle-most number or two numbers didn't change, the median won't change.<\/li>\r\n \t<li>What would happen though, if instead of Employee 2 receiving the raise, Employee 1 had received it instead? What would the new median be?\r\n<ul>\r\n \t<li>The January median of the data set [latex]3, 3, 4, 5, 6, 6[\/latex] is the mean of [latex]4[\/latex] and [latex]5[\/latex] in thousands of dollars, or [latex]\\$4,500[\/latex]<\/li>\r\n \t<li>Changing one of the salaries from [latex]6[\/latex] thousand to [latex]8[\/latex] thousand didn't affect the middle two numbers.<\/li>\r\n \t<li>But changing the [latex]4[\/latex] to an [latex]8[\/latex] would require the reordering of the values.\r\n<ul>\r\n \t<li>[latex]3, 3, 5, 6, 6, 8[\/latex] now yields a median of [latex]5.5[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow let's consider a slightly different question.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\n[ohm_question hide_question_numbers=1]241083[\/ohm_question]\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 data set. A key idea to take from this activity is that, while the median stays relatively fixed in a data set 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 data set.\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 data sets.<\/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 data set by hand.<\/a><\/li>\n<li><a href=\"#CalcMean\">Calculate the mean of a data set 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>Calculating Mean and Median of a Data Set: What to Know<\/em><\/a>, you had a chance to calculate the mean and median of a small data set 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<h3>Recall<\/h3>\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 data sets listed below. 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<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;\">[latex]4[\/latex]<\/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;\">[latex]6[\/latex]<\/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;\">[latex]3[\/latex]<\/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;\">[latex]5[\/latex]<\/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;\">[latex]6[\/latex]<\/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;\">[latex]3[\/latex]<\/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 data set lists the monthly salaries (in thousands of dollars) for the same six employees during the month of February. Can you locate which employee got the raise?<\/p>\n<div style=\"margin: auto;\">\n<table style=\"height: 215px;\">\n<tbody>\n<tr style=\"height: 131px;\">\n<td style=\"text-align: center; height: 131px; width: 102.625px;\"><strong>Employee<\/strong><\/td>\n<td style=\"height: 131px; width: 287.375px;\">\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: 102.625px;\"><strong>Employee 1<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 2<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 3<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 4<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 5<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 6<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241070\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241070&theme=oea&iframe_resize_id=ohm241070\" width=\"100%\" height=\"150\"><\/iframe><\/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 data set 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\">Calculating Median<\/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 data set 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\u00a0[latex]1, 2, 3 4[\/latex].<\/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\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 that using the process to find the median of a data set with an even number of values.<\/p>\n<p>The middle two numbers are\u00a0[latex]2[\/latex] and\u00a0[latex]3[\/latex]. 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><iframe loading=\"lazy\" id=\"ohm241071\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241071&theme=oea&iframe_resize_id=ohm241071\" width=\"100%\" height=\"150\"><\/iframe><\/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? Now consider what the median implies about the data. Remember that we think of the median as the 50th percentile.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241072\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241072&theme=oea&iframe_resize_id=ohm241072\" width=\"100%\" height=\"150\"><\/iframe><\/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\u00a0[latex]50[\/latex]<sup>th<\/sup> percentile, and splits the data in half.<\/div>\n<\/div>\n<\/div>\n<h3 id=\"CalcMean\">\u00a0Calculating Mean<\/h3>\n<div class=\"textbox examples\">\n<h3>Recall<\/h3>\n<p>Do you recall how to calculate the mean of a data set?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q330489\">Core Skill: Calculating a Mean<\/span><\/p>\n<div id=\"q330489\" class=\"hidden-answer\" style=\"display: none\">\n<p>The formula to calculate the mean is [latex]\\bar{x}=\\dfrac{\\sum{x}}{n}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241073\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241073&theme=oea&iframe_resize_id=ohm241073\" width=\"100%\" height=\"150\"><\/iframe><\/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>Calculating Mean and Median: What to Know<\/em>.<\/div>\n<\/div>\n<\/div>\n<p>Repeat the steps above to locate the mean in the tool under Descriptive Statistics then compare your answer to the technology calculation. Did they match?<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241076\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241076&theme=oea&iframe_resize_id=ohm241076\" width=\"100%\" height=\"150\"><\/iframe><\/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<p>Now, consider what the mean implies about the data. See the recall box at the top of the page for a hint.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241078\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241078&theme=oea&iframe_resize_id=ohm241078\" width=\"100%\" height=\"150\"><\/iframe><\/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 data set.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241084\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241084&theme=oea&iframe_resize_id=ohm241084\" width=\"100%\" height=\"150\"><\/iframe><\/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<p>We saw that the median and the mean employee salaries for January were the same. What understanding can we take from that information?<\/p>\n<div class=\"textbox exercises\">\n<h3>Interactive Example<\/h3>\n<p>What can we understand about the median and mean employee salaries for January being the same? Fill in the blanks to answer the following questions.<\/p>\n<ol>\n<li>The median of the data set implies that ____________ made more than\u00a0[latex]\\$4,500[\/latex] in January and _________ made less.<\/li>\n<li>The mean of the data set implies that if the January salaries had been added up and evenly distributed across all six employees, each person would have received ________________.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q968607\">Show Answer<\/span><\/p>\n<div id=\"q968607\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Half the employees made more than\u00a0[latex]\\$4,500[\/latex] and half made less.<\/li>\n<li>Each person would have received\u00a0[latex]\\$4,500[\/latex]. That is, the average salary was\u00a0[latex]\\$4,500[\/latex] for January.<\/li>\n<\/ol>\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 data set? Let&#8217;s move on to questions 8 &#8211; 10 to find out.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3 id=\"CompareMeanMedian\">Comparing Mean and Median<\/h3>\n<p>What happens to the mean and median if we change one of the values in the data set?<\/p>\n<p>Recall that the data set of employee salaries from February includes a big raise from one employee. First calculate the median of this set to answer Question 8 below, then consider how we might expect the mean of the February salaries compares to the mean of the January salaries.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q845599\">Here is the February salary table again for convenience.<\/span><\/p>\n<div id=\"q845599\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"height: 215px;\">\n<tbody>\n<tr style=\"height: 131px;\">\n<td style=\"text-align: center; height: 131px; width: 102.625px;\"><strong>Employee<\/strong><\/td>\n<td style=\"height: 131px; width: 287.375px;\">\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: 102.625px;\"><strong>Employee 1<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 2<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 3<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 4<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 5<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 102.625px;\"><strong>Employee 6<\/strong><\/td>\n<td style=\"text-align: center; height: 14px; width: 287.375px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>Before you get the median and the mean from the technology, or before you calculate the mean by hand, first think about what you think will be true about the February mean compared to the January mean and why.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241079\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241079&theme=oea&iframe_resize_id=ohm241079\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241080\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241080&theme=oea&iframe_resize_id=ohm241080\" width=\"100%\" height=\"150\"><\/iframe><\/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><iframe loading=\"lazy\" id=\"ohm241082\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241082&theme=oea&iframe_resize_id=ohm241082\" width=\"100%\" height=\"150\"><\/iframe><\/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=\"textbox exercises\">\n<h3>Interactive example<\/h3>\n<p>Was the mean you calculated for February salaries higher, lower, or similar? What do you think caused that to be true? Click below for a discussion\u00a0<strong>after<\/strong> you enter your answers to Questions 8 &#8211; 10.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q476661\">Here are a couple of good questions to ask about it.<\/span><\/p>\n<div id=\"q476661\" class=\"hidden-answer\" style=\"display: none\">The mean is now higher than the median. They were identical in January.<\/p>\n<ul>\n<li>Did the increase in one salary cause the mean to rise?<\/li>\n<li>Would that always happen if a data value increases?<\/li>\n<li>How could we predict mathematically how much the mean would increase under the increase of a single value?\n<ul>\n<li>We could predict the increase in mean mathematically by taking the difference in the January salary and the February salary then distributing that difference out evenly among the employees.<\/li>\n<li>Ex. One salary increased by [latex]$2,000[\/latex]. If we divide the\u00a0[latex]$2,000[\/latex] across all six employees, we&#8217;ll have the amount by which the new mean is higher.\n<ul>\n<li>[latex]\\dfrac{$2,000}{6}=\\$333.33[\/latex]<\/li>\n<li>For January, [latex]\\bar{x}=$4,500[\/latex] and for February,\u00a0[latex]\\bar{x}=$4,833.33[\/latex]. The mean increased by $333.33.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>But why did the median stay the same? Would the median always be roughly the same if a data value changes?<\/p>\n<ul>\n<li>If the middle-most number or two numbers didn&#8217;t change, the median won&#8217;t change.<\/li>\n<li>What would happen though, if instead of Employee 2 receiving the raise, Employee 1 had received it instead? What would the new median be?\n<ul>\n<li>The January median of the data set [latex]3, 3, 4, 5, 6, 6[\/latex] is the mean of [latex]4[\/latex] and [latex]5[\/latex] in thousands of dollars, or [latex]\\$4,500[\/latex]<\/li>\n<li>Changing one of the salaries from [latex]6[\/latex] thousand to [latex]8[\/latex] thousand didn&#8217;t affect the middle two numbers.<\/li>\n<li>But changing the [latex]4[\/latex] to an [latex]8[\/latex] would require the reordering of the values.\n<ul>\n<li>[latex]3, 3, 5, 6, 6, 8[\/latex] now yields a median of [latex]5.5[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p>Now let&#8217;s consider a slightly different question.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241083\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241083&theme=oea&iframe_resize_id=ohm241083\" width=\"100%\" height=\"150\"><\/iframe><\/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 data set. A key idea to take from this activity is that, while the median stays relatively fixed in a data set 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 data set.<\/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":175116,"menu_order":46,"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-339","chapter","type-chapter","status-publish","hentry"],"part":1252,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/339","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/users\/175116"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/339\/revisions"}],"predecessor-version":[{"id":1259,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/339\/revisions\/1259"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/parts\/1252"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/339\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/media?parent=339"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapter-type?post=339"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/contributor?post=339"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/license?post=339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}