{"id":860,"date":"2023-03-20T19:17:37","date_gmt":"2023-03-20T19:17:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/interpreting-the-mean-and-median-of-a-data-set-fresh-take\/"},"modified":"2023-12-07T00:50:36","modified_gmt":"2023-12-07T00:50:36","slug":"interpreting-the-mean-and-median-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/interpreting-the-mean-and-median-fresh-take\/","title":{"raw":"Interpreting the Mean and Median: Fresh Take","rendered":"Interpreting the Mean and Median: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Name the features of the distribution of a data set using statistical language<\/li>\r\n\t<li>Describe the connection between the distribution of a data set and its mean and median<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox recall\">Recall that we think of the <strong>mean<\/strong> as the \"average\" data value and the <strong>median<\/strong> as the [latex]50[\/latex]<sup>th<\/sup> percentile, the value that splits the data in half.\u00a0<\/section>\r\n<section class=\"textbox example\">Let's say the mean of a data set is given as [latex]10.5[\/latex] and the median as [latex]11[\/latex]. Which of the following statements are true? Explain.\r\n\r\n<ol>\r\n\t<li>The median tells us a typical value for this data set. That is, if we took all the values and spread them evenly about, each value would be about [latex]11[\/latex].<\/li>\r\n\t<li>About half the data values fall below [latex]11[\/latex] and half fall above.<\/li>\r\n\t<li>The most common data value appearing is [latex]10.5[\/latex].<\/li>\r\n\t<li>A typical data value for this set is [latex]10.5[\/latex]. That is, if we distributed the sum of all the values evenly, each value would be about [latex]10.5[\/latex].<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"679737\"]Show solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"679737\"]<\/p>\r\n<ol>\r\n\t<li><strong>False.<\/strong> The median represents the [latex]50[\/latex]<sup>th<\/sup> percentile, with about half the values falling above [latex]11[\/latex] and half below.<\/li>\r\n\t<li><strong>True.<\/strong> The median is [latex]11[\/latex].<\/li>\r\n\t<li><strong>Cannot be determined.<\/strong> The mode tells us the most common data value. Neither the mean nor the median gives us that information.<\/li>\r\n\t<li><strong>True.<\/strong> The mean is [latex]10.5[\/latex], which we can consider to be the \"average\" data value.<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Recall the data set about employee salaries.<\/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 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.<br \/>\r\nThe second data set lists the monthly salaries (in thousands of dollars) for the same six employees during the month of February.<\/p>\r\n<div 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<\/div>\r\n<section class=\"textbox example\">We saw that the median and the mean employee salaries for January were the same. What can we understand from that information?\r\n\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<p>[reveal-answer q=\"968607\"]Show solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"968607\"]<\/p>\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.[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p>It was interesting that the mean and the median were identical values. This tells us that the the salaries were <strong>evenly distributed<\/strong> among high and low values and the distribution was <strong>symmetrical<\/strong>, without skew.<\/p>\r\n<p>But what happens if we change one of the values in the data set?<\/p>\r\n<h2 id=\"CompareMeanMedian\">Comparing Mean and Median<\/h2>\r\n<p>Let's look at the data set of employee salaries from February which includes a big raise for one employee. How will the mean of the February salaries compares to the mean of the January salaries?<\/p>\r\n<p>[reveal-answer q=\"845599\"]Here is the February salary table again for convenience.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"845599\"]<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2065[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2066[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2067[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Was the mean you calculated for February salaries higher, lower, or similar? What do you think caused that to be true?<br \/>\r\n[reveal-answer q=\"476661\"]Here are a couple of good questions to think about.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"476661\"]The mean is now higher than the median. They were identical in January.\r\n\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\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\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\n<p>But why did the median stay the same? Would the median always be roughly the same if a data value changes?<\/p>\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\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\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<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Now let's consider a slightly different question.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2068[\/ohm2_question]<\/section>\r\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>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Name the features of the distribution of a data set using statistical language<\/li>\n<li>Describe the connection between the distribution of a data set and its mean and median<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\">Recall that we think of the <strong>mean<\/strong> as the &#8220;average&#8221; data value and the <strong>median<\/strong> as the [latex]50[\/latex]<sup>th<\/sup> percentile, the value that splits the data in half.\u00a0<\/section>\n<section class=\"textbox example\">Let&#8217;s say the mean of a data set is given as [latex]10.5[\/latex] and the median as [latex]11[\/latex]. Which of the following statements are true? Explain.<\/p>\n<ol>\n<li>The median tells us a typical value for this data set. That is, if we took all the values and spread them evenly about, each value would be about [latex]11[\/latex].<\/li>\n<li>About half the data values fall below [latex]11[\/latex] and half fall above.<\/li>\n<li>The most common data value appearing is [latex]10.5[\/latex].<\/li>\n<li>A typical data value for this set is [latex]10.5[\/latex]. That is, if we distributed the sum of all the values evenly, each value would be about [latex]10.5[\/latex].<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q679737\">Show solution<\/button><\/p>\n<div id=\"q679737\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li><strong>False.<\/strong> The median represents the [latex]50[\/latex]<sup>th<\/sup> percentile, with about half the values falling above [latex]11[\/latex] and half below.<\/li>\n<li><strong>True.<\/strong> The median is [latex]11[\/latex].<\/li>\n<li><strong>Cannot be determined.<\/strong> The mode tells us the most common data value. Neither the mean nor the median gives us that information.<\/li>\n<li><strong>True.<\/strong> The mean is [latex]10.5[\/latex], which we can consider to be the &#8220;average&#8221; data value.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Recall the data set about employee salaries.<\/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 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.<br \/>\nThe second data set 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 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<\/div>\n<section class=\"textbox example\">We saw that the median and the mean employee salaries for January were the same. What can we understand from that information?<\/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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q968607\">Show solution<\/button><\/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.<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<p>It was interesting that the mean and the median were identical values. This tells us that the the salaries were <strong>evenly distributed<\/strong> among high and low values and the distribution was <strong>symmetrical<\/strong>, without skew.<\/p>\n<p>But what happens if we change one of the values in the data set?<\/p>\n<h2 id=\"CompareMeanMedian\">Comparing Mean and Median<\/h2>\n<p>Let&#8217;s look at the data set of employee salaries from February which includes a big raise for one employee. How will the mean of the February salaries compares to the mean of the January salaries?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q845599\">Here is the February salary table again for convenience.<\/button><\/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<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2065\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2065&theme=lumen&iframe_resize_id=ohm2065&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2066\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2066&theme=lumen&iframe_resize_id=ohm2066&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2067\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2067&theme=lumen&iframe_resize_id=ohm2067&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Was the mean you calculated for February salaries higher, lower, or similar? What do you think caused that to be true?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q476661\">Here are a couple of good questions to think about.<\/button><\/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<\/section>\n<p>Now let&#8217;s consider a slightly different question.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2068\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2068&theme=lumen&iframe_resize_id=ohm2068&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\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","protected":false},"author":13,"menu_order":19,"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":"fresh_take","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\/860"}],"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":3,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/860\/revisions"}],"predecessor-version":[{"id":4546,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/860\/revisions\/4546"}],"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\/860\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=860"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=860"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=860"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}