{"id":469,"date":"2022-07-11T18:39:55","date_gmt":"2022-07-11T18:39:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/?post_type=chapter&p=469"},"modified":"2022-08-09T20:06:40","modified_gmt":"2022-08-09T20:06:40","slug":"interpreting-the-mean-and-median-of-a-data-set-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/interpreting-the-mean-and-median-of-a-data-set-background-youll-need-3\/","title":{"raw":"Interpreting the Mean and Median of a Data Set: Background You'll Need 3","rendered":"Interpreting the Mean and Median of a Data Set: Background You’ll Need 3"},"content":{"raw":"We saw that the median and the mean employee salaries for January were the same. What understanding can we take from that information?\r\n
\r\n

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
    \r\n \t
  1. 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
  2. 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
      \r\n \t
    1. Half the employees made more than\u00a0[latex]\\$4,500[\/latex] and half made less.<\/li>\r\n \t
    2. 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\r\n

      Comparing Mean and Median<\/h2>\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\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Employee<\/strong><\/td>\r\n\r\n

      Monthly Salary in February<\/strong><\/p>\r\n

      (in thousands of dollars)<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n

      Employee 1<\/strong><\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
      Employee 2<\/strong><\/td>\r\n[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n
      Employee 3<\/strong><\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
      Employee 4<\/strong><\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
      Employee 5<\/strong><\/td>\r\n[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n
      Employee 6<\/strong><\/td>\r\n[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
      \r\n

      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
      \r\n

      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 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
      \r\n

      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
      \r\n

      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\u00a0after<\/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
        \r\n \t
      • Did the increase in one salary cause the mean to rise?<\/li>\r\n \t
      • Would that always happen if a data value increases?<\/li>\r\n \t
      • How could we predict mathematically how much the mean would increase under the increase of a single value?\r\n
          \r\n \t
        • 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
        • 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 \t
          • [latex]\\dfrac{$2,000}{6}=\\$333.33[\/latex]<\/li>\r\n \t
          • 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
              \r\n \t
            • If the middle-most number or two numbers didn't change, the median won't change.<\/li>\r\n \t
            • 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 \t
              • 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
              • 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
              • But changing the [latex]4[\/latex] to an [latex]8[\/latex] would require the reordering of the values.\r\n
                  \r\n \t
                • [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
                  \r\n

                  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.","rendered":"

                  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

                  \n

                  Interactive Example<\/h3>\n

                  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

                    \n
                  1. The median of the data set implies that ____________ made more than\u00a0[latex]\\$4,500[\/latex] in January and _________ made less.<\/li>\n
                  2. 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
                    Show Answer<\/span><\/p>\n
                    \n
                      \n
                    1. Half the employees made more than\u00a0[latex]\\$4,500[\/latex] and half made less.<\/li>\n
                    2. 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

                      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’s move on to questions 8 – 10 to find out.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                      Comparing Mean and Median<\/h2>\n

                      What happens to the mean and median if we change one of the values in the data set?<\/p>\n

                      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

                      Here is the February salary table again for convenience.<\/span><\/p>\n
                      \n\n\n\n\n\n\n\n\n\n
                      Employee<\/strong><\/td>\n\n

                      Monthly Salary in February<\/strong><\/p>\n

                      (in thousands of dollars)<\/strong><\/p>\n<\/td>\n<\/tr>\n

                      Employee 1<\/strong><\/td>\n[latex]4[\/latex]<\/td>\n<\/tr>\n
                      Employee 2<\/strong><\/td>\n[latex]8[\/latex]<\/td>\n<\/tr>\n
                      Employee 3<\/strong><\/td>\n[latex]3[\/latex]<\/td>\n<\/tr>\n
                      Employee 4<\/strong><\/td>\n[latex]5[\/latex]<\/td>\n<\/tr>\n
                      Employee 5<\/strong><\/td>\n[latex]6[\/latex]<\/td>\n<\/tr>\n
                      Employee 6<\/strong><\/td>\n[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                      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

                      \n

                      question 8<\/h3>\n