Let's compare the mean and median of data sets. Exercise \(\PageIndex{1}\): Heights of Presidents Here are two dot plots. The first dot plot shows the heights of the first 22 U.S. presidents. The second dot plot shows the heights of the next 22 presidents. Based on the two dot plots, decide if you agree or disagree with each of the following statements. Be prepared to explain your reasoning. Exercise \(\PageIndex{2}\): The Tallest and the Smallest in the World Your teacher will provide the height data for your class. Use the data to complete the following questions.
Exercise \(\PageIndex{3}\): Mean or Median?
Are you ready for more? Most teachers use the mean to calculate a student’s final grade, based on that student’s scores on tests, quizzes, homework, projects, and other graded assignments. Diego thinks that the median might be a better way to measure how well a student did in a course. Do you agree with Diego? Explain your reasoning.
Both the mean and the median are ways of measuring the center of a distribution. They tell us slightly different things, however. The dot plot shows the weights of 30 cookies. The mean weight is 21 grams (marked with a triangle). The median weight is 20.5 grams (marked with a diamond). Figure \(\PageIndex{2}\)The mean tells us that if the weights of all cookies were distributed so that each one weighed the same, that weight would be 21 grams. We could also think of 21 grams as a balance point for the weights of all of the cookies in the set. The median tells us that half of the cookies weigh more than 20.5 grams and half weigh less than 20.5 grams. In this case, both the mean and the median could describe a typical cookie weight because they are fairly close to each other and to most of the data points. Here is a different set of 30 cookies. It has the same mean weight as the first set, but the median weight is 23 grams. In this case, the median is closer to where most of the data points are clustered and is therefore a better measure of center for this distribution. That is, it is a better description of a typical cookie weight. The mean weight is influenced (in this case, pulled down) by a handful of much smaller cookies, so it is farther away from most data points. In general, when a distribution is symmetrical or approximately symmetrical, the mean and median values are close. But when a distribution is not roughly symmetrical, the two values tend to be farther apart.
Definition: Median The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order. For the data set 7, 9, 12, 13, 14, the median is 12. For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. \(6+8=14\) and \(14\div 2=7\). PracticeExercise \(\PageIndex{4}\) Here is a dot plot that shows the ages of teachers at a school. Which of these statements is true of the data set shown in the dot plot? Figure \(\PageIndex{4}\)
Exercise \(\PageIndex{5}\) Priya asked each of five friends to attempt to throw a ball in a trash can until they succeeded. She recorded the number of unsuccessful attempts made by each friend as: 1, 8, 6, 2, 4. Priya made a mistake: The 8 in the data set should have been 18. How would changing the 8 to 18 affect the mean and median of the data set?
Exercise \(\PageIndex{6}\) In his history class, Han's homework scores are: \(100\qquad 100\qquad 100\qquad 100\qquad 95\qquad 100\qquad 90\qquad 100\qquad 0\) The history teacher uses the mean to calculate the grade for homework. Write an argument for Han to explain why median would be a better measure to use for his homework grades. Exercise \(\PageIndex{7}\) The dot plots show how much time, in minutes, students in a class took to complete each of five different tasks. Select all the dot plots of tasks for which the mean time is approximately equal to the median time. Figure \(\PageIndex{5}\): Five dot plots labeled A, B, C, D, and E. On each dot plot, the numbers 0 through 60, in increments of 5, are indicated. The data for dot plot A are as follows: 2, 1 dot. 3, 2 dots. 7, 1 dot. 8, 1 dot. 10, 1 dot. 11, 2 dots. 13, 2 dots. 15, 1 dot. 18, 1 dot. 19, 2 dots. 20, 1 dot. 22, 1 dot. 29, 2 dots. 36, 2 dots. 37, 1 dot. 38, 1 dot. 40, 1 dot. 57, 1 dot. 60, 1 dot. The data for dot plot B are as follows: 31, 1 dot. 32, 2 dots. 33, 3 dots. 34, 2 dots. 35, 5 dots. 36, 3 dots. 37, 2 dots. 38, 2 dots. 39, 2 dots. 40, 1 dot. 41, 1 dot. 44, 1 dot. The data for dot plot C are as follows: 4, 3 dots. 5, 3 dots. 6, 1 dot. 7, 3 dots. 8, 6 dots. 9, 2 dots. 10, 1 dot. 11, 4 dots. 12, 2 dots. The data for dot plot D are as follows: 2, 2 dots. 3, 2 dots. 4, 3 dots. 7, 1 dot. 9, 2 dots. 11, 1 dot. 12, 2 dots. 14, 2 dots. 15, 1 dot. 19, 1 dot. 20, 1 dot. 21, 1 dot. 22, 1 dot. 31, 1 dot. 43, 1 dot. 45, 1 dot. 49, 1 dot. 60, 1 dot. The data for dot plot E are as follows: 3, 4 dots. 4, 4 dots. 6, 1 dot. 7, 1 dot. 9, 4 dots. 10, 3 dots. 14, 1 dot. 16, 1 dot. 18, 1 dot. 20, 1 dot. 27, 1 dot. 36, 1 dot. 44, 1 dot.Exercise \(\PageIndex{8}\) Zookeepers recorded the ages, weights, genders, and heights of the 10 pandas at their zoo. Write two statistical questions that could be answered using these data sets. (From Unit 8.1.2) Exercise \(\PageIndex{9}\) Here is a set of coordinates. Draw and label an appropriate pair of axes and plot the points. \(A=(1,0), B=(0,0.5), C=(4,3.5), D=(1.5, 0.5)\) (From Unit 7.3.2) |