# Illustrative Mathematics Grade 6, Unit 8, Lesson 14: Comparing Mean and Median

Learning Targets:

• I can determine when the mean or the median is more appropriate to describe the center of data.
• I can explain how the distribution of data affects the mean and the median.

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Illustrative Math

#### Lesson 14: Comparing Mean and Median

Let’s compare the mean and median of data sets.

Illustrative Math Unit 6.8, Lesson 14 (printable worksheets)

#### Lesson 14 Summary

The following diagram shows how both the mean and the median are ways of measuring the center of a distribution. #### Lesson 14.1 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.

1. The median height of the first 22 presidents is 178 centimeters.
2. The mean height of the first 22 presidents is about 183 centimeters.
3. A typical height for a president in the second group is about 182 centimeters.
4. U.S. presidents have become taller over time.
5. The heights of the first 22 presidents are more alike than the heights of the second 22 presidents.
6. The MAD of the second data set is greater than the MAD of the first set.

#### Lesson 14.2 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.

1. Find the mean height of your class in centimeters.
2. Find the median height in centimeters. Show your reasoning.
3. Suppose that the world’s tallest adult, who is 251 centimeters tall, joined your class.
a. Discuss the following questions with your group and explain your reasoning.
• How would the mean height of the class change?
• How would the median height change?
b. Find the new mean.
c. Find the new median.
d. Which measure of center—the mean or the median—changed more when this new person joined the class? Explain why the value of one measure changed more than the other.
1. The world’s smallest adult is 63 centimeters tall. Suppose that the world’s tallest and smallest adults both joined your class.
a. Discuss the following questions with your group and explain your reasoning.
• How would the mean height of the class change from the original mean?
• How would the median height change from the original median?
b. Find the new mean.
c. Find the new median.
d. How did the measures of center—the mean and the median—change when these two people joined the class? Explain why the values of the mean and median changed the way they did.

#### Lesson 14.3 Mean or Median?

1. Your teacher will give you six cards. Each has either a dot plot or a histogram. Sort the cards into two piles based on the distributions shown. Be prepared to explain your reasoning.
2. Discuss your sorting decisions with another group. Did you have the same cards in each pile? If so, did you use the same sorting categories? If not, how are your categories different?
Pause here for a class discussion.
3. Use the information on the cards to answer the following questions.
a. Card A: What is a typical age of the dogs being treated at the animal clinic?
b. Card B: What is a typical number of people in the Irish households?
c. Card C: What is a typical travel time for the New Zealand students?
d. Card D: Would 15 years old be a good description of a typical age of the people who attended the birthday party? e. Card E: Is 15 minutes or 24 minutes a better description of a typical time it takes the students in South Africa to get to school?
f. Card F: Would 21.3 years old be a good description of a typical age of the people who went on a field trip to Washington, D.C.?
4. How did you decide which measure of center to use for the dot plots on Cards A–C? What about for those on Cards D–F?

#### 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.

The mean score can be influenced by a few low scores (pull down the mean) or a few high scores (pull up the mean).
The median would not be affected as much.

#### Lesson 14 Practice Problems

1. 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?
A. The mean is less than the median.
B. The mean is approximately equal to the median.
C. The mean is greater than the median.
D. The mean cannot be determined.
2. 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?
A. The mean would decrease and the median would not change.
B, The mean would increase and the median would not change.
C, The mean would decrease and the median would increase.
D, The mean would increase and the median would increase.
3. In his history class, Han’s homework scores are:
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.
4. 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.
5. 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.
6. 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)

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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