14.1: Heights of Presidents (5 minutes)
In this warm-up, students review ways to interpret and compare data shown on a dot plot. The discussion on each given statement gives the teacher an opportunity to hear how students reason about the median, mean, typical value, spread, balance point, and MAD. This discussion will be helpful in upcoming activities, as students compare median and mean values for different data sets.
Give students 2 minutes of quiet work time. Follow with a whole-class discussion.
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.
- The median height of the first 22 presidents is 178 centimeters.
- The mean height of the first 22 presidents is about 183 centimeters.
- A typical height for a president in the second group is about 182 centimeters.
- U.S. presidents have become taller over time.
- The heights of the first 22 presidents are more alike than the heights of the second 22 presidents.
- The MAD of the second data set is greater than the MAD of the first set.
For each statement, ask students to indicate if they agree or disagree. If all students agree or all students disagree, ask a couple of students to explain their reasoning. If the class is divided on a statement, ask students on both sides to share their reasoning until the class comes to an agreement. As students share, record and display their responses for all to see. If possible, record their reasoning on the dot plots to highlight important terms students use.
To help facilitate the discussion, consider asking:
- “Who can restate ___’s reasoning in a different way?”
- “Did anyone have the same reasoning but would explain it differently?”
- “Did anyone reason about the statement in a different way?”
- “Does anyone want to add on to _____’s reasoning?”
- “Do you agree or disagree? Why?”
14.2: The Tallest and the Smallest in the World (15 minutes)
In this lesson, students begin to notice how the distribution of data affects the mean and median of a data set. Using their class height data, they examine how both measures of center are affected when a value that is far from the center is added to a data set. Students find that adding these unusually large or small values pulls the mean up or down while having little or no effect on the median. They begin to see that for data sets with some far-off values, the median might be a better choice for describing a typical value because the sizes of those extreme values (whether very large or very small) do not affect the median as much as they do the mean.
Consider preparing a large-scale dot plot that shows student height data and displaying it for all to see.
Arrange students in groups of 3–4. Provide each student with the data on students' heights (collected in the first lesson of the unit) and access to calculators. Give groups 8–10 minutes to complete the first three questions, and then ask them to pause for a brief class discussion before moving on to the last set of questions.
During this discussion, compare the mean that students calculated and the median they found, and solicit students' ideas on why the mean changed more than the median if the tallest person in the world joined the class. (If a class dot plot that shows student heights is made, add the point that represents the tallest person in the world.)
At this time students should begin to see that the value of the additional data point greatly affects the mean because finding the mean entails redistributing data values so that they are all the same (or moving the balance point toward the new data point to keep the distribution balanced). The new data point does not affect the median as much in this case because finding the median entails finding the point in a distribution that divides the data in half, and the numbers around the middle of the distribution are likely very close.
Give groups another few minutes to complete the rest of the task.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Design Principle(s): Maximize meta-awareness
Your teacher will provide the height data for your class. Use the data to complete the following questions.
- Find the mean height of your class in centimeters.
- Find the median height in centimeters. Show your reasoning.
Suppose that the world’s tallest adult, who is 251 centimeters tall, joined your class.
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?
- Find the new mean.
- Find the new median.
- 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.
The world’s smallest adult is 63 centimeters tall. Suppose that the world’s tallest and smallest adults both joined your class.
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?
- Find the new mean.
- Find the new median.
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.
When calculating the mean after a new person joined the class, some students might enter individual heights into a calculator once again, rather than using the sum from their original mean calculation to save time (e.g. by adding 251 to the sum of heights of students and then dividing by a class size that includes one additional student). Urge them to think about how they might use the previous calculation to make the process more efficient.
Use the whole-class discussion to further explore how unusually high or low values affect the mean and the median. Invite several students to share the new mean and median, and to explain how these measures changed if the world's shortest person joined the class. (If a class dot plot that shows student heights is made, add the point that represents the smallest person in the world.) Discuss:
- “What affect does the smallest person in the world have on the mean? Why?”
- “Which would affect the mean more: the height of the tallest person, or the height of the smallest person? Why?”
- “Suppose a new student who has a height close to the mean joined the class. Would her height affect the mean? Why or why not?”
- “Does adding two values—one unusually high and one unusually low—affect the median? Why or why not?”
Students should see that adding a data point to each end does not change the median much, if at all; the original middle value would still be the middle value as the halfway point is unchanged. Conversely, the mean will change greatly due to values that are extremely greater or less than most of the data.
14.3: Mean or Median? (15 minutes)
In the previous activity, students analyzed the effects of unusually high or low values on the mean and median. Here they study distributions (displayed using dot plots and a histogram) for which the mean and median can be the same, close, or far apart, and make conjectures about how the distributions affect the mean and median (MP7). Along the way, students recognize that the mean and median are equal or close when the distribution is roughly symmetrical and are farther apart when the distribution is non-symmetrical.
Arrange students in groups of 3–4. Provide each group with a cut-up set of cards from the blackline master. Give groups 4–5 minutes to take turns sorting the cards and completing the first two problems. Then, pause the activity to discuss the sorting decisions and observations of the class.
Ask a few groups how they sorted the cards. If not mentioned by students, highlight that in three of the distributions, the mean and median of the data are approximately equal. In the other three distributions, the mean and median are quite different. Discuss:
- “What do you notice about the shape and features of distributions that have roughly equal mean and median?” (They are roughly symmetrical and each have one peak in the middle, with roughly the same number of values to the left and right. They may have gaps, but the gaps are somewhat evenly spaced out.)
- “What about the shape and features of a distribution that has very different mean and median?” (They are not at all symmetrical. They may have one peak, but it is off to one side, or they don't really show any peaks. They may have gaps or data values that are unusually high or low. There is more variability in these data sets.)
- “In the second group, why might the mean and the median be so different?” (The mean is pulled toward the direction of unusually large or small values. The median simply tells us where the middle of the data lies when sorted, so it is not as affected by these values that are far from where most data points are.)
Afterwards, give students another 3–4 minutes to answer and discuss the remaining questions with their group.
Supports accessibility for: Language; Organization
Design Principle(s): Support sense-making
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.
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.
Use the information on the cards to answer the following questions.
- Card A: What is a typical age of the dogs being treated at the animal clinic?
- Card B: What is a typical number of people in the Irish households?
- Card C: What is a typical travel time for the New Zealand students?
- Card D: Would 15 years old be a good description of a typical age of the people who attended the birthday party?
- 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?
- 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.?
- 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.
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Use the whole-class discussion to reinforce the idea that the distribution of a data set can tell us which measure of center best summarizes what is typical for the data set. Briefly review the answers to the statistical questions, and then focus the conversation on the last questions (how students knew which measure of center to use in each situation). Select a couple of students to share their responses. Discuss:
- “For data sets with non-symmetrical distributions, why does the median turn out to be a better measure of center for non-symmetrical data sets?” (Non-symmetrical data sets often have unusual values that pull the mean away from the center of data. The median is less influenced by these values.)
- “Does it matter which measure we choose to describe a typical value? For example, in Card F, would it matter if we said that a typical age for the people who went on the field trip to D.C. was about 21 years old?” (Yes, it does matter in some cases. In that example, it wouldn’t really make sense to say that 21 years is a typical age because the vast majority of the people on the trip were teenagers.)
We see in the lesson that sometimes the two measures of center could be the same or very close, but other times they could be very different.
- “When are the mean and median likely to be close together?” (When the distribution is approximately symmetrical.)
- “When are they likely to be different?” (When the distribution is not roughly symmetrical or has unusually high or low values that are far from others.)
- “Why might the median be a more useful measure of center when the distribution is not symmetrical?” (Values far from the middle tend to have a greater influence on the mean than the median, so individual values can have a greater impact.)
- “In the situations we saw today, did it matter which measure we choose to describe a typical value?” (Yes, it did matter in some cases. For the 8-year-old's birthday party, it would not make sense to say that 15 years is a typical age for the partygoers.)
- “A student reports that 7 is a typical number of pets that students in her class has. Do you think she used the mean number of pets or the median? How do you know?” (The mean. That number seems too high to be a typical number of pets. The data may have included one or more students who have a tank of fish or other small animals that get counted individually, which would pull the mean up.)
- “Can you think of other real-world situations where reporting the mean or median can be misleading?” (Example: Salaries at a company with many low-level workers and one executive who gets paid a lot more than anyone else would be better reported with a median. If they had a job opening and reported the mean, it might might make people think they will make more than they probably will.)
14.4: Cool-down - Which Measure of Center to Use? (5 minutes)
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Student Lesson Summary
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).
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.