The aim of this activity is to expose the limits of the mean in summarizing a data set that has gaps and values far from the center, and to motivate a need to have another measure of center. Students first use a table of values and a dot plot to estimate a typical value for a data set. Then they calculate the mean and notice that it does not lie near the center of the data. A closer look helps them see that when a data set contains values that are far away from the bulk of the data, or when there are gaps in the data set, the mean can be a little or a lot higher or lower than what we would consider typical for the data. 

Because previous lessons have used the mean as the best way to find a typical value, some students may go directly to that method from the beginning. Although this is valid at this stage, encourage them to look at the dot plot and think about what a typical value should be.

Select a few students to share their estimate for a typical number of siblings. Consider asking students:

• “When you looked at the table of values, what was your sense of a typical number of siblings for the ten students in Tyler's class?” (Around 2 because most students have 0, 1, or 2 siblings, but there are a couple of students with a lot.)
• “When you looked at the dot plot, did your estimate change?” (It was easier to see where most of the data are.)

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “Do you think the mean summarizes the data set well?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback: 

• “_____ makes sense, but what do you mean when you say . . . ?”
• “Can you describe that another way?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

After Stronger and Clearer Each Time, discuss how the calculated mean compared to their estimates. Draw students’ attention to the idea that the mean may not always represent a typical value for a data set. Discuss:

• “We have learned that the mean is a way to measure the center of a distribution. How did your calculated mean compare to your estimate of what was typical for the data set?” (The mean is greater than my estimate for a typical value.)
• “Why do you think the mean was higher than your estimate?” (Only two of the points are above the mean of 2.4 and both are quite far above it, and seven points are below 2.4, so the mean might not paint an accurate picture of what is typical in this situation.)
• “If the mean does not always reflect what is typical in a data set, should we always rely on it as the best way to describe the center? If not, in what other ways might we measure the center of a data set?” (Maybe we could look for the most common values or pick a number close to where most of the data are.)

This activity introduces students to the term median. They learn that the median describes the middle value in an ordered list of data, and that it can capture what we consider typical for the data in some cases.

Students learn about the median through a kinesthetic activity. They line up in order of the number of letters in their name. Then, those at both ends of the line count off and sit down simultaneously until one or two people in the middle remain standing. If one person remains standing, that person has the median number of letters. If two people remain standing, the median is the mean or the average of their two values. 

Students then practice identifying the median of other data sets, by analyzing both tables of values and dot plots.

When determining the median, students might group multiple data points that have the same value and treat it as a single point, instead of counting each one separately. Remind them that when they lined up to find the median number of letters in their names, every student counted off, even if their name had the same number of letters as their neighbor’s name. 

Select a few students to share their responses to the questions about number of siblings and Elena's travel times. Focus the discussion on the median as another measure of the center of a data set and whether it captures what students would estimate to be a typical value for each data set.

Emphasize to students that the median is a value and not an individual. For example, if the last person standing in the class has 10 letters in their name, the median is the number 10 and not the person standing. If there is another student who has 10 letters in their name, they might have switched places with the last person standing when lining up initially. Although the person standing could change, the median remains the same value of 10.

At this point, it is unnecessary to compare the mean and the median. Students will have many more opportunities to explore the median and think about how it differs from the mean in the upcoming activities.