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.

Provide technology such as a spreadsheet or calculator that can quickly find the mean and median. The focus of the activity should be on interpreting why the measures differ with outliers rather than on calculation.

When calculating the mean after a new person joins 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 (for example, 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 would change 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 effect does the tallest person in the world have on the mean? Why?” (The mean is pulled higher by the inclusion of this value. In terms of fair share, more would need to be taken from that value so that all of the heights are equal, so the mean would be greater.)
• “Suppose a new student who has a height close to the mean joins the class. Would that person’s height affect the mean? Why or why not?” (Because every value is included, it would affect the mean in some way unless it was exactly the same as the mean. But, because the person’s height is close to the mean, it should not change much.)
• “Does adding two values—one unusually high and one unusually low—affect the median? Why or why not?” (It does not. The middle value on the list does not change when one extreme value is added on each side.)

In this lesson, students study distributions 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.

Students sort different distributions during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and to make connections (MP2, MP7).

As students work, encourage them to refine their descriptions of the distributions using more precise language and mathematical terms (MP6).

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?” (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.)

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the question, “When are median and mean likely to be close?”, by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:
• “They’re the same when most of the points are in the middle.”
• Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Display and review these criteria: 
• Specific words and phrases: “symmetric” or “approximately symmetric” distributions
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.