Lesson 10

This warm-up prompts students to think about what variables they may use to analyze a situation. Then, students describe data displays they may use to compare two sets of data. Choosing variables and planning a process for comparing data sets engage students in aspects of mathematical modeling (MP4).

Listen for groups that choose a variable other than the number of wins to determine the top players in the game and groups that select different data displays or ways of comparing data sets to share with the whole group.

Arrange students in groups of 2. Tell students to think quietly about their answers to the questions for about 1 minute before discussing with their partner and then sharing with the whole group.

Select previously identified students to share their solutions. If it does not come up in discussion, ask students how they might interpret a situation in which a small group of players has significantly greater number of wins or players defeated than the rest of the group. It might mean that most of the players are not very good and there are a few who are or it might mean that there are a few dominant players who are much better than average players.

The mathematical purpose of this activity is to help students understand how measures of center for distributions with different shapes are impacted by changes in the data. Students will create a dot plot, then describe the shape of the distribution and find measures of center. They will investigate how the measures of center change when the data set changes. Students create and investigate a data set from a given a set of parameters including the shape of the distribution. Monitor for students using the correct terminology to describe the shape of the distribution.

This activity works best when each student has access to technology that computes measures of center and displays dot plots or histograms easily because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting the distributions would be helpful during the launch.

Students may have difficulty using technology to create dot plots so you may need to demonstrate how to use the technology. Students may confuse mean and median. Ask them to refer to previous work in which they calculated each measure of center. After students input the additional values as directed, they may use the wrong \(n\) when calculating the new mean. Remind students to complete detailed calculations.

The goal is to make sure that students understand that the median is the preferred measure of center when a distribution is skewed or if there are extreme values, and the the mean is the preferred measure of center when a distribution is symmetric and there are no extreme values. Here are some questions for discussion.

• “What do you notice and wonder about the mean and median for each of these distributions?” (I noticed that sometimes the median did not change and the mean did. I wondered what would happen if I added a value of 1,000 to the data set.)
• “For which distributions does it look like the mean best represents what is typical in the data?” (The symmetric distributions.)
• “When is the median a better statistic to describe typical values?” (The skewed distributions.)
• “Why is the median a better statistic for skewed distributions?” (When you add extreme values to a data set they tend to have a greater effect on the mean than the median.)

The mathematical purpose of this activity is to recognize the relationship between measures of center and the shape of the distribution by creating and describing distributions with given measures of center. Listen for students using the terms symmetric, uniform, and skewed. When students create a dot plot with a given mean and median using technology they are engaging in MP7 because they are using their understanding of the structure of the distribution to adjust individual data values to change the measures of center. Making a spreadsheet available gives students an opportunity to choose appropriate tools strategically (MP5).

Keep students in their groups. 

The purpose of this discussion is for students to understand why the median is the preferred measure of center when a distribution is skewed or if there are extreme values, and the mean is the preferred measure of center when a distribution is symmetric and there are no extreme values.

For each description, select 2–3 groups to share their dot plots.

Here are some questions for discussion.

• “For the first and second dot plot, what do the distribution shapes have in common? Why do we choose the mean as the more appropriate measure of center?” (Symmetric. The mean of a set of data gives equal importance to each value to find the center, so it is a preferred measure of center when it accurately represents typical values for data.)
• “What do the shapes of the dot plots have in common when the mean is greater than the median?” (Skewed right.)
• “What information does the shape of the skewed distributions tell you about the median and mean?” (When distributions are skewed right they will likely have a mean that is greater than the median because the values to the right disproportionately impact the mean. When distributions are skewed left they will likely have a mean that is less than the median because the values to the left disproportionately impact the mean.)