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 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.

In the digital version of the activity, students use an applet to do much of the visualization and calculation of statistics. The applet allows students to focus on the impact of the values in the data set rather than creating visualizations or doing calculations themselves. Use the digital version if it is available so that students can dynamically see the impact of different values on various statistics.

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 that 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 on the median.)

In this activity students 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." By using technology to create a dot plot with a given mean and median, students are using their understanding of the structure of the distribution (MP7) 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).

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 affect 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 affect the mean.)

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  when calculating the new mean. Remind students to complete detailed calculations.