If students struggle to use symmetry as a method for finding the mean, consider asking them to find the mean for the values: 1, 2, 3, 4, 5.

Much discussion takes place between partners.

Direct students' attention to the reference created using Collect and Display. Ask students to share how they found their matches. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. (For example, “The distribution is skewed right because there is a longer tail on that side.”)

After all groups have completed the matching, discuss the following:

• “Which matches were tricky? Explain why.” (The box plot labeled F was tricky because I had to use a process of elimination to figure out that it was the one that was uniform.)
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?” (Yes. I realized that I thought incorrectly that skewed left meant that most of the data were on the left. However, I learned that skewed left means that there are data to the left of where most of the data are located.)
• “Can you determine the median using only a histogram? Why or why not?” (No, but you can determine the interval that contains the median.)
• “Can you determine if a distribution is uniform from a box plot? Why or why not?” (No. You can determine that the data distribution could possibly be symmetric based on the distribution of the five-number summary, but beyond that you would not be able to know, using only a box plot, that the data is uniform.)

The purpose of the second part of the activity is to discuss the relationship between mean and median based on the shape of the distribution and to make the connection to measures of variability. Ask:

• “If the mean is the appropriate measure of center, should we use the MAD or the IQR to measure variability?” (MAD)
• “If the median is the appropriate measure of center, should we use the MAD or the IQR to measure variability?” (IQR)

Students may have forgotten what variability means or which statistic to use to determine the variability in a data set. Refer them to previous work, or ask them what measure is useful in determining a data set's tendency to have different values.

The purpose of this discussion is to make the connection between the shape of the distribution and the use of either IQR or MAD to quantify variability. Another goal is to make sure students understand that a greater value for IQR or MAD means greater variability.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they arranged the distributions. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. 

Display the box plots in order of variability with the IQR included, and then display the dots plots in order of variability with the MAD included.

The IQR for the data shown in box plots A through E are {40, 60, 50, 40, 20} and the MAD for the data shown in dot plots A through E are approximately {1.56, 1.10, 2.68, 2.22, 0}. Here are some questions for discussion:

• “What are the IQR and MAD measuring?” (They are measuring the spread, or variability, of the data)
• “Which plots were the most difficult to arrange?” (The dot plots were more difficult because it was easy to find the IQR for the box plots.)
• “Does the order given when arranged by the IQR and MAD match your order?” (Yes, except for the box plots A and D which had the same IQR, and I didn’t know how to arrange them.)
• “What do you notice about the values for IQR and MAD?” (The values for the MAD were higher than I thought except for distribution J. I did not know that the MAD could be equal to zero.)
• “What advantages are offered by using IQR and MAD versus visual inspection?” (The IQR and MAD are values that can be easily sorted.)

If some students already arranged the plots using IQR or MAD, you should ask them, “Why did you choose to arrange the plots by IQR or MAD?” (I knew that IQR and MAD were measures of variability so I used them.)