In this activity, students learn the term mean absolute deviation (MAD) as a way to quantify variability and calculate it by finding distances between the mean and each data value. Students compare data sets with the same mean but different MADs and interpret the variability in context.

While this process of calculating MAD involves taking the absolute value of the difference between each data point and the mean, this formal language is downplayed here. Instead, the idea of “finding the distance,” which is always positive, is used. This is done for a couple of reasons. One reason is to focus students' attention on the statistical work rather than on terminology or symbolic work. Another reason is that finding these differences may involve operations with signed numbers, which are not expected in this course. 

Students may recall an earlier lesson about thinking of the mean as a balance point and think that the MAD should always be zero because the left and right distances should be equal. Remind them that distances are always positive, so the average of these distances to the mean can be zero only if all the data points are exactly at the mean.

During discussion, highlight that finding how far away, on average, the data points are from the mean is a way to describe the variability of a distribution. Discuss:

• “What can we say about a data set whose data points have very small distances from the mean?” (That means most of the points are near the mean, so there is not much spread.)
• “What about a data set with points that show large distances from the mean?” (That means most of the points are not close to the mean, so there is a large spread.)
• “Does a data set with smaller distances (and therefore smaller average distances) show less or more variability?” (There is less variability when the points have smaller distances from the mean.)

This activity allows students to practice calculating MAD and to build a better understanding of what it tells us. Students compare data sets with the same mean but different MADs and interpret what these differences imply in the context of the situation. During the discussion, they select a student to be on their team based on the comparison.

Expect students to choose different players to be on their team, but be sure they support their preferences with a reasonable explanation (MP3).

Select a couple of students to share their responses to the first set of questions about how they matched the dot plots to the players and how they knew.

Then, display a completed table and the MAD for the second set of questions. Give students a moment to check their work. To facilitate discussion, help students connect MAD and the spread of data, and enable them to make a comparison. Consider displaying all three dot plots at the same scale and using a line segment to represent the MAD on each dot plot, as shown here.

Invite a few students to share their observations about how the means and MADs of Noah and the eighth-grade student compare. Discuss:

• “How are the distributions of points related to the mean? How are they related to the MAD?” (The mean is usually near the center of the distribution and the MAD describes the spread of the distribution.)
• “Which might be more desirable for a basketball team: a lower mean or a higher mean number of baskets made? Why?” (A greater mean is preferable. It means that the person typically makes more baskets, which is usually good for a player on your team.)
• “Which might be more desirable: a lower MAD or a higher MAD? Why?” (Usually a lower MAD is better. It means that the player is more consistent, so you can more accurately estimate how the player will perform.)
• “Of the three students, which one would you want on your team? Why?” (I would want the eighth-grade student on my team because that player has the greatest mean and the least MAD. That player consistently scores about 6 out of 10 baskets.)

Students should walk away understanding that, in this context, a higher MAD indicates more variability and less consistency in the number of shots made.

In this activity, students continue to practice interpreting the mean and the MAD and to use them to answer statistical questions. A new context is introduced, but students should continue to consider both the center and variability of the distribution as ways of thinking about what is typical for a set of data and how consistent the data tends to be.

Display the dot plots for all to see. Invite a student to add the means and MADs to the plots. Then invite several students to share their comparison of the distributions. Here are some discussion questions:

• “What can you say about the size of the team? Has it changed?” (The team in 1986 had 14 swimmers and the team in 2016 has 22 swimmers, so there are a lot more swimmers on the later team. Maybe there are more specialty swimmers who swim in only one or two races rather than in several.)
• “Overall, has the team gotten older, younger, or stayed about the same? How do you know?” (Because the mean age is greater on the 2016 team, the typical swimmer is older on that team than the typical swimmer on the 1984 team was.)
• “Has the team become more diverse in ages, in general? Or have the swimmers become more alike in their age? How do you know?” (The swimmers on the 2016 team are more diverse in age because the MAD is greater.)