Lesson 11

The purpose of this warm-up is for students to first reason about the mean of a data set without calculating and then practice calculating mean. The context will be used in an upcoming activity in this lesson so this warm-up familiarizes students with the context for talking about deviation from the mean. 

In their predictions, students may think that Elena will have the highest mean, because she has a few very high scores (7, 8, and 9 points). They may also think that Lin and Jada will have very close means because they each have 5 higher scores than one another and their other scores are the same. Even though each player has the same mean, all of these ideas are reasonable things for students to consider when looking at the data. Record and display their predictions without further questions until they have calculated and compared the mean of their individual data sets. 

Arrange students in groups of 3. Display the data sets for all to see. Ask students to predict which player has the largest mean and which has the smallest mean. Give students 1 minute of quiet think time and then poll students on the player who they think has the largest and smallest mean. Ask a few students to share their reasoning.

Tell each group member to calculate the mean of the data set for one player in the task, share their work in the small group, and complete the remaining questions. 

Ask students to share the mean for each player's data set. Record and display their responses for all to see. After each student shares, ask the class if they agree or disagree and what the mean tells us in this context. If the idea that the means show that all three students make, on average, half of the 10 attempts to get the basketball in the hoop does not arise, make that idea explicit. 

If there is time, consider revisiting the predictions and asking how the mean of Elena's data set can be the same as the others when she more high scores?

In this activity, students continue to develop their understanding of what could be considered typical for a group as well as variability in a data set. Students compare distributions with the same mean but different spreads and interpret them in the context of a situation. The context given here (basketball score) prompts them to connect the mean to the notion of how “well” a player plays in general, and deviations from the mean to how “consistently” that player plays. 

They encounter the idea of calculating the average absolute deviation from the mean as a way to describe variability in data.

Arrange students in groups of 3–4. Give groups 6–7 minutes to answer the questions and follow with a whole-class discussion.

The purpose of the discussion is to highlight that the center of the distribution is not always the only consideration when discussing data. The variability or spread can also influence how we understand the data.

There are many ways to answer the second set of questions. Invite students or groups who have different interpretations of “playing well” and “playing consistently” to share their thinking. Allow as many interpretations to be shared as time permits. Discuss:

• “How might we use the given data to quantify ‘playing well’ and ‘playing consistently?’”
• “Is there a way to describe variability and consistency in playing precisely and in an objective way (rather than using broad, verbal descriptions)?”

Explain that we can describe variability more formally and precisely—using a number to sum it up; we will look at how to do so in the next activity. 

In the previous activity, students evaluated the performance of three students based on the mean and variability. Here they 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 variations 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 grade 6. 

Remind students in the last lesson they found distance between each data point and the mean, and found that the sum of those distances on the left and the sum on the right were equal, which allows us to think of the mean as the balancing point or the center of the data. Explain that the distance between each point and the mean can tell us something else about a distribution. 

Arrange students in groups of 2. Give students 4–5 minutes to complete the first set of questions with their partner, and then 4–5 minutes of quiet time to complete the remaining questions. Follow with a whole-class discussion.

Students may recall the previous lesson about thinking of the mean as a balance point and think that the MAD should always be zero since the left and right distances should be equal. Remind them that distances are always positive, so finding the average of these distances to the mean can only be zero 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?”
• “What about a data set with points that show large distances from the mean?”
• “Does a data set with smaller distances (and therefore smaller average distances) show less or more variability?”
• “What do MAD values of 2, 1.5, and 1 mean in this context?”

This optional activity uses a game to help students develop the idea of variability. Use this activity if students could benefit from more concrete experiences around the idea of distance from the mean. Students will draw from a standard deck of playing cards and find the sum. The player with the least mean distance from 22 wins the round.

Students in groups of 2–3. A deck of playing cards per group. Play 4–6 rounds.

Ask students to think about how average distance from a number can be used to summarize variability and invite a couple of students to share their thinking.

In the game, we can think of the player with the least average distance from 22 as having cards that are, on the whole, closest to 22 or the “least different” from 22. By the same token, a player with the greatest average distance from 22 can be seen as having cards that are, on the whole, farthest away from 22 or the “most different” from 22. Connect this to the idea that a data set with a large MAD means it has many values that vary from what we could consider a typical member of the group.