In this activity, students are asked to compare the heights of two groups of people. The wording of the questions allows for multiple interpretations and any reasonable answer should be accepted as long as the argument is supported (MP3). This activity also provides an opportunity to remind students of how to analyze dot plots as well as how to calculate the measures of center and variability of the data.

The purpose of this discussion is for students to think about ways to approach comparing two groups as well as have an opportunity to review dot plots, measures of center, and measures of variation from prior grades.

Ask, “What are some ways we can compare groups of things?” At this stage, students are only expected to informally compare the groups. Although a consistent general rule for comparing groups will be introduced in later lessons, this activity is about getting a general idea that some groups (like the gymnastics and volleyball teams) have a rather clear difference, while others (like the tennis and badminton teams) may be more alike.

Ask students about the distribution of the data shown in the dot plots. Make sure to highlight the shape, center, and spread. Review how to find the mean as a measure of the center of a data set. Review how to calculate the mean absolute deviation (MAD) as a measure of the variability of a data set. Students may mention median and interquartile range (IQR) as other ways to measure center and variability. Although median and IQR are not needed in this activity, it may be useful to review how to calculate those values as well. Both measures of center and variability will be used later in the unit.

Introduce the idea of judging how much two data sets overlap.

In this activity, students use a measure of center and a measure of variability to compare two groups more formally. In the discussion following the activity involving describing the difference of the measures of center as a multiple of the variability, students are shown one quantifiable method of determining whether the two groups are relatively close or relatively very different (MP2). The important idea for students to grasp from this activity is that the measures of center and measures of variability of the groups work together to give an idea of how similar or different the groups are.

Monitor for groups who use different strategies to compare the heights. Here are some strategies from less to more formal:

1. Create dot plots of the data and look at the overlap visually.
2. Compute measures of center to compare the groups numerically.
3. Also compute measures of variability to compare the measures of center for each group accounting for the spread of values.

Allow students access to calculators to compute measures of center and measures of variability so that students can use those values without using too much time calculating in other ways.

The purpose of this discussion is for students to begin to be more formal in their comparison of data from different groups. In particular, a general rule is established that will be used in this unit.

Invite previously selected groups to share how they compared the groups. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “How does the inclusion of a numerical value for a measure of center help the argument that Jada’s family tends to be taller?” (By including calculated numbers, it can reduce bias about how different the groups are.)

• “In addition to the measure of center, how does a measure of variability help show how different the groups are?” (Along with the measure of center, it gives a number to how much overlap we expect from the groups.)

Explain that the difference between the means is not enough information to know whether or not the data sets are very different. One way to express the amount of overlap is to divide the difference in means by the (larger) mean absolute deviation.

Demonstrate for students how to do this calculation for the volleyball and gymnastics teams: 

• The difference in means is more than 6 times the measure of variability: .

Leave the calculation for the two teams displayed. Ask students to do the calculation for Jada’s and Noah’s families. 

• The difference in means is a little more than 1 time the measure of variability: .

As a general rule, we will consider it a large difference between the data sets if the difference in means is more than twice the mean absolute deviation. If the mean absolute deviation is different for each group, use the larger one for this calculation.

For students who ask why twice the MAD is used rather than some other value, defer the question for later in the unit.

Students begin by matching information about a set of data to its dot plot and calculating the mean and mean absolute deviation for the remaining dot plots. Then they compare the data sets pairwise using the mean and MAD values (MP3). In the discussion, students are introduced to a way to add extra information to dot plots to visualize the general rule given in the previous activity.

The purpose of the discussion is to help students visualize the calculations they performed.

Ask, “Before calculating any of the values, would you have guessed that dot plots B and C would not be very different, but A would be very different from the other two? Explain your reasoning.” (Yes, since there is some overlap between the data of dot plots B and C, but the data in A does not overlap very much with B and not at all with C.)

Demonstrate a technique using Dot Plot A for students to see the general rule on the dot plots. Mark the mean of the data set on the dot plot with a triangle. Then draw a line segment from the triangle so that its length is 2 MADs in each direction. On Dot Plot A, draw a triangle at 5.57 and a line segment from 4.69 () to 6.45 () as in the image here.

Tell students to draw a triangle for the mean and a segment representing 2 MADs in each direction on all three dot plots. Ask students, “How can you use these markings to see the general rule from these pictures?” (If any of the means are within 2 MADs of the mean of another group, then there is not a large difference between the two groups.)