Lesson 15

The mathematical purpose of this activity is for students to compare different distributions using shape, measures of center, and measures of variability. This warm-up prompts students to compare four distributions representing recent bowling scores for potential teammates. It gives students a reason to use language precisely (MP6) and gives you the opportunity to hear how they use terminology and talk about characteristics of the images in comparison to one another. 

Arrange students in groups of 2–4.

Ask each group to share one bowler they would choose and their reasoning. If none of the groups select a certain player, ask why this player was not chosen or to give reasons why another team may want this player on their team. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain how they used the statistics given as well as the histograms.

The mathematical purpose of this activity is for students to compare measures of center and measures of variability in context. Monitor for students who

1. Determine the slower age group by using an informal description of the shift in data
2. Determine the slower age group by using a numerical estimate for the mean or median for measures of center
3. Determine variability from the range of values.
4. Use a numerical estimate for IQR or standard deviation as measures of variability.

The purpose of this discussion is for students to understand how to compare data sets using measures of center and measures of variability.

Select students to share their answers and reasoning in a sequence that moves from less formal reasoning to more definite values and statistics.

After several estimates for measure of center and measure of variation are mentioned, display the actual values for these data sets.

Ages 30–39

• Mean: 294.5 minutes
• Standard deviation: 41.84 minutes
• Median: 288.5 minutes
• IQR: 65 minutes
• Q1: 260 minutes
• Q3: 325 minutes

Ages 40–49

• Mean: 340.6 minutes
• Standard deviation: 59.93 minutes
• Median: 332 minutes
• IQR: 92 minutes
• Q1: 289 minutes
• Q3: 381 minutes

Ask students:

• “What measure of center and measure of variability are most appropriate to use with the distributions? Explain your reasoning.” (The median is the most appropriate because it is closer to where the majority of the data is. Since I used the median as the measure of center, I used the IQR as a measure of variability.)
• “Which points show values that are most likely to be outliers?” (The slowest runners in each group, on the right end of the dot plot, might be outliers.)
• “Based on the displayed information, are there any outliers in these data sets?” (No. For the 30–39 age group, outliers would need to be values greater than 422.5 minutes, but the slowest times are around 380 minutes. For the 40–49 age group, outliers would need to be greater than 534 minutes, but the longest times are around 450 minutes.)

In this activity, students take turns with a partner determining the best measure of center and the best measure of variability for several data sets. Students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3). Students also determine which data set has a greater measure of center and which has a greater measure of variability.

Arrange students in groups of 2. Tell students that for each data display or description of a data set in column A, one partner determines the appropriate measure of center and measure of variability and explains why they think it is appropriate. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next data display or description of a data set in column B, the students swap roles. If necessary, demonstrate this protocol before students start working. The last item has a column C. Students can work together to determine the best measures for set C. Once an agreement is reached for each group of data sets, students will determine which data set has the greatest measure of center, and which data set has the greatest measure of variability.

For the situations described in words, students may think there is not enough information to answer the question. Ask these students, "What do you think the distributions might look like for the situations described?" Tell them to use their distributions to answer the question and be prepared to explain their reasoning.

Select students to share how they determined whether to use the mean or the median, and how they figured out which data set showed greater variability.

• “What were some ways you handled the last two problems.” (I reasoned about what a dot plot might look like to imagine where the center might be and how spread out the data might look.)
• “Describe any difficulties you experienced and how you resolved them.” (I forgot what information was in a box plot, so I asked my partner.)
• “How did you decide whether to use the mean or the median?” (I recalled what I learned about shape from the previous lesson. When the distribution was symmetric or close to it, I used the mean. When it was skewed, I used the median.)
• “How did you decide which data set showed greater variability?” (It was easy for the box plot, I calculated the IQR. For the dot plots, I looked at which one was more spread apart. For the problem contexts, I tried to see which one would have data that would be more varied.)