In this activity, students compare and contrast different measures of center and variability for data sets that have gaps and are not symmetrical. They interpret mean, MAD, median, and IQR in the context of a situation. Unlike many of the data sets that students have seen so far, this one shows values that could roughly divide into three parts: the days when there is little or no homework, the days when there is a moderate number of homework problems, and the days when the assignment is relatively large. Because of this distribution, finding a typical number of homework problems (or whether it would be helpful to identify a typical number) is not obvious, prompting students to interpret measures of center and spread more carefully (MP2).

As students work and discuss, identify at least one student or group that decides that the mean and MAD are appropriate measures of center and spread and can explain their reasoning, and another that decides to go with the median and IQR and could support their choice. Invite them to share during a whole-class discussion.

In the digital version of the activity, students use an applet to calculate the IQR and MAD. The applet allows students to focus on finding the less calculation-heavy statistics. Use the digital version if students need more work on interpreting the statistics in context rather than on calculating their values.

Briefly discuss students’ interpretations of the measures they just calculated:

• “What do the mean of 9 and MAD of 7.4 tell us? How can we interpret them in this context?” (The mean tells us that she could’ve done 9 problems per day and completed the same total number of problems. The MAD being close to the value of the mean means that there is a large spread to the number of problems due each day.)
• “What do the median of 7.5 and IQR of 14 tell us?” (The median tells us that half of the days had fewer than 7.5 problems assigned and half had more. The IQR means that the middle half of the distribution includes values that are up to 14 problems apart.)

Then, select two or more previously identified students to share their responses about which measures of center and spread are appropriate for summarizing the data set. After each person shares, briefly poll the class to see if others reasoned the same way about the measures. Summarize by asking:

• “Now that you have two pairs of measures of center and spread, how would you respond if someone asked you for a typical number of homework problems that Mai did over these 2 weeks. Is the question easier to answer now?” (We have some values to say, but neither does a great job describing Mai’s homework. The most important feature about this distribution is the wide spread.)

Students should walk away with increased awareness that, in some cases, measures of center and spread do not always paint a full picture of what the actual data set entails, and that the measures should be interpreted with care. 

In this activity, students show a data set using three different kinds of graphical representations—a dot plot, a box plot, and histograms using different bin sizes—and decide which are more useful or more appropriate for communicating the distribution. 

As students work and discuss, identify those who draw clear graphical displays, those who notice that the different displays offer different insights about the data distribution, and those who advocate for using different representations to display Jada's data. Ask them to share with the class later.

In the digital version of the activity, students use an applet to create the graphical representations. The applet allows students to focus on interpreting the information. Use the digital version if students need more work on interpreting the statistics in context rather than on drawing the representations.

Invite previously identified students to share their dot plot, box plot, and histograms. Display their drawings for all to see. Then, select several students or groups to share their response to their choice of which representation is best and their explanation. If not already mentioned by students, discuss the different insights that each display offers, or different challenges that each poses. For instance, consider asking these questions about each data display:

• “What information can we get from this display?” (The dot plot shows all of the values and is good for estimating the center and spread. The box plot clearly shows the five-number summary. A histogram can show the shape of the distribution, but loses some information about the exact data values.)
• “Does it give us a meaningful snapshot of the distribution?” (Each display could be useful for different purposes. In this case, students may defend their choice using arguments similar to those listed in the response section.)
• “What characteristics of a different data set would make this representation more useful?” (Dot plots are good when there are not many different values. Box plots are good for when there is a lot of data or when the median and IQR are useful. The histograms are useful when there is a lot of data and the shape of the distribution is important.)

Help students see that, in this case, none of the representations here are ill-suited to represent the data set, but a couple of them allow us to describe the distribution of the data set more easily because of how they summarize the data values in some ways.

In this culminating activity, students use what they have learned in the unit to answer statistical questions about a species of fish in the Great Lakes region. They use a histogram to represent the given data distribution, decide on appropriate measures of center and variability, and use their analyses to draw conclusions about a certain fish population.

To allow all groups a chance to present, consider putting 2–3 groups together and asking them to present their work to each other. Groups that are not the first to present should focus on sharing new insights that have not been mentioned by the preceding groups. Invite students who are not presenting to attend carefully to the reasoning of the presenting group and to ask clarifying questions. 

If time permits, highlight some conclusions that students draw about whether the fish in the sample are young, adult age, or old, and whether researchers should be worried.

Tell students that several years after the students at Rufus King High School participated in the research, newer samples of yellow perch showed more favorable length-age distributions: More of the the fish are smaller or younger.

This optional activity provides students the opportunity to work with a large data set that would not be reasonable to work with by hand.

Throughout much of the unit, students have worked with smaller data sets so that they could check their understanding by hand and get a sense of the data as a whole. Sometimes, though, this makes the calculation of statistics seem unnecessary because many of them could be clearly seen. The power of these statistics comes out with larger data sets when the values are less obvious.

Due to the nature of the task, only a digital option is available.

Invite previously selected groups to share their visual displays. Encourage students to ask questions about the choice of data display and measures of center and variability that went into creating the display. Here are questions for discussion, if not already mentioned by students:

• “Which measure of center did you use to describe the distribution and why? What does it mean in this situation?” (We selected the median because the distribution didn’t appear to be almost symmetric. It means that half of the butterflies have a wingspan of 38.9 cm or longer and half have wingspans that are shorter.)
• “Which measure of variability did you use to describe the distribution, and what does it mean in this situation?” (Because we used the median for the measure of center, we used the IQR to measure the variability. In this situation it means that the middle half of the butterfly wingspans are within 21.8 cm of one another.)