In this activity, students reason abstractly and quantitatively (MP2) about numerical data sets to match them with questions that are likely to produce the data. They see that some survey questions lead to responses that are expected to vary when posed to different people, but others produce responses that are likely to be the same. Students categorize data sets based on whether more than one different value is present and make use of this structure (MP7) to define variability.

As students match questions with data sets, look out for different plausible explanations for their choices. Their matches are reasonable if they can explain why the given data could be responses to a question. Identify one student to share the response to each question. Also notice students who offer different but equally reasonable explanations for the same data set, and invite them to share later.

Some students may have trouble matching questions and data sets because they do not attend carefully to the range of possible solutions. For example, they may not notice that a data set with 11 as a data value cannot be a response to the first question about flipping a coin 10 times. Ask them to study the questions and data values more closely, and to look for values that seem unlikely or impossible for a given context.

The purpose of this discussion is for students to define variability and recognize when it is present.

Select previously identified students to share their choices and explanations. Briefly poll the class after each explanation to see if others made the same choice for the same reason. If not, invite students with different explanations to share.

Discuss how the question about grade level and the one about number of inches in a foot are different from the others. If not mentioned by students, highlight the idea of variability. Explain that we use the term variability to describe data sets in which not every data value is the same. Data sets B and D are unlike the other sets because they show no variability.

In this activity, students use both the experience of reasoning about questions and the idea of variability to define statistical questions as questions that can be answered by using data in which variability is expected.

For example, the question, “What is the favorite subject of students in my class?” is a statistical question because we need data about favorite subjects and we can expect students to have different preferences. The question, “What is the counselor's favorite subject?”, is not a statistical question because it can be answered by collecting a single data value. Even if multiple responses were collected, the responses are not expected to show variability.

As students analyze and discuss examples and non-examples of statistical questions, listen for groups who distinguish the two in terms of the data needed to answer the questions. For example, some questions may require collecting data that will probably show some variability while other questions may have only a single response. Invite them to share later.

Invite students to share their responses to the second set of questions. Be sure that students understand that a question is statistical if we need data to answer it and that the data are expected to have variability. 

Determining whether a question is statistical or not may depend on the situation in which it is being asked. Here are two examples to ask students:

• “Who is the tallest student in the class?”

• “How long is the longest river in the United States?”

While the tallest student may be obvious in some classrooms, it is helpful to remember that this is not true in all classrooms. The students in a class are often close (but not identical) in height, and finding out who is tallest requires collecting data about different heights then analyzing it to determine who is the tallest. So the question, “Who is the tallest person in the class?”, is generally a statistical question, but there may be specific situations in which it is not.

Likewise, while the longest river in the country can be easily researched, it is helpful to remember that this was not always the case. The answer may be considered a fact now, but the question was once a statistical question—at some point, lengths of rivers were collected and compared in order to answer it.  

To tell statistical questions from non-statistical ones, it is useful to look closely at the context of the questions and what it takes to answer them.

This optional activity provides additional practice in determining what it means for a question to be statistical in nature.

Students develop a deeper understanding of statistical questions by studying a wider range of examples and non-examples. Students sort a variety of questions and explain why they are or are not statistical. During the card sort, they explain their reasoning, critique the reasoning of others, and attempt to persuade one another (MP3). Students also begin writing statistical questions and think about the data that might be used to answer the questions.

As students work, encourage them to refine their descriptions of statistical questions using more precise language and mathematical terms (MP6).

Most of the discussions happen in small groups. Bring the class together to discuss any remaining disagreements or questions.

Direct students' attention to the reference created using Collect and Display. Ask students to share how they decided which cards went into each group. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. (For example, “We decided that the height of the door is not statistical because it requires only a single measurement with no variability.”)

Ask the class:

• “Do you all agree with the list of the statistical questions?”
• “If not, which one(s) are harder to distinguish? Why?”
• “Now that you have seen more examples of statistical questions, what new insights do you have about them?”

Ensure that students understand the difference between statistical questions and survey questions. A survey question is what is used to collect data. A statistical question is one that is answered using collected data. 

For example, the question, “Are most residents of this building older or younger than 30?” is a statistical question, because answering it requires collecting and analyzing the ages of the residents. It is not a question you would ask people directly, though, so it is not a survey question.

The related survey question is “How old are you?” because it can be used to gather data about the ages of people in a group being studied. On its own, though, it is not a statistical question because there is a single response without any variability when it is asked.