Chapter 13 · Statistics and data displays
Statistical Questions: Expecting Data That Vary
Learn to spot questions that need data you expect to vary, and practice writing statistical questions of your own.
Here are two questions about backpacks.
- How heavy is your backpack right now?
- How heavy are the backpacks in your class right now?
You can answer the first question with one weighing. You get one number of pounds, and you are done.
The second question is different. You would weigh every backpack, so you would collect many numbers, and you would not expect them all to be the same. One weighing cannot answer it.
In this lesson you will learn to tell these two kinds of questions apart. Then you will write the second kind yourself.
Data are the pieces of information you collect by counting, measuring or asking. Each piece is one data value. A data value can be a number, such as pounds, or a word, such as a favorite color. All the data values you collect for one question form a set of data.
When the values in a set of data are not all the same, the data vary. Some values can match. The data still vary as long as the values are not all the same. Variability is the name for the differences among the values.
You can often expect variability before you collect anything. You have not weighed a single backpack, yet you already expect the weights to vary, because students pack different things. To expect something ahead of time is to anticipate it.
Key concept
A statistical question anticipates variability
A statistical question is a question you answer by collecting data whose values you expect to vary. In other words, it anticipates variability. Test any question with two checks:
- What data would I collect to answer it?
- Do I expect those data values to vary?
If you would collect many values and you expect them to vary, the question is statistical. If you would collect only one value, or look up one known fact, or if every value must be the same, it is not statistical.
Because the values vary, the answer to a statistical question must take all of them into account. It describes the whole set of data, not just one of its values.
| Question | Data you would collect | Do you expect the values to vary? | Statistical? |
|---|---|---|---|
How heavy is your backpack right now? | One weight | No. There is only one value. | No |
How heavy are the backpacks in your class right now? | One weight for each backpack | Yes. Students pack different things. | Yes |
How many days are in a week? | None. It is a known fact: days. | No. The answer is always days. | No |
How long did the students in your class sleep last night? | One number of hours for each student | Yes. Students go to bed and get up at different times. | Yes |
How many sides does each square on a worksheet have? | One count for each square | No. Every square has sides. | No |
Look at the last row of the table. You would collect many counts there, one for each square, but every count must be . Collecting many data values is not enough. You must also expect the values to vary. That is why the test has two checks.
Worked example
Test two questions about letters
Decide whether each question is statistical. Question A: How many letters are in the word "triangle"? Question B: How many letters are in the first name of each student in your class?
Start with Question A and ask what data you would collect. You would count the letters in one word, t-r-i-a-n-g-l-e, so you would collect one count.
Ask whether you expect the data to vary. There is only one value, 8. One value cannot vary, because there is no other value for it to differ from. So Question A is not statistical.
Now take Question B and ask what data you would collect. You would count the letters in every student's first name, one count for each student. For example, the names Ana, Priya and Marcus give the counts 3, 5 and 6.
Ask whether you expect the counts to vary. You do, because first names come in different lengths. You would collect many values and you expect them to vary, so Question B is statistical.
Answer
Question A is not statistical, because you would collect only one count, . Question B is statistical, because you would collect one count for each student and you expect those counts to vary.
Now see what happens when a club collects real data. A reading club has members. The club asks: How many books did each member read last month?
To answer it, every member reports one whole number of books. That number is the member's data value.
The table shows the results. Each row stands for one exact number of books, not a range of numbers. The count beside it tells how many members gave that number. The table lists every whole number of books from to , even , which no member gave.
| Number of books | Number of members |
|---|---|
Worked example
Check a question against its data
The reading club asks, "How many books did each member read last month?" Use the table of results above. Is the club's question statistical? What kind of answer fits its data?
Run the two checks on the question. To answer it, you need one number of books from each of the 16 members. You expect those numbers to vary, because people read at different speeds and have different amounts of free time. So the question is statistical.
Decide how to count the data values in the table. Each count is a number of members, and each member gave exactly one number, so the number of data values is the sum of the counts. Do not count the rows that have members instead. There are 5 such rows, but one row can stand for several members.
Add the counts one at a time, from the top row down. The total is 16, which matches the 16 members. So the table holds one data value for each member.
See whether the data vary, as you expected. Members gave 0, 1, 2, 3 and 5 books. Some members share a value, but the values are not all the same, so the data vary.
Build an answer that takes all 16 values into account. One member's number cannot stand for the club. One way to describe the set is to give the lowest value, the highest value and the value that the most members gave. The lowest is 0 and the highest is 5. The largest count is 6, so the value given most often is 2.
Answer
The club's question is statistical. Its data values vary, so a fitting answer describes the whole set: members read from to books last month, and books was the most common amount, read by members.
Now write statistical questions yourself. To write one, choose two things.
- A group to collect from. The group can be many people, many objects or many times, such as the nights in a month. Each one is a member of the group.
- One thing to record for each member. Record the same thing, in the same unit, for every member.
Then run the two checks on your question. If you would collect only one value, widen the group. If every value must be the same, change the thing you record. Keep the question only when you expect the values to vary.
Worked example
Turn one question into two statistical questions
Lena asks, "How many minutes did I spend on homework last night?" Her question is not statistical. Write two statistical questions about minutes of homework: one about a group of students, and one about Lena alone.
Find what keeps Lena's question from being statistical. It asks about one student on one night, so you would collect only one number of minutes. One value cannot vary.
Widen the group to many people. Choose the sixth graders at Lena's school as the group. Record the same thing for each member: minutes of homework last night. This gives the question: How many minutes did each sixth grader at Lena's school spend on homework last night?
Run the two checks. You would collect one number of minutes for each sixth grader. You expect those numbers to vary, because students have different assignments and work at different speeds. The question is statistical.
Now keep Lena and widen the group in time instead. Choose the school nights in this month as the group. Record the same thing for each night: minutes of homework. This gives the question: How many minutes did Lena spend on homework on each school night this month?
Run the two checks again. You would collect one number of minutes for each school night. You expect those numbers to vary, because the homework is not the same every night. The question is statistical, even though it is about one person.
Answer
Two statistical questions are "How many minutes did each sixth grader at Lena's school spend on homework last night?" and "How many minutes did Lena spend on homework on each school night this month?" Each needs many data values that you expect to vary.
Takeaway. A statistical question is one you answer by collecting data whose values you expect to vary. To identify one, ask what data you would collect and whether you expect those values to vary. To write one, choose a group and one thing to record for each member, then run the same two checks. Because the data vary, the answer must describe the whole set of values.
Next, you will look more closely at variability. You will learn to describe a whole set of data: where its middle is, how spread out it is and what shape it makes.