From Curiosity to a Statistical Question
A statistical investigation starts with a question. The question helps determine what information to collect and what the resulting data can tell us. A curiosity such as “Are students getting enough sleep?” is a useful starting point, but it is too vague to guide data collection: it does not say which students, what “enough” means, or what information would be recorded.
In What Is a Variable in Statistics, you learned to identify the individuals and the characteristics recorded about them. In Identifying the Observational Unit, you practiced naming what one observation describes. Use those ideas here: a well-formed statistical question makes clear what cases are being considered and what variable or variables will be observed.
Variability is the key idea. If a question asks about many students’ sleep times, the recorded answers will likely differ from student to student. If it asks for one specific student’s sleep time on one particular night, the answer is a single value. That single value might not be known yet, but not knowing an answer does not by itself make a question statistical.
A statistical question does not have to ask for a complicated calculation or a formal conclusion. It can ask what values occur, how much they vary, or how a category is distributed among cases. The important point is that its answer depends on data that can vary.
What Makes a Question Statistical?
A useful way to test a question is to imagine the data table that could answer it. What would one row represent? What variable or variables would appear in the columns? Would you expect the entries to be identical, or could they differ? As in Reading a Data Table: Rows, Columns, and Cases, the cases and variables should make sense together.
For instance, “How many minutes did Maya spend reading last night?” asks about one person on one occasion. It calls for one value. By contrast, “How many minutes did students in the reading club spend reading last night?” calls for a value for each student. Those values may vary, so the second question is statistical.
A statistical question can concern a categorical variable as well as a quantitative one. If a club asks each member for a preferred reading format, the answers may fall into categories such as print, audio, or digital. The responses can vary even though they are not measurements or counts. The distinction between categorical and quantitative variables, introduced in Categorical Versus Quantitative Variables, helps you describe what kind of data the question would produce.
The check is a guide, not a rule that every question must include a long list of details. A question may be statistical but still need clarification before data are collected. For example, “How long do people wait?” anticipates different waiting times, but it does not identify which people, where they are waiting, or what counts as the start and end of the wait.
Worked Example: Revising a Question About Sleep
A student is curious about sleep among students at a school and proposes, “Are students getting enough sleep?” Decide whether this is ready to use, then revise it into a statistical question.
Identify what is unclear: “Students” does not say which group is being considered. “Enough” has no stated definition. The question also does not say what information would be recorded. As written, different people might interpret it in different ways, so their answers might not be comparable.
Choose the cases and variable: For a focused investigation, suppose the cases will be students in one grade at the school. Record each student’s number of hours of sleep on the night before the survey. Sleep time is a quantitative variable, and students may report different values.
Write the statistical question: “How many hours of sleep did students in Grade 10 at this school get on the night before the survey?” This question identifies the cases, the recorded variable, and the occasion. It anticipates a range of answers rather than one value.
Answer: The original wording expresses a curiosity but is too vague to guide consistent data collection. The revised question is statistical because it asks for sleep-time data from multiple students, whose answers may vary.
Statistical and Non-Statistical Questions
A non-statistical question generally has one fixed answer, even if that answer has to be looked up or measured. “What time does the library open on Saturday?” asks for one schedule time. “How many pages are in this particular book?” asks about one book. Collecting those facts may take effort, but the questions do not expect answers to vary across a set of cases.
A statistical question, in contrast, asks about data from a collection of cases or occasions where differences are expected. It might ask how many pages are in each of several books, or what opening times are listed for different library branches. The wording makes the variation part of what the question is about.
Sometimes a small change in wording changes the type of question. “What is the height of this sunflower?” asks for one value. “How do the heights of sunflowers in this garden vary?” asks for measurements across multiple plants. The first may be answered by measuring one plant; the second requires data for a group and anticipates different measurements.
Do not decide based only on whether the question contains a number. A question about one person’s age is non-statistical in this sense, even though the answer is quantitative. A question about the favorite type of music across a group can be statistical, even though the answers are categories. What matters is whether the question expects variation in the data.
Worked Example: Questions About Reusable Bottles
A school sustainability club wants to learn about reusable-bottle habits. Classify each proposed question as statistical or non-statistical, and explain what data it would require.
Question A: “How many reusable bottles does Jordan own?” This asks about one person and one count. It calls for one value, so it is non-statistical as written.
Question B: “How many reusable bottles do students in the club own?” This asks for a count for each club member. The counts could differ, so it is statistical. The cases are club members, and the variable is number of reusable bottles owned.
Question C: “What color is the bottle on the club table?” This asks about one particular bottle. It has one answer, so it is non-statistical, even though color is a categorical characteristic.
Question D: “What colors are the reusable bottles owned by club members?” This asks for bottle-color information across members. The categories may differ, so it is statistical. If some members own more than one bottle, the question would need clarification about whether to record a color for every bottle or one response per member.
Answer: Questions B and D are statistical because they seek information across multiple cases and allow answers to vary. Questions A and C concern one person or object and ask for a single value.
Make a Vague Curiosity Specific
A broad curiosity often becomes more useful when you make three choices: identify the cases, name the variable, and specify the context or occasion. These choices do not guarantee that an investigation is complete, but they make the question easier to understand and answer consistently.
Say what one observation describes, such as one student, one garden plant, one bus trip, or one day.
State the characteristic to record for each case, such as travel time, plant height, preferred snack, or number of books borrowed.
Clarify a relevant place, group, time period, or event so readers know what the question covers.
Ask whether the answers could differ among the cases or occasions. If they could, the question can be statistical.
For example, “Are buses late?” leaves the relevant buses and meaning of “late” unclear. One revision is: “How many minutes late did each city bus on Route 6 arrive at the central stop during the morning commute this week?” It identifies the trips as cases, arrival delay as the variable, and the route, stop, and time period as context. The delays may vary from trip to trip.
Precision should fit the question. You do not need to add details that have no bearing on the data. But terms that affect how a value is recorded should be clear. If “late” means any arrival after the posted time, say so when that definition matters. If you ask about “screen time,” specify the device or time period if different interpretations could lead to different recorded values.
Worked Example: A Question About Community Garden Plants
A volunteer asks, “Do plants grow well here?” Turn this broad curiosity into a statistical question that could be answered with data from plants in a garden.
Identify the issue: “Plants” could mean many species, and “grow well” is not a clearly stated measurement. One person might mean height, another might mean number of leaves, and another might mean whether a plant survives.
Choose cases and a variable: Suppose each case is one tomato plant in a particular garden bed. Record each plant’s height, in centimeters, on a specified date. Height is a quantitative measurement, and plants may have different heights.
Write the question: “What are the heights, in centimeters, of the tomato plants in the east garden bed on June 15?” This is statistical because it requests measurements across multiple plants and anticipates variation in those measurements.
Consider another valid version: If the volunteers care about leaf counts instead, they could ask, “How many leaves does each tomato plant in the east garden bed have on June 15?” This is also statistical: it uses the same type of cases but a different quantitative variable. The question must match the information the volunteers actually want.
Answer: The vague question becomes answerable when “plants” and “grow well” are replaced by clearly identified cases and a recordable variable. The chosen variable shapes what the resulting data can address; plant heights alone do not answer every possible meaning of “grow well.”
Common Mistakes and AP Exam Tips
- Calling any unanswered question statistical. Not knowing a fact does not mean it anticipates variability. “What is the height of this one tree?” still asks for one value.
- Assuming a statistical question must use numbers. A question about favorite fruit or travel method can be statistical if it asks for categorical answers across cases and those answers may differ.
- Leaving the cases unclear. “How long do people exercise?” does not explain which people or what counts as exercise. A stronger response names the cases and gives enough context to understand the data being requested.
- Using an undefined phrase as the variable. Words such as “healthy,” “popular,” and “successful” can have several meanings. State what information would be recorded to represent the idea.
- Confusing many possible answers with many cases. A question about one person could have several possible answers, but it still asks for one value. A statistical question expects variation across a set of cases or occasions.
- Writing only “it varies.” A full-credit explanation says what varies and across what cases or occasions. For example: “This is statistical because travel times may differ among the bus trips on this route during the stated week.”
On an AP response, support your classification with the question’s data structure. Name the cases, identify the variable or information being recorded, and explain whether different cases could give different answers. If a question is vague, identify what needs clarification and offer a specific revision.
Check Your Understanding
For each item, decide whether the question is statistical or non-statistical. If it is vague, suggest what detail would make it clearer.
- “How many minutes did one particular student spend walking to school yesterday?” Explain why the answer does or does not involve expected variation across cases.
- “How long do students in the school’s chess club spend practicing each week?” Name the cases and the variable, and explain whether the question is statistical.
- “What is the most common way people travel?” Identify one detail that should be specified before collecting data.
- Rewrite “Are the neighborhood parks busy?” as a statistical question that identifies cases, a variable, and a relevant time context.
- Explain why “Which music genre does Priya prefer?” is not statistical as written, then revise it into a statistical question about a group.