Identifying Statistical Questions - Sixth Grade Mathematics

Statistics is the mathematical discipline devoted to collecting, organizing, analyzing, and interpreting numerical data to address questions characterized by inherent variability.

What is a Statistical Question?

A statistical question is an inquiry that anticipates variability in the data collected to answer it. Instead of having a single, deterministic answer, a statistical question generates a distribution of varying numerical or categorical responses.

The Core Distinction:
+-----------------------------------+-----------------------------------+
| Statistical Question              | Non-Statistical Question          |
+-----------------------------------+-----------------------------------+
| - Anticipates VARIABILITY.        | - Expects a SINGLE, FIXED fact.   |
| - Generates a collection of data. | - Has only one definitive answer. |
| - Answers describe a group/trend. | - No variation in the data.       |
+-----------------------------------+-----------------------------------+

Examples Contrasted:

Non-Statistical: "How old is our math teacher?"
Reason: The teacher has one specific age. There is zero variability.

Statistical: "How old are the students in our school?"
Reason: Students have many different ages (11, 12, 13, 14...). There is natural variability.

Non-Statistical: "How tall is Mount Everest?"
Reason: It has a single fixed geographic elevation.

Statistical: "How tall are the mountains in Colorado?"
Reason: Different peaks have distinct heights across a wide range.

Categorical vs. Numerical Data

When investigating a statistical question, the responses collected fall into two primary classifications:

1. Numerical Data (Quantitative):
   - Values that are counts or measurements represented by numbers.
   - You can meaningfully compute mathematical operations (average, median).
   - Examples: Heights, test scores, hours of sleep, number of pets.

2. Categorical Data (Qualitative):
   - Values that describe qualities, traits, or groups sorted into categories.
   - Represented by words, labels, or codes.
   - Examples: Favorite ice cream flavor, eye color, mode of transportation to school.

Crafting Well-Defined Statistical Questions

To conduct meaningful statistical research, a question must define:
1. The population of interest (who or what is being studied).

2. The specific variable being observed (what measurement or category is recorded).

3. The context and units of measurement.

Refining a Question:
Vague / Poor: "Do kids like sports?"
Well-Defined Statistical Question: "How many hours per week do sixth-grade students at Oak Middle School spend participating in organized sports?"

Exercises

Exercise 1. Determine whether each question is a statistical question. Explain your reasoning: (a) How many letters are in the English alphabet? (b) How many books did each sixth-grade student read over summer vacation? (c) What is the capital of France? (d) What shoe sizes are worn by the players on the school basketball team? (e) How many minutes does it take students to travel from home to school?

Exercise 2. For each statistical question, identify whether the expected data is numerical or categorical: (a) What is the favorite cafeteria meal among middle school students? (b) How many pets live in the households of students in our class? (c) What are the resting heart rates of seventh-grade athletes? (d) What brand of smartphone do teenagers prefer?

Exercise 3. Rewrite each non-statistical question into an effective statistical question: (a) How much did Jordan pay for his backpack? (b) How many inches of snow fell yesterday in Chicago?

Solutions

Solution to Exercise 1. For (a), it is not a statistical question because there is exactly one fixed answer (26 letters) with no variability. For (b), it is a statistical question because different students read different numbers of books, producing variability in the data. For (c), it is not a statistical question because Paris is the single unchanging capital of France. For (d), it is a statistical question because basketball players wear a distribution of varying shoe sizes. For (e), it is a statistical question because travel times vary widely among students depending on distance and transportation mode.

Solution to Exercise 2. For (a), cafeteria meals are descriptive food categories (pizza, tacos, salad), so the data is categorical. For (b), number of pets is a numerical count (0, 1, 2, 3...), so the data is numerical. For (c), heart rate is a measured quantity in beats per minute, so the data is numerical. For (d), smartphone brands are manufacturer names (Apple, Samsung, Google), so the data is categorical.

Solution to Exercise 3. For (a), a rewritten statistical question could be: "How much money do sixth-grade students typically spend on their school backpacks?" For (b), a rewritten statistical question could be: "How many inches of snow fall on each day during the month of January in Chicago?"