1.5: Exploring Statistical Questions
- Page ID
- 58857
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Thinking Like a Statistician – A Guided Experience
Now that you've seen the building blocks of a statistical investigation — data types, sampling, randomness, and bias — it's time to bring it all together. In this section, you'll engage with a real question and work through the full cycle of a statistical study.
This will give you practice using the language and structure you've learned so far, and prepare you for more complex analyses in future chapters.
The Six-Step Statistical Process
There are six major steps that guide statistical thinking. Every good study — from a school science project to a published research paper — follows some version of this cycle.
Choose a Question to Investigate
Below are five sample statistical questions from different real-world domains. Select one that interests you most — then use the guided steps and the AI reflection tool that follows to work through your thinking.
- Education: Do students who participate in after-school tutoring programs perform better on standardized math tests?
- Environment: Is there a relationship between air quality and asthma-related hospital visits in major cities?
- Health: How does average daily screen time differ between high school students who sleep 6 or fewer hours and those who sleep 8 or more?
- Business: Does the size of a restaurant's menu affect the average customer rating on review platforms?
- Sports: Are baseball players with higher batting averages more likely to have higher on-base percentages?
Work Through the Six Steps
Take your chosen question and apply each step below. Use a notebook, a worksheet, or a small group discussion to organize your thinking. Some ideas or concepts below may be unfamiliar — take time with each step. Throughout the book we will reflect back on this process, gradually filling in all the details.
Step 1: Ask a Clear Question
Rephrase the topic into a well-formed statistical question.
- Does your question involve variability?
- Can it be answered by collecting real data (not just opinions)?
- What exactly are you comparing or examining?
Step 2: Collect Accurate and Relevant Data
Imagine or sketch out a plan to collect meaningful data.
- Who or what will you study (your population)?
- Will you take a sample, and if so, how will you select it?
- Where will the data come from — a survey, experiment, database, or sensors?
- Could any forms of bias affect your results?
Step 3: Organize and Summarize the Data
Think about how to make sense of what you collect.
- Would tables, averages, or percentages be useful?
- What types of graphs or displays might help explain the data?
- How will you account for trends, groups, or outliers?
Step 4: Analyze the Data
What relationship or pattern are you looking for?
- Are you comparing two groups? Looking for a correlation?
- Will you be calculating differences, percent changes, or association measures?
- Would a statistical test or confidence interval help support your findings later?
Step 5: Interpret the Results in Context
What do your calculations and patterns actually mean?
- What is your overall takeaway?
- How confident are you in your result? What limitations exist?
- Could there be alternative explanations?
Step 6: Communicate Clearly and Effectively
Imagine explaining this to someone unfamiliar with the data.
- How would you explain your findings in plain language?
- What visuals or summaries would support your answer?
- What key information must be included to avoid misinterpretation?
Looking Ahead: Designing a Statistical Study
Next we will look at how to synthesize the ideas of questions, sampling, and data collection into the framework of a complete statistical study.