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Exploring one-variable data and collecting data practice problems AP Statistics, Unit 1

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Describing distributions, z-scores and the normal model, sampling methods and bias, and the design of experiments. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for exploring one-variable data and collecting data

x̄ = Σx / nmean
IQR = Q3 − Q1interquartile range
Q3 + 1.5 × IQRupper outlier fence
z = (x − μ) / σz-score

The formulas and tables are provided in the exam. Answers within 2.5% are marked correct (5% for probabilities), so values from a table or a calculator both count. The full sheet is in our AP Statistics formula sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
Find the mean of 4, 7, 7, 10 and 12.

Hint

Add them up and divide by how many there are.

Worked solution
  1. Sum = 40, and there are 5 values.
  2. Mean = 40/5 = 8.
Q2Warm-up
A data set has Q1 = 12 and Q3 = 20. Above what value is a data point an outlier, by the 1.5 × IQR rule?

Hint

Find the IQR first.

Worked solution
  1. IQR = 20 − 12 = 8.
  2. Upper fence = Q3 + 1.5 × IQR = 20 + 12 = 32.
Q3Warm-up
A school tests two teaching methods by randomly assigning 10 whole classes to each method. What are the experimental units?

Hint

The experimental unit is what the treatment is assigned to.

Worked solution
  1. The treatment is assigned to whole classes, not to individual students.
  2. So the experimental units are the classes: B.

Exam-style questions

Questions at the level of the AP Statistics exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
Test scores have mean 70 and standard deviation 8. Find the z-score of a score of 82.

Hint

z = (x − mean)/SD.

Worked solution
  1. z = (82 − 70)/8 = 12/8.
  2. z = 1.5: the score is 1.5 standard deviations above the mean.
Q5Exam-style
Adult heights are approximately normal with mean 170 cm and standard deviation 6 cm. What proportion are taller than 182 cm?

Hint

Standardise, then use the normal table or normalcdf.

Worked solution
  1. z = (182 − 170)/6 = 2.
  2. P(Z > 2) = 1 − 0.9772.
  3. Proportion = 0.0228, about 2.3%.
Q6Exam-style
A researcher divides a city into neighbourhoods by income level, then takes a simple random sample from each income level. What kind of sample is this?

Hint

Is every group sampled, or are whole groups chosen?

Worked solution
  1. The population is split into groups (strata) of similar individuals, and a random sample is taken from every group.
  2. That is a stratified random sample: B.
Q7Exam-style
SAT-style scores are approximately normal with mean 500 and standard deviation 100. What score is at the 90th percentile?

Hint

Find the z-score with 0.90 of the area below it, then unstandardise.

Worked solution
  1. invNorm(0.90) gives z = 1.2816.
  2. x = 500 + 1.2816 × 100 = 628.
Q8Challenge
An observational study finds that people who drink more coffee live longer. Why can't we conclude that coffee causes a longer life?

Hint

What does an experiment have that an observational study does not?

Worked solution
  1. Without random assignment of coffee drinking, coffee drinkers may differ in other ways that affect lifespan.
  2. Those confounding variables mean we cannot conclude cause and effect: B.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

Experimental design style · 10 points · about 22 minutes

A teacher wants to know whether a new revision app improves test scores more than revising from notes. Forty students volunteer. Their scores on last term's test vary widely.

  1. (a) Identify the treatments and the response variable.
  2. (b) Describe how to carry out a completely randomised design with the 40 students.
  3. (c) Explain why blocking by last term's score might be better, and describe how to do it.
  4. (d) Can the results be generalised to all students at the school? Explain.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

Treatments: revising with the app, and revising from notes. Response: the score on the next test.

Part (b)

Number the students 1 to 40. Use a random number generator to pick 20 different numbers: those students use the app, the other 20 revise from notes. Compare mean test scores.

Part (c)

Last term's score is likely related to the response, so blocking reduces variability. Pair students with similar scores, and randomly assign one of each pair to the app.

Part (d)

No. The students volunteered, so they may not represent all students; there was no random sample from the school.

Your mark: 0 of 10

Common mistakes in exploring one-variable data and collecting data

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Statistics past papers, with official scoring guidelines. · All AP Statistics units