Sampling Variability
A free Statistics and Data Analysis lesson from the “Inference and Conclusions” unit, with a worked example and practice problems including step-by-step solutions.
Different random samples from the same population usually produce different statistics. Inference works by understanding and measuring that sampling variability. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Explain why samples vary
- Distinguish sample statistic from population parameter
- Describe variability across repeated samples
Worked example
Problem. Two random samples from the same school give slightly different sample proportions. What idea explains the difference?
- Worked Example: First identify exactly what the question is asking: Two random samples from the same school give slightly different sample proportions. What idea explains the difference?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Different random samples usually give different statistics.
- That sample-to-sample change is sampling variability.
Answer: sampling variability
Practice problems
1. Practice case A: One sample estimates 54% and another estimates 58% from the same population. What idea explains this?
Choices: sampling variability · sampling bias · placebo effect · causation
Show solution
- Warm-up: First identify exactly what the question is asking: One sample estimates 54% and another estimates 58% from the same population. What idea explains this?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Samples vary because they include different cases.
- That variation is expected, even with random sampling.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: sampling variability
2. Practice case B: Why use a sample mean in inference?
Choices: proves the parameter exactly · removes all uncertainty · assigns treatments · estimates a population parameter
Show solution
- Warm-up: First identify exactly what the question is asking: Why use a sample mean in inference?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- A statistic comes from a sample.
- It is used to estimate an unknown population value.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: estimates a population parameter
3. Practice case C: If sample size increases, sampling variability usually:
Choices: becomes impossible · proves causation · decreases · increases automatically
Show solution
- Warm-up: First identify exactly what the question is asking: If sample size increases, sampling variability usually:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Larger samples tend to be more stable.
- That reduces sample-to-sample variation.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: decreases
4. Practice case D: A poll estimate differs from the true population percent. What is the best interpretation?
Choices: the result is useless · random samples naturally vary · the study must be an experiment · the variable must be categorical
Show solution
- Warm-up: First identify exactly what the question is asking: A poll estimate differs from the true population percent. What is the best interpretation?
- For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
- Sampling variability is normal.
- Inference accounts for that uncertainty.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: random samples naturally vary
5. Practice case E: Which value is fixed but usually unknown?
Choices: the true value for the whole population · the value from one sample · the treatment assignment · the response option order
Show solution
- Warm-up: First identify exactly what the question is asking: Which value is fixed but usually unknown?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The parameter describes the population.
- A sample statistic estimates it.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: the true value for the whole population
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