Random Sampling
AP StatisticsΒ· 12 min read
1. Core Random Sampling Method Definitionsβ β ββββ± 15 min
All valid random sampling techniques use a known, non-zero probability to select every individual in the target population, eliminating systematic favoritism that skews results. No random sampling method guarantees a perfectly representative sample, but it removes predictable bias from the selection process.
Simple Random Sample (SRS)
A sampling method where every possible group of n individuals from the population has an equal chance of being selected.
Example:
Assign every student in a school a unique number, then draw 50 numbers out of a hat to select survey participants.
Stratified Random Sampling: Divide the population into homogeneous subgroups (strata) that share a key characteristic, then take an SRS from each stratum.
Cluster Sampling: Divide the population into naturally occurring heterogeneous subgroups (clusters), then randomly select entire clusters to include in your sample.
Systematic Random Sampling: Select every kth individual from a complete sampling frame, starting from a randomly generated initial position.
A researcher wants to survey 200 residents of a town with 10,000 total residents. Describe how to run a systematic random sample for this study.
- 1
First calculate the sampling interval k by dividing total population size by desired sample size:
- 2
- 3
Randomly generate a starting integer between 1 and 50, for example 17.
- 4
Select the 17th resident on the town resident list, then every 50th resident after that: 17, 67, 117, ..., up to the 200th selected resident.
Test your understanding of basic sampling definitions
Which sampling method divides the population into groups with similar characteristics before sampling?
A) SRS
B) Stratified sampling
C) Cluster sampling
D) Systematic sampling
Reveal answer
B βStratified sampling sorts the population into homogeneous strata to reduce sampling variability for key subgroups.
2. Common Sources of Sampling Biasβ β β βββ± 12 min
Bias in sampling occurs when some members of the target population are systematically less likely to be selected than others, making your results unrepresentative even if you intended to run a random sample.
Undercoverage bias: Your sampling frame fails to include key subsets of the target population, for example using a landline phone list to survey all voters, which excludes people who only use mobile phones.
Voluntary response bias: You invite participants to opt in to your survey, which overrepresents people with strong positive or negative opinions on the topic.
Convenience sampling: You select participants who are easy to reach, for example surveying people outside a shopping mall to estimate average household income for the entire city.
A high school principal wants to survey student opinions on new after-school programs, and posts a survey link on the school's Instagram page for students to complete voluntarily. Identify the bias in this approach, and explain how it skews results.
- 1
First identify the sampling method used: this is a voluntary response sample, not a random sample.
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The bias introduced is voluntary response bias.
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Results will overrepresent students who have very strong positive or negative feelings about after-school programs, and underrepresent students who do not check the school Instagram page or have neutral opinions.
- 4
The principal cannot generalize the survey results to the full student population.
Exam tip:
AP graders will deduct points if you name a bias but fail to explain how it impacts results for the specific context given in the question.
3. Selecting the Right Sampling Method for Contextβ β β βββ± 10 min
Use this framework to select the correct sampling method for any given study scenario on the exam.
Simple Random Sample
Best for small, well-defined populations with no obvious subgroups of interest.
+ Pros: Easy to explain, no extra grouping steps required.
β Cons: Can underrepresent small key subgroups, less statistically efficient than stratified sampling.
Stratified Random Sample
Best when you want to guarantee representation for small, distinct subgroups in the population.
+ Pros: Reduces overall sampling variability, allows for separate analysis of each stratum.
β Cons: Requires pre-existing data on each individual's stratum membership, can be more logistically complex.
Cluster Sample
Best for geographically dispersed populations where you cannot easily access every individual.
+ Pros: Dramatically reduces travel and data collection costs.
β Cons: Higher sampling variability if clusters are not representative of the full population.
A school district with 12 different elementary schools wants to survey 300 4th grade students across the district. Explain why cluster sampling is more practical than SRS for this scenario.
- 1
First identify the natural clusters: each individual elementary school is a pre-existing heterogeneous cluster of 4th grade students.
- 2
For SRS, you would need to travel to all 12 schools to find the randomly selected students, which is logistically expensive.
- 3
For cluster sampling, you can randomly select 6 entire elementary schools, then survey 50 4th grade students at each selected school.
- 4
This approach cuts down on travel time and data collection costs while still producing a representative sample.
4. AP Exam Sampling FRQ Best Practicesβ β β β ββ± 8 min
5. Common Pitfalls
Wrong move:
Confusing stratified and cluster sampling by saying you select entire strata groups
Why:
Strata are homogeneous, so you never select the entire stratum, you sample from it
Correct move:
Remember: stratified = sample from every group, cluster = randomly select entire groups
Wrong move:
Calling a convenience sample a simple random sample
Why:
Convenience sampling has no random selection step, so it is not a random sampling method at all
Correct move:
Only label a method random if every individual in the population has a known non-zero chance of being selected
Wrong move:
Stating that random sampling eliminates all sampling variability
Why:
Random sampling reduces bias but cannot remove natural variation between different samples
Correct move:
Clarify that random sampling eliminates systematic bias, not random sampling error
Wrong move:
Forgetting to link bias to the specific context in FRQ answers
Why:
AP graders will not award full points for generic definitions of bias that do not reference the scenario given
Correct move:
Always add one sentence explaining exactly which group is overrepresented or underrepresented in the given study
Wrong move:
Using the terms population and sample interchangeably
Why:
The population is the full group you want to generalize to, the sample is only the subset you collect data from
Correct move:
Explicitly distinguish the two terms every time you reference them in your response
6. Quick Reference Cheatsheet
Sampling Method | Key Feature | Best Use Case |
|---|---|---|
Simple Random Sample | All groups of size n have equal selection chance | Small, well-defined populations |
Stratified Random Sample | Sample from every homogeneous subgroup | Guarantee representation for small key groups |
Cluster Sample | Randomly select entire heterogeneous groups | Geographically dispersed populations |
Systematic Sample | Select every kth individual | Large ordered sampling frame |
7. Frequently Asked
What is the difference between a census and a random sample?
A census collects data from every single individual in the target population, while a random sample only collects data from a randomly selected subset. A census is often impractical, expensive, or impossible to run for large populations, making random sampling the standard approach for most statistical studies.
Can random sampling be used to prove causation?
No. Random sampling only ensures your sample is representative of the broader population, allowing you to generalize results. To prove causation, you need random assignment of treatments to subjects in a controlled experiment, not just random sampling.
When this came up on past exams
AI-estimated based on syllabus patterns β cross-check with official past papers for accuracy. Use only as revision-focus signals.
- 2023 Β· Paper 1
Identify appropriate sampling method
- 2022 Β· Paper 2
Compare stratified vs cluster sampling
- 2021 Β· Paper 1
Explain sampling bias impact
What's Next
Mastering random sampling is the foundation for understanding experimental design, statistical inference, and generalizing results from your sample to the full population. This skill is tested in nearly every AP Stats exam's free response section, often paired with questions on bias and study type. Next, you will learn how to distinguish random sampling from random assignment, a critical distinction that determines whether you can draw causal conclusions from study results. You will also practice identifying observational studies vs controlled experiments, and build context for sampling distributions.