# Random Sampling

> AP Statistics · AP Stats 2024-2026
> Source: https://www.owlsprep.com/study/ap-statistics-u10-random-sampling/

We break down core random sampling methods, distinguish them from flawed non-random approaches, spot common sampling bias, and select the correct technique to earn full points on AP exam FRQs.

**Prerequisites:** [Definition of population vs sample](https://www.owlsprep.com/study/ap-statistics-u10-intro-to-data-collection/); [Overview of basic study design principles](https://www.owlsprep.com/study/ap-statistics-u10-intro-to-studies/)

## Learning objectives

- Distinguish between simple random, stratified, cluster, and systematic random sampling methods
- Identify sources of bias introduced by poorly designed non-random sampling techniques
- Select the appropriate random sampling method for a given study context
- Justify why random sampling allows for generalization of results to a larger target population

## Core Random Sampling Method Definitions

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.

**Worked example:** 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. $$k = 10000 / 200 = 50$$
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.

**Check your understanding**

Test your understanding of basic sampling definitions

1. Which sampling method divides the population into groups with similar characteristics before sampling?

   - A) SRS
   - B) Stratified sampling
   - C) Cluster sampling
   - D) Systematic sampling

   *Why:* Stratified sampling sorts the population into homogeneous strata to reduce sampling variability for key subgroups.

## Common Sources of Sampling Bias

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.

**Worked example:** 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.
2. The bias introduced is voluntary response bias.
3. 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.

## Selecting the Right Sampling Method for Context

**Comparing methods**

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.
  - 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.
  - 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.
  - Cons: Higher sampling variability if clusters are not representative of the full population.

**Worked example:** 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.

## AP Exam Sampling FRQ Best Practices

**Exam command terms**

AP exam questions use specific command terms that define exactly what you need to include to earn full points.

- **Identify** — Name the sampling method or bias with no extra explanation required. *("Identify the type of bias in the sampling approach described.")*

- **Describe** — Walk through the full step-by-step process of running the sampling method for the given context. *("Describe how you would implement a stratified random sample for this study.")*

- **Explain** — Connect the sampling choice to its impact on the study's results or bias level. *("Explain why the sampling method described will produce biased results.")*

> **Easy Memory Hook**
>
> Use the mnemonic SSC-S to remember the four core random sampling methods: Simple, Stratified, Cluster, Systematic.

## Common pitfalls

- **Wrong:** Confusing stratified and cluster sampling by saying you select entire strata groups
  - Why it fails: Strata are homogeneous, so you never select the entire stratum, you sample from it
  - Correct: Remember: stratified = sample from every group, cluster = randomly select entire groups
- **Wrong:** Calling a convenience sample a simple random sample
  - Why it fails: Convenience sampling has no random selection step, so it is not a random sampling method at all
  - Correct: Only label a method random if every individual in the population has a known non-zero chance of being selected
- **Wrong:** Stating that random sampling eliminates all sampling variability
  - Why it fails: Random sampling reduces bias but cannot remove natural variation between different samples
  - Correct: Clarify that random sampling eliminates systematic bias, not random sampling error
- **Wrong:** Forgetting to link bias to the specific context in FRQ answers
  - Why it fails: AP graders will not award full points for generic definitions of bias that do not reference the scenario given
  - Correct: Always add one sentence explaining exactly which group is overrepresented or underrepresented in the given study
- **Wrong:** Using the terms population and sample interchangeably
  - Why it fails: The population is the full group you want to generalize to, the sample is only the subset you collect data from
  - Correct: Explicitly distinguish the two terms every time you reference them in your response

## 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 |

## 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.

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