Topic 3.3 Notes – Random Sampling and Data Collection

Why Random Sampling Matters

Everything starts with the difference between a population and a sample.

• Population: the entire group you want information about (all seniors at your school).
• Sample: the smaller group you actually collect data from.
• Sampling frame: the list or method used to identify members of the population (a school roster, customer database, etc.).

If selection is not random, certain types of individuals can be overrepresented or underrepresented. That creates bias, and once bias is built into the data collection, you cannot fix it later with fancy statistics.

Random sampling uses a chance process so each individual (or group) has a known probability of being selected. That’s what allows you to generalize your results to the population.

Sampling With and Without Replacement

• Without replacement: once selected, an individual cannot be chosen again.
  → This is how most real surveys work.
• With replacement: individuals can be selected more than once.

Without replacement slightly changes probabilities after each selection, but in large populations the effect is tiny. On AP questions, if they describe drawing names and not putting them back, that’s without replacement.

Types of Random Sampling Methods

On quizzes and the AP exam, you’re usually asked to identify the method from a description or explain whether it makes sense in context.

Simple Random Sample SRS

A simple random sample (SRS) is a sample where every group of size n has an equal chance of being chosen.

That means:

• Every individual has an equal chance.
• Every possible sample of size n is equally likely.

Common ways to get an SRS:

• Number individuals 1 to N and use a random number generator.
• Use a random number table.
• Draw names/cards without replacement.

This is the gold standard because it’s fair and conceptually simple. The limitation is practical. You need a complete list of the population.

Stratified Random Sample

Here, you divide the population into strata based on a shared characteristic, then take an SRS within each stratum and combine them.

Key structure:

• Strata are homogeneous within (similar inside each group).
• Random sampling happens inside each stratum.

Why do this?

• Guarantees representation from important subgroups.
• Reduces variability and often gives more precise estimates.

If a study divides students by grade level and randomly samples from each grade, that’s stratified.

Cluster Sample

A cluster sample also divides into groups, but the logic is different.

• Clusters should be heterogeneous within (mini-populations).
• Clusters should look similar to each other.
• Randomly select some clusters.
• Collect data from all individuals in chosen clusters.

The key contrast: in stratified sampling, individuals are randomly selected from each group. In cluster sampling, entire groups are selected and everyone in the chosen groups is included.

Cluster sampling is often used for efficiency. For example, randomly choosing 5 classrooms and surveying every student in those rooms.

If you sample everyone in selected groups, it’s cluster.

Systematic Random Sample

A systematic random sample uses a random start and a fixed interval.

If population size is $N$ and desired sample size is $n$:

$$k = \frac{N}{n}$$

Procedure:

1. Compute $k$.
2. Randomly choose a starting number from 1 to $k$.
3. Select every $k$th individual from the list.

Example: If $N = 800$ and $n = 100$, then $k = 8$. Choose a random start between 1 and 8, then take every 8th name.

It’s quick and spreads the sample across the list. The risk comes if the list has a hidden pattern that matches the interval.

Census

A census collects data from every individual in the population.

• No sampling variability.
• Often too expensive or time-consuming.
• Still possible to have measurement bias.

Choosing the Right Method

When you’re asked if a method is appropriate, think about:

• Is random selection actually used?
• Is a full list of the population available?
• Are there key subgroups that must be represented?
• Is cost or geography an issue?

For example:

• Use SRS when a complete list exists and fairness is the priority.
• Use stratified when subgroup differences matter.
• Use cluster when logistics and cost dominate.
• Use systematic when you have an ordered list and no harmful pattern.

When you justify your answer, tie it to the actual population in the question. Say the group name. That’s what earns full credit.