Sampling Methods

Why Sample?

A census (measuring every individual in the population) is often impractical. Instead, we take a sample and use statistics to make inferences about the population.

The key requirement: the sample must be representative of the population.

Simple Random Sample (SRS)

Every possible sample of size $n$ has an equal chance of being selected.

How to select an SRS:

1. Assign a number to each individual in the population
2. Use a random number generator (or table) to select $n$ numbers
3. Include the corresponding individuals in the sample

An SRS ensures that every individual has an equal probability of being selected, and every pair, triple, etc. of individuals is equally likely.

Stratified Random Sampling

1. Divide the population into strata (groups of similar individuals)
2. Take a separate SRS from each stratum
3. Combine the results

When to use: When the population has distinct subgroups and you want to ensure each subgroup is represented proportionally.

Example: Stratify by grade level (9, 10, 11, 12) and sample from each.

Cluster Sampling

1. Divide the population into clusters (naturally occurring groups, often geographic)
2. Randomly select entire clusters
3. Survey all individuals in the selected clusters

When to use: When the population is spread out geographically and it's costly to reach individuals.

Example: Randomly select 5 schools from a district and survey all students at those schools.

Systematic Sampling

1. Select every $k$th individual from a list
2. Start with a randomly chosen individual from the first $k$

Example: Survey every 10th customer entering a store.

Comparison Summary

| Method | Pros | Cons | |--------|------|------| | SRS | Unbiased, simple | May not represent subgroups | | Stratified | Ensures subgroup representation | Requires knowledge of strata | | Cluster | Practical for large/spread populations | Higher variability | | Systematic | Easy to implement | Risk of hidden patterns |

AP Tip: Know the difference between strata and clusters. Strata are homogeneous groups (sample FROM each); clusters are heterogeneous groups (sample entire clusters).