Stratified Random Sampling
In short: Stratified random sampling divides the population into similar groups called strata, then draws a simple random sample from every stratum, so each group is guaranteed a place in the final sample.
AP StatisticsStatisticsAP Statistics Unit 1
What is Stratified Random Sampling?
Stratified random sampling begins by sorting the population into non-overlapping groups, called strata, whose members share a trait related to the question being studied: grade level, region, or age band.
Next, a separate simple random sample is selected within each stratum, and the pieces are combined. With proportional allocation, a stratum making up 30 percent of the population supplies 30 percent of the sample. The design shields against a sample missing a key group by bad luck, and when strata are internally similar, it produces more precise estimates than a plain SRS of the same size.
How to spot it
- A description says the population was first divided by grade, gender, region or income group before stratified random sampling began.
- A researcher worries that a small but important group might be missed or underrepresented.
- Separate random selections are made within each group, and the sizes match group proportions.
- Responses are expected to differ more between groups than within them.
The method
- For stratified random sampling, choose a stratifying variable related to the response, and list the strata with their sizes.
- Decide the total sample size \(n\) and the allocation; for proportional allocation, stratum sample = \(n \times \frac{\text{stratum size}}{N}\).
- Take a simple random sample of that size independently within each stratum.
- Combine the subsamples and analyze, weighting by stratum size if allocation was not proportional.
Stratified Random Sampling: solved example
Problem
A school district has 2,400 elementary, 1,200 middle and 400 high school students on its bus routes. An administrator wants a proportionally allocated stratified sample of 80 riders. How many come from each level?
Solution
The total is \(N = 4{,}000\). Elementary share: \(\frac{2400}{4000} = 0.60\), so \(0.60 \times 80 = 48\). Middle: \(\frac{1200}{4000} = 0.30\), so 24. High: \(\frac{400}{4000} = 0.10\), so 8.
Take a simple random sample of 48 elementary, 24 middle and 8 high school riders; \(48 + 24 + 8 = 80\).
Try it
Question 1
A sample of 60 is drawn from a population that is 55% urban and 45% rural, stratified proportionally. How many from each?
Show the answer
Urban: \(0.55 \times 60 = 33\). Rural: \(0.45 \times 60 = 27\).
Question 2
Why can stratifying beat a plain SRS when responses differ a lot between groups?
Show the answer
It removes between-group variation from the sampling error and guarantees each group is represented, so estimates are usually more precise.
Stratified Random Sampling vs. Simple Random Sample
A plain SRS may, by chance, over- or under-represent a group. Stratifying fixes the group counts ahead of time. Stratified random sampling is also different from cluster sampling, where whole groups are selected and everyone inside is surveyed; in stratified sampling, every group contributes some members.
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