Key takeaway: Stratified random sampling divides a population into homogeneous groups called strata, then takes a separate simple random sample from every stratum so that each group is represented.
Grade 12 – AP StatisticsAP Statistics Unit 1Statistics and Data
Definition of Stratified Random Sampling
Stratified random sampling begins by splitting the population into strata, groups whose members share a characteristic that could affect the answer, such as grade level, region or age band. A random sample is then drawn independently within each stratum.
Combining those samples guarantees every group a voice and usually reduces variability compared with one big random draw. With proportional allocation, each stratum supplies a share of the sample equal to its share of the population.
Formula and notation
If a stratum contains \(N_k\) of the \(N\) individuals and the total sample size is \(n\), its proportional sample size is
\[ n_k=n\cdot\frac{N_k}{N}. \]
Strata should be similar inside and different from each other. This differs from cluster sampling, where whole groups are chosen.
Key facts
- Strata must not overlap, and together they must cover the whole population.
- Every stratum is sampled, unlike cluster sampling, where only some groups are used.
- It works best when the strata differ on the variable being measured.
- Rounded stratum sizes should add up to the intended total sample size.
Stratified Random Sampling example
A school of 1,200 students has 360 ninth graders, 330 tenth graders, 300 eleventh graders and 210 twelfth graders. A survey will use proportional stratified random sampling for 120 students. How many come from each grade?
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The sampling fraction is \(\frac{120}{1200}=\frac{1}{10}\).
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Ninth: \(360\cdot\frac{1}{10}=36\). Tenth: \(330\cdot\frac{1}{10}=33\).
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Eleventh: \(300\cdot\frac{1}{10}=30\). Twelfth: \(210\cdot\frac{1}{10}=21\).
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Check: \(36+33+30+21=120\).
Select 36, 33, 30 and 21 students by simple random sampling within grades 9, 10, 11 and 12.
Common mistake
Watch out
To be fair, a researcher draws exactly 30 students from each of the four grades.
Why it is wrong
Equal counts over-represent the small twelfth grade and under-represent the large ninth grade, so a school-wide estimate would be skewed unless it is weighted. Proportional allocation, 36, 33, 30 and 21, keeps each grade’s share the same in the sample as in the school.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data