Stratified random sample

Strata are non-overlapping groups that together cover the whole population, built so the individuals inside one stratum are alike on a trait related to what is being measured, and unlike the individuals in the other strata. The randomness then acts inside every stratum: a separate simple random sample is drawn from each one and the pieces are combined into a single sample. No stratum can be missed, because the design reaches into all of them.

Take 1,200 students at one school, 300 in each of grades 9 through 12, with a plan that draws an SRS of 20 students per grade. The sample is $4 \times 20 = 80$ students and every grade is guaranteed 20 seats in it. Each student's chance of selection is $20/300 = 1/15$, which matches the overall rate $80/1200$. Equal rates across strata is called proportional allocation, and it is a choice rather than part of the definition; strata of unequal size and unequal sampling rates are still strata.

Here is the sentence to catch: "I randomly picked 4 of the school's advisory groups and surveyed everyone in them, so the sample is stratified." Dividing the population first is not what makes a design stratified. What makes it stratified is that the randomness selects individuals within every group. Randomness that selects whole groups, with everyone inside the chosen ones measured, is a cluster sample. Some of every group is stratified, all of a few groups is cluster.

Stratifying pays off only when the strata differ from one another on what you are measuring. If ninth and twelfth graders answer the question about the same way, stratifying by grade leaves you with an estimate about as variable as an SRS of 80, so it costs nothing and buys nothing. The gain comes from taking the between-stratum differences out of the sampling error.

Strata are not blocks. You stratify a population before selecting a sample, while blocking groups experimental units before assigning treatments. Sampling methods are Unit 1, topic 1.11 Random Sampling.