Random Assignment vs Stratified Random Sample

Random assignment means a chance device, not the experimenter, decides which unit gets which treatment, and the device's probabilities do not depend on anything about the unit. That is the whole requirement. It does not demand equal group sizes and it does not promise that the finished groups will look alike. What it buys is that every variable other than the treatment was spread across the groups by that same chance mechanism, so a difference in the response has only two explanations left, the treatment or chance, and the p-value measures the second one.

Balance is a tendency, not a guarantee. Take 20 subjects, 10 of them women, split into two groups of 10 by shuffling names. The expected split is 5 and 5, but the chance of landing exactly there is $\frac{\binom{10}{5}^2}{\binom{20}{10}} = \frac{63504}{184756} = 0.344$. A group holding 8 or more of the 10 women turns up about 2.3 percent of the time, and all 10 landing together about once in 92,000 assignments.

So "the treatment group came out older on average, so the randomization failed" reads the wrong thing. Random assignment is judged on the procedure used, not on the split it produced, and imbalance of exactly that size already sits inside the reference distribution the p-value comes from. Redrawing until the groups look even destroys that: the assignment is no longer random and the stated error rate no longer holds.

The boundary is who is in the study at all, and random assignment says nothing about it. Forty volunteers randomly assigned support a causal claim about people like those volunteers and about no one else. Widening the audience takes random selection, a separate act on a separate list; see scope of inference.

Topic 1.13 lists random assignment beside comparison, replication, and direct control as the four elements of a well-designed experiment.

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.