Stratified Sampling

Stratified sampling partitions a population of size $N$ into $H$ mutually exclusive, collectively exhaustive strata of sizes $N_1,\dots ,N_H$, and draws an independent sample of size $n_h$ from each stratum $h$, typically via simple random sampling within each stratum. The overall population mean is estimated by the stratum-weighted combination ${\bar{x}}_{st}={\sum }_{h=1}^H{W}_h{\bar{x}}_h$, where $W_h=N_h/N$ is the stratum's population weight and ${\bar{x}}_h$ is the sample mean within stratum $h$; because each stratum's estimate only carries its own within-stratum variance, and between-stratum variance is eliminated entirely (the weights $W_h$ are known population quantities, not estimated), ${\bar{x}}_{st}$ has smaller variance than the SRS mean whenever the strata differ from each other on the outcome of interest.

• Stratified sampling can never be less precise than SRS with proportional allocation and a large enough sample, and is typically strictly more precise when the strata differ meaningfully on the outcome, this is why national statistical agencies stratify by region, industry, or firm size as a near-universal default.
• Neyman allocation, which sets $n_h\propto N_h{\sigma }_h$ (larger and more variable strata get proportionally more sample), minimizes the variance of ${\bar{x}}_{st}$ for a fixed total sample size, but requires advance knowledge or estimates of each stratum's variance ${\sigma }_h$, which proportional allocation avoids needing.
• The stratification variable must be known for every unit in the sampling frame before sampling begins, a variable measured only after the fact (like a survey response) cannot be used to stratify; this is what distinguishes stratification, a design-time choice, from post-stratification, a weighting adjustment applied after data collection.