Stratified Sampling: Building Representativeness into Your Study

In the experimental workflow: for how stratified sampling is used when designing and running experiments, see Sampling.

Stratified sampling is a probability-sampling technique in which the population is divided into homogeneous sub-groups (strata)—such as age bands, geographic regions, or company sizes—and random samples are then drawn within each stratum.
By controlling the composition of the sample up front, researchers can:

Boost precision (smaller sampling error)
Guarantee coverage of small but important sub-groups
Enable valid subgroup analysis without massive overall n

Why Stratify?
Step-by-Step Workflow
Proportional vs. Disproportional Allocation
Sample-Size & Weighting Formulas
Comparing Stratified, Cluster, and Simple Random Sampling
When Not to Use Stratification
Common Pitfalls & How to Avoid Them
Key Takeaways
Further Reading

Why Stratify?

Research Need	How Stratification Helps
Increase precision	If strata are internally homogeneous, sampling error (SE) ↓ vs. SRS.
Assure subgroup n	Guarantees at least k Indigenous respondents for intersectional analysis.
Correct frame imbalance	Over-sample rural counties to offset under-coverage in the frame, then weight.
Cost efficiency	Sampling high-cost strata (remote areas) at lower rates can minimize total survey cost for a fixed variance target.

Step-by-Step Workflow

Define Target Population
Clear inclusion/exclusion criteria (e.g., “all public high-school teachers in Ontario”).
Choose Stratification Variables
Select variables strongly correlated with the study outcome or sampling cost (gender, age, region, school size).
Construct Sampling Frame with Stratum Flags
Each unit gets coded (e.g., Province = AB, BC; School Size = Small, Large).
Decide Allocation
- Proportional (nₕ ∝ Nₕ)
- Neyman optimal (nₕ ∝ Nₕ·σₕ)
- Equal-size for analytic power.
Randomly Select Within Strata
Simple or systematic random sampling inside each stratum list.
Apply Weights (if disproportional)
Weight wₕ = Nₕ / nₕ for unbiased population estimates.
Document Everything
Store codebook with stratum counts, selection probabilities, and replicable random seeds.

Proportional vs. Disproportional Allocation

Allocation	n in Stratum (nₕ)	Pros	Cons
Proportional	nₕ = n·(Nₕ/N)	Population estimates unbiased without weights; easiest weighting.	Small strata may have tiny (n_h).
Equal (Self-Weighting)	All strata same n	Power for subgroup comparisons	Requires weights for overall estimates; inflates cost.
Neyman (Optimal)	nₕ ∝ Nₕ·σₕ	Minimises SE for fixed n	Need stratum SD guesses; complex.
Cost-Based	nₕ ∝ Nₕ·σₕ/√cₕ	Balances variance & fieldwork cost	Requires cost estimates by stratum.

Sample-Size & Weighting Formulas

Overall sample size for mean with margin of error (MOE) d in stratified design

n = ((Σ_{h=1}^H N_h S_h)^2) / (N^2 d^2 + Σ_{h=1}^H N_h S_h^2)

Where Sh = SD in stratum _h. (Neyman allocation assumed.)

Design weight

`w_{hi} = N_h / n_h`

Post-stratification adjustment

`w'_{hi} = w_{hi} × (P_h / P̂_h)`

Comparing Stratified, Cluster, and Simple Random Sampling

Feature	Stratified	Cluster	SRS
Primary goal	Precision & subgroup control	Cost & logistics	Simplicity
SE relative to SRS	↓ if strata homogeneous	↑ if clusters homogeneous	Baseline
Design effect (DEFF)	Usually < 1	Often > 1	1
Weights needed?	Only if disproportional	Yes	No

When Not to Use Stratification

No reliable frame data for stratification variables.
Outcome weakly related to candidate strata → little or no variance gain.
Strata counts too small causing disclosure risks or unstable weights.
Cost & complexity outweigh benefits—SRS may suffice for pilot studies.

Common Pitfalls & How to Avoid Them

Pitfall	Fix
Misclassified units (frame errors)	Cross-validate frame with external sources; imputation flags.
Weight inflation (large wₕ)	Collapse tiny strata; trim/extreme-weight adjust.
Ignoring finite-population correction (fpc)	Apply fpc when n/N > 0.05 per stratum.
Confusing strata vs. clusters	Remember: strata → sample within each; clusters → sample some clusters then all/part of their members.

Key Takeaways

Stratified sampling improves precision and subgroup representation by randomizing within homogeneous slices of the population.
Choice of allocation rule (proportional, equal, Neyman) shapes both analytic power and weighting needs.
Proper weights & variance estimators (Taylor series, replicate weights) are essential for correct inferences.
Weigh the admin cost vs. variance gain; stratify only on variables you can code accurately and that matter empirically or logistically.

Methodology

Methodology: This topic is part of the experimental workflow covered in Experimentology. For the relevant methods and how they are evaluated in real experiments, see Sampling.