Stratified Sampling

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Ensuring Representation Through Deliberate Division

Stratified sampling divides the population into distinct, non-overlapping subgroups (strata) based on one or more relevant characteristics, and then draws a random sample from each stratum. This technique ensures that important subgroups are adequately represented in the sample, often producing more precise estimates than simple random sampling of the same size.

The Logic Behind Stratification

Imagine surveying employee satisfaction at a company with 1,000 employees: 700 in operations, 200 in sales, and 100 in management. A simple random sample of 100 employees might, by chance, include only 2-3 managers — too few for meaningful analysis of this group. Stratified sampling guarantees proportional (or deliberately disproportional) representation of each group:

• Operations: 70 employees sampled
• Sales: 20 employees sampled
• Management: 10 employees sampled

Every subgroup is represented, and the overall sample reflects the population's composition.

When to Use Stratified Sampling

Stratified sampling is most beneficial when:

1. Key subgroups exist that may differ on the variable of interest. If you suspect job satisfaction varies by department, stratify by department.
2. You need subgroup estimates — not just overall population estimates but also reliable estimates for each subgroup.
3. Population is heterogeneous across strata but relatively homogeneous within strata. Stratification reduces variance by grouping similar members together.
4. A stratification variable is available in the sampling frame. You can only stratify by characteristics you know before sampling (gender, department, region — not attitudes or beliefs).

Types of Stratified Sampling

Proportionate Stratified Sampling

Sample from each stratum in proportion to its size in the population. If the population is 40% male and 60% female, the sample maintains this ratio.

Example: A university population of 10,000 students:

• First year: 3,000 (30%) → Sample 30% of 400 = 120
• Second year: 2,800 (28%) → Sample 28% of 400 = 112
• Third year: 2,500 (25%) → Sample 25% of 400 = 100
• Fourth year: 1,700 (17%) → Sample 17% of 400 = 68
• Total sample: 400

Disproportionate Stratified Sampling

Deliberately over-sample from smaller strata to ensure adequate representation for subgroup analysis. The sample no longer mirrors population proportions, but weighting during analysis corrects for this.

Example: Studying health outcomes across ethnic groups in a region where the population is 85% majority ethnicity and 15% minorities. Proportionate sampling of 300 would yield only 45 minority respondents — insufficient for reliable subgroup analysis. Disproportionate sampling might sample 200 from the majority and 150 from minorities, with weights applied during analysis to produce unbiased population estimates.

Optimal (Neyman) Allocation

Allocates sample sizes to strata based on both their size and their variability. Strata with larger variance receive more samples because they need more observations to be estimated precisely. Formula:

n_h = n × (N_h × S_h) / Σ(N_h × S_h)

Where n_h = sample from stratum h, N_h = population size of stratum h, S_h = standard deviation within stratum h.

This produces the most precise estimates for a given total sample size but requires advance knowledge of within-stratum variability.

Step-by-Step Implementation

1. Define strata: Choose stratification variable(s) that divide the population into groups expected to differ on your outcome variable. Common stratification variables: gender, age group, geographic region, organizational level, income bracket.

1. Determine population sizes within strata: From the sampling frame, count how many members belong to each stratum.

1. Calculate stratum sample sizes: Using proportionate, disproportionate, or optimal allocation.

1. Draw random samples within each stratum: Apply simple random sampling independently within each stratum.

1. Combine strata samples: The full sample consists of all randomly selected members across all strata.

Precision Gain Calculation

The design effect (DEFF) compares the variance of a stratified sample to that of a simple random sample of the same size:

DEFF = Var(stratified) / Var(SRS)

When DEFF < 1 (as it typically is when strata are homogeneous internally but heterogeneous between themselves), stratification has improved precision. A DEFF of 0.7 means the stratified sample is as precise as an SRS 43% larger (1/0.7 = 1.43).

Practical Example with Calculations

Research question: What is the average monthly savings among Indian working professionals?

Population strata (by income level):

• Low income (< ₹30,000/month): N₁ = 40,000; estimated SD₁ = ₹2,000
• Middle income (₹30,000-₹80,000): N₂ = 35,000; estimated SD₂ = ₹8,000
• High income (> ₹80,000): N₃ = 25,000; estimated SD₃ = ₹20,000

Using optimal allocation with total n = 500:

• n₁ = 500 × (40,000 × 2,000) / (40,000×2,000 + 35,000×8,000 + 25,000×20,000)
• n₁ = 500 × 80,000,000 / (80,000,000 + 280,000,000 + 500,000,000)
• n₁ = 500 × 80M / 860M = 47
• n₂ = 500 × 280M / 860M = 163
• n₃ = 500 × 500M / 860M = 291

Notice: the high-income stratum, despite being the smallest in population, receives the most samples because its variability in savings is greatest. This allocation produces the most precise overall estimate of average savings.

Common Mistakes

• Choosing stratification variables unrelated to the outcome variable (stratifying by hair color for a study of academic performance adds complexity without benefit)
• Using too many strata (creating strata so small that random sampling within them becomes impractical)
• Forgetting to apply weights when using disproportionate allocation in analysis

Conclusion

Stratified sampling offers a powerful enhancement over simple random sampling by ensuring key subgroups are represented and reducing sampling variability. It requires advance knowledge of population composition and a sampling frame that includes the stratification variable, but when these conditions are met, it produces more precise estimates, enables reliable subgroup comparisons, and guarantees demographic representativeness that simple random sampling can only achieve probabilistically.