# Stratified Random Sampling

Stratified random sampling refers to a sampling method that has the following properties.

• The population consists of N elements.
• The population is divided into H groups, called strata.
• Each element of the population can be assigned to one, and only one, stratum.
• The number of observations within each stratum Nh is known, and N = N1 + N2 + N3 + ... + NH-1 + NH.
• The researcher obtains a probability sample from each stratum.

In this tutorial, we will assume that the researcher draws a simple random sample from each stratum.

Stratified sampling offers several advantages over simple random sampling.

• A stratified sample can provide greater precision than a simple random sample of the same size.
• Because it provides greater precision, a stratified sample often requires a smaller sample, which saves money.
• A stratified sample can guard against an "unrepresentative" sample (e.g., an all-male sample from a mixed-gender population).
• We can ensure that we obtain sufficient sample points to support a separate analysis of any subgroup.

Compared to simple random sampling, stratified sampling has two main disadvantages. It may require more administrative effort than a simple random sample. And the analysis is computationally more complex.

## Proportionate Versus Disproportionate Statification

All stratified sampling designs fall into one of two categories, each of which has strengths and weaknesses as described below.

• Proportionate stratification. With proportionate stratification, the sample size of each stratum is proportionate to the population size of the stratum. This means that each stratum has the same sampling fraction.
• Proportionate stratification provides equal or better precision than a simple random sample of the same size.
• Gains in precision are greatest when when values within strata are homogeneous.
• Gains in precision accrue to all survey measures.
• Disproportionate stratification. With disproportionate stratification, the sampling fraction may vary from one stratum to the next.
• The precision of the design may be very good or very poor, depending on how sample points are allocated to strata. The way to maximize precision through disproportionate stratification is discussed in a subsequent lesson (see Statistics Tutorial: Sample Size Within Strata).
• If variances differ across strata, disproportionate stratification can provide better precision than proportionate stratification, when sample points are correctly allocated to strata.
• With disproportionate stratification, the researcher can maximize precision for a single important survey measure. However, gains in precision may not accrue to other survey measures.

Recommendation. If costs and variances are about equal across strata, choose proportionate stratification over disproportionate stratification. If the variances or costs differ across strata, consider disproportionate stratification.