Sample Size Determination - Stratified Sample Size

Stratified Sample Size

With more complicated sampling techniques, such as stratified sampling, the sample can often be split up into sub-samples. Typically, if there are k such sub-samples (from k different strata) then each of them will have a sample size n_i, i = 1, 2, ..., k. These n_i must conform to the rule that n₁ + n₂ + ... + n_k = n (i.e. that the total sample size is given by the sum of the sub-sample sizes). Selecting these n_i optimally can be done in various ways, using (for example) Neyman's optimal allocation.

There are many reasons to use stratified sampling: to decrease variances of sample estimates, to use partly non-random methods, or to study strata individually. A useful, partly non-random method would be to sample individuals where easily accessible, but, where not, sample clusters to save travel costs.

In general, for H strata, a weighted sample mean is

with

The weights, W(h), frequently, but not always, represent the proportions of the population elements in the strata, and W(h)=N(h)/N. For a fixed sample size, that is n=Sum{n(h)},

which can be made a minimum if the sampling rate within each stratum is made proportional to the standard deviation within each stratum: .

An "optimum allocation" is reached when the sampling rates within the strata are made directly proportional to the standard deviations within the strata and inversely proportional to the square roots of the costs per element within the strata:

or, more generally, when