quality:correction, information:filtering, linguistic:voice_change
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 H such sub-samples (from H different strata) then each of them will have a sample size n, h = 1, 2,..., H. These n must conform to the rule that n + n +... + n = n (i.e., that the total sample size is given by the sum of the sub-sample sizes). Selecting these n 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 {\displaystyle W_{h}}, frequently, but not always, represent the proportions of the population elements in the strata, and W h = N h / N {\displaystyle W_{h}=N_{h}/N}. For a fixed sample size, that is n = ∑ n h {\displaystyle n=\sum n_{h}},

which can be made a minimum if the sampling rate within each stratum is madeproportional to the standard deviation within each stratum: n h / N h = k S h {\displaystyle n_{h}/N_{h}=kS_{h}}, where S h = Var (x ¯ h) {\displaystyle S_{h}={\sqrt {\operatorname {Var} ({\bar {x}}_{h})}}} and k {\displaystyle k} is a constant such that ∑ n h = n {\displaystyle \sum {n_{h}}=n}.

An "optimum allocation" is reached when the sampling rates within the strataare made directly proportional to the standard deviations within the strataand inversely proportional to the square root of the sampling cost per elementwithin the strata, C h {\displaystyle C_{h}}:

where K {\displaystyle K} is a constant such that ∑ n h = n {\displaystyle \sum {n_{h}}=n}, or, more generally, when