Stratified sampling

In the stratified random sampling (also: stratified random sample ) in the statistics the corresponding population is divided into several smaller groups, which are called layers. Then you pull separately from each grouping a simple random sample. These samples are then applied to the population according to the sizes of the individual layers, which must be known, weighted at the end. By layering is obtained at cheaper selection more accurate results, but at least as good results as in the simple random sample. In Monte Carlo simulations can be employed stratified random drawings as a means of variance reduction. The stratification characteristics (parameter data) must be known in advance.

Based on parameter

  • Average: with q = 1 .. L layers; the probability of selection and the layer extent
  • Variance with the variance within strata and variance between the layers
  • Unit value

Estimator of the population parameter

  • Estimator for the mean:
  • Estimator for the variance:
  • Estimator of the unit value

Stratification types

There are various types of lamination:

  • Equal division ( same selection): in layers
  • Proportional selection: If the sample drawn according to their size in the respective proportions of the layers in the population, refers to the selection as " stratified proportional ".
  • Disproportionate selection: time the number of cases, however, the current sample of the proportions of the population, saith one of " disproportionate stratified selections ". Optimal selection: The Optimal selection takes into account the individual variations in the layers. More elements are selected from a layer with greater variance.
  • Cost-optimized selection


Stratification is the division of the population into layers. This results in two subproblems:

The aim is to combine the two sub-problems to solve so that the estimates are more accurate. But this will usually prior information about the population (such as the official statistics or previous studies ).

Above-mentioned A solution Problem is about the Stratifikationsmodell after Dalenius including the corresponding approximate solutions such as the cum- rule or the equal aggregate rule dar.