Stichprobenraum

Stichprobenraum, in statistics and survey sampling, is the set of all possible samples that can be drawn from a given population under a specified sampling design. The term mirrors the concept of a sample space in probability theory but is applied to the collection of samples rather than individual outcomes. The Stichprobenraum is a theoretical construct used to reason about the properties of estimators and sampling distributions.

If the population contains N units and a sample of size n is drawn without replacement, the number of possible samples equals the binomial coefficient C(N, n) if samples are treated as unordered sets, or N!/(N-n)! if the order of selection matters. If sampling is with replacement, the number of ordered samples is N^n (and with certain equivalence relations, the count for unordered samples with replacement is C(N+n-1, n)).

In practice, the Stichprobenraum provides the framework for calculating or enumerating the sampling distribution of an estimator, such as the sample mean or sample proportion, under the given design. It also clarifies the influences of design choices—such as replacement, stratification, clustering—on the variability of estimators.

Relation to broader concepts: The Stichprobenraum is distinct from the population (the set of units from which samples are drawn) and from the probability space (Wahrscheinlichkeitsraum) used to model random mechanisms. It is primarily a combinatorial or probabilistic space defined by the sampling method.