Finite Population

Most statistical theory is premised on an underlying infinite population. By contrast, survey sampling theory and practice are built on a foundation of sampling from a finite population. This basic difference has myriad ramifications, and it highlights why survey sampling is often regarded as a separate branch of statistical thinking. On a philosophical level, the theory brings statistical theory to a human, and thus necessarily finite, level.

Before describing the basic notion of finite population sampling, it is instructive to explore the analogies and differences with sampling from infinite populations. These analogies were first described in Jerzy Neyman's seminal articles in the 1930s and are discussed in basic sampling theory textbooks such as William Cochran's in the 1970s. In the general framework of finite population sampling, we consider samples of size n from a finite population of size N, that is, a population with N elements or members.

The bridge of finite to infinite population sampling is also seen in terms of a finite population correction (fpc) that applies to the variances under most sampling designs. Finite population sampling typically begins with simple random sampling (SRS), the simplest form of sampling design, which can be considered with replacement or without replacement. For SRS designs, the fpc may be expressed as 1 − n/N, or 1 = f, where f is the sampling fraction or the sampling rate, f = n/N. Clearly, the fpc does not materially affect variances when sampling from large populations, particularly when the sample is not too large itself.

Finite population corrections are applicable for estimation but may not be necessary for many inferential uses such as statistical testing (e.g. comparisons between subgroups). In many instances, it is more sensible to consider an underlying infinite population when comparing subgroup parameters. In general, an infinite population approach to sampling has been developed that is based on superpopulation models. The super-population approach treats the value associated with a population unit as the realization of a random variable rather than as a fixed number.