Finite Population Correction (FPC) Factor

The finite population correction (fpc) factor is used to adjust a variance estimate for an estimated mean or total, so that this variance only applies to the portion of the population that is not in the sample. That is, variance is estimated from the sample, but through the fpc it is used to assess the error in estimating a mean or a total, which is due to the fact that not all data from the finite population are observed.

This concept is found throughout sample survey statistics, but this entry concentrates on the simplest of design-based sample survey statistics, simple random [Page 285]sampling (without replacement). A sample of n observations for a data element of interest, say, pairs of shoes sold, are randomly selected from the N members of the universe, say, of all shoe stores or dwellings, respectively, in a geographic region. (This can also be done by strata in stratified random sampling. Other strategies can be more complex. Also, this concept can be applied to ratios of totals, such as price per unit.) An estimated mean or total will be found by extrapolating from the sum of the n observations in the sample, to an estimate, , of the sum of these values for the universe where this total is estimated for the part of the population not in the sample. (If represents an estimate of the total, T, then we can write This will be considered later.)

Therefore, there is an error associated with making this leap, and that is the sampling error. There are nonsam-pling errors to consider, such as poorly constructed survey measures, data processing errors, and reporting errors, but here we concentrate on the error due to the fact that not all data were observed, only the data for members of the sample. (Note also that there is a model-based analogy to this, but the fpc is considered to be part of a design-based approach.) Nonresponse by members of the sample can be handled in more than one way, but again, here we concentrate on simple random sampling, without regard to nonsampling error, or nonresponse. This applies straightforwardly to stratified random sampling where simple random sampling is accomplished within each stratum (group). Other designs become more complicated.

Consider the estimation of a total, T, as previously shown. (Means and ratios follow from there. Here, totals are discussed.) For a stratified random sample design, survey weights are used=often adjusted to calibrate for auxiliary information or in some other way—and the finite population total is estimated within each stratum by adjusting from the sample total within that stratum, to account for the data not collected. We can consider one stratum at a time and, therefore, consider simple random sampling.

To estimate the variance of an estimated total, we use the estimated variance within the sample, and accepted practice is to apply it only to the part of the population that was not in the sample, This may seem odd at first, but it has a certain logic, if we ignore variance due to nonsampling error to some extent. If we can estimate variance for data within a population, it must be based on the sample data, as those are all the data available. If we consider a finite population, then the variance of the estimate of a total is due to the data that are not in the sample. In other words, error in estimating a finite population data element total will be considered as being due to failure to observe all data, and instead, estimating for some of it. Thus the estimated variance is applied only to the part of the population not sampled, assuming that the variability of the data available is the same as would be found in the data not collected. Therefore, any variance estimate for a finite population total has to be adjusted downward, because the data observed are considered to have no contribution to the variance of that estimated finite population total. Thus we regard the n observations made for a given data element (say, pairs of shoes sold) to be completely known, so the variance of the estimated total will only be derived from the N = n cases in the subtotal shown previously, that are not known.

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