Sampling Fraction

A sampling fraction, denoted f, is the proportion of a universe that is selected for a sample. The sampling fraction is important for survey estimation because in sampling without replacement, the sample variance is reduced by a factor of (1 − f), called the. finite population correction or adjustment.

In a simple survey design, if a sample of n is selected with equal probability from a universe of N, then the sampling fraction is defined as f = n/N. In this case, the sampling fraction is equal to the probability of selection. In the case of systematic sampling, f = 1/i where i is the sampling interval.

The sampling fraction can also be computed for stratified and multi-stage samples. In a stratified (single-stage) sample, the sampling fraction, fh is computed separately for each of the h strata. For a stratified sample fh = nh/Nh, where nh is the sample size for stratum h and nh is number of units (in the universe of na that belong to stratum h. Because many samples use stratification to facilitate oversam-pling, the probabilities of selection may differ among strata, in which case the fh values will not be equal.

For multi-stage samples, the sampling fraction can be computed at each stage, assuming sampling is with equal probability within the stage. A two-stage sample could include selection of na primary sampling units from a universe of Na, and within the ath primary sampling unit, selecting nba out of Nba units (e.g. households or businesses). In this case, fa = (na/Na) and fab = (nba/Nba) and where ba and ba are the mean values of nba and Nba; the overall sampling fraction would then be f = fa(fb∗)

However, many if not most multi-stage samples use selection with probability proportional to size, which makes computing sampling fractions at each stage problematic.