Systematic sampling

In many contexts it is difficult or impossible to select a simple random sample. For example, the number of units in the finite population, N, may not be known in advance, or it may not be feasible to assign labels to all units in the population and to select an SRS from these labels (e.g., crabs within boxes on a fishing vessel). Instead, one may select a random start, r, on the integers 1 through k and then select that unit and every kth unit thereafter for inclusion in the sample. This selection method, called linear systematic sampling, results in an extremely restricted randomization—there are only k possible linear systematic samples—compared to the typically large number [N!/(N-n)!n!] of possible samples of size n that can be selected from N by SRS. If units are in random order, then linear systematic sampling with mean-per-unit estimation will have sampling variance comparable to SRS with mean-per-unit estimation. But if there is a trend of increase or decrease in unit-specific y value with unit label or location, then sampling variance of a mean-per-unit estimator for a linear systematic design may be substantially less than for an SRS design. Circular and fractional interval systematic sampling designs are also presented. The disadvantage of these systematic sampling designs is that the highly restricted randomizations generally rule out unbiased estimation of sampling variance from a single systematic sample. Several approaches for variance estimation are considered.