Systematic sampling of discrete and continuous populations: sample selection and the choice of estimator.

Systematic sampling is easy, efficient, and widely used, though it is not generally recognized that a systematic sample may be drawn from the population of interest with or without restrictions on randomization. The restrictions or the lack of them determine which estimators are unbiased, when using the sampling design as the basis for inference. We describe the selection of a systematic sample, with and without restriction, from populations of discrete elements and from linear and areal continuums (continuous populations). We also provide unbiased estimators for both restricted and unrestricted selection. When the population size is known at the outset, systematic sampling with unrestricted selection is most likely the best choice. Restricted selection affords estimation of attribute totals for a population when the population size - - for example, the area of an areal continuum - - is unknown. Ratio estimation, however, is most likely a more precise option when the selection is restricted and the population size becomes known at the end of the sampling. There is no difference between restricted and unrestricted selection if the sampling interval or grid tessellates the frame in such a way that all samples contain an equal number of measurements. Moreover, all the estimators are unbiased and identical in this situation.