Systematic Sampling

This paper gives an account of the results of an investigation into one-dimensional systematic sampling, i.e. the sampling of sequences of quantitative values by the use of sampling points equally spaced along the sequence. New methods, using what are termed partial systematic samples, are evolved for estimating the systematic sampling error from short sections of sequences of completely enumerated numerical material. This gets over the difficulty, which previously existed, that the only estimates of the systematic sampling error of a numerical sequence, even when completely enumerated, were those provided by the actual deviations of the systematic samples of the whole sequence. Such deviations are few in number and by no means independent. Simple end-corrections are proposed for eliminating the errors, due to trend, which are otherwise inherent in randomly located systematic samples. It is demonstrated that it is impossible to make any fully reliable estimate of the sampling error from the systematic sampling results themselves, though if the continuous components of variation are not too marked, the sum of sets of terms taken alternately positive and negative, with suitable end adjustments, will provide a moderately satisfactory estimate, which will always be an overestimate provided there are no periodicities. This estimate is substantially better than the customary estimate based on successive differences. In other cases supplementary sampling is required to furnish an estimate of error, and methods are described whereby estimates can be derived from supplementary samples at half-spacing, or at half and quarter spacing. The performance of systematic sampling is investigated theoretically for certain mathematical functions, and also by the numerical analysis of certain numerical sequences. The mathematical functions investigated are (1) the two-valued function,/ ( a?) = 0 or 1, corresponding to sampling for attributes, (2) the normal error function, which corresponds to sampling for density with material normally distributed about a point in a line, and (3) the one-term autoregressive function yr+1=by?+a?? In the case of the two-valued function the relative performance of systematic and random samples is shown to depend on the lengths of the intervals of the function relative to the sampling interval. If these are small all forms of sampling are about of equal accuracy, but if they are large, systematic sampling is on the average twice as accurate as ra