Some Remarks on Systematic Sampling

Consider a finite population consisting of N elements y₁, y₂, ⋯, yN. Throughout the paper we will assume that $N = nk$. A systematic sample of n elements is drawn by choosing one element at random from the first k elements y₁, ⋯, yₖ, and then selecting every kth element thereafter. Let yᵢⱼ = yi + (j - 1)k(i = 1, ⋯, k; j = 1, ⋯, n); obviously systematic sampling is equivalent to selecting one of the k "clusters" Cᵢ = ᵢⱼ; j = 1, ⋯, n\ at random. From this it follows that the sample mean yᵢ = 1/n ∑ⁿj = 1 yᵢⱼ is an unbiased estimate for the population mean y = 1/N ∑ᵏi = 1 ∑ⁿj = 1 yᵢⱼ and that Var yᵢ = 1/k ∑ᵏi = 1 ( yᵢ - y)². We will denote this variance by V⁽¹⁾sy indicating by the superscript that only one cluster is selected at random. V⁽¹⁾sy can be written as {equation*}{1}V⁽¹⁾sy = S^2 - 1/k ∑^ki = 1 S^2_i, where S^2 = 1/N ∑^ki = 1 ∑^nj = 1 (yᵢⱼ - y)^2,{equation*} \\ {equation*} S^2_i = 1/n ∑^nj = 1 (yᵢⱼ - y_i)^2.{equation*} It is natural to compare systematic sampling with stratified random sampling, where one element is chosen independently in each of the n strata ₁, ⋯, yₖ\, + 1, ⋯, y₂ₖ\, ⋯, and with simple random sampling using sample size n. The corresponding variances of the sample mean will be denoted by V⁽¹⁾ₛₜ V⁽ⁿ⁾ᵣₐₙ respectively. We consider now the following generalization of systematic sampling which appears to have been suggested by J. Tukey (see [3], p. 96, [4], [5]). Instead of choosing at first only one element at random we select a simple random sample of size s (without replacement) from the first k elements and then every kth element following those selected. In this way we obtain a sample of $ns$ elements and, if i₁, i₂, ⋯, iₛ are the serial numbers of the elements first chosen, the sample mean 1/s( yi₁ + ⋯ + yiₛ) can be used as an estimate for the population mean. This sampling procedure is clearly equivalent to drawing a simple random sample of size s from the k clusters Cᵢ(i = 1, ⋯, k). It therefore follows (see, for example, [2], Chapter 2.3 to 2.4) that the sample mean is an unbiased estimate for the population mean and that its variance, which we denote by V⁽ˢ⁾sy, is given by begin{equation*} {2}V⁽ˢ⁾sy = k - s/ks 1/k - 1 ∑^ki = 1 ( y_i - y)^2 = 1/s k - s/k - 1 V⁽¹⁾sy.{equation*} Again, it is natural to compare this sampling procedure with stratified random sampling, where a simple random sample of size s is drawn independently in each of the n strata ₁, ⋯, yₖ\, + 1, ⋯, y₂ₖ\, ⋯ or with simple random sampling employing sample size $ns$. We denote the corresponding variances of the sample mean (which in both cases is an unbiased estimate for the population mean) by V⁽ˢ⁾ₛₜ ,V⁽ⁿˢ⁾ᵣₐₙ respectively. From well-known variance formulae (see, for example, [2], Chapters 2.4 and 5.3) it follows that {equation*}{3}V⁽ˢ⁾ₛₜ = 1/s k - s/k - 1 V⁽¹⁾ₛₜ,\\ V⁽ⁿˢ⁾ᵣₐₙ = N - ns/s(N - n) V⁽ⁿ⁾ᵣₐₙ = 1/s k - s/k - 1 V⁽ⁿ⁾ᵣₐₙ. {equation*} Thus the relative magnitudes of the three variances V⁽ˢ⁾sy, V⁽ˢ⁾ₛₜ, V⁽ⁿˢ⁾ᵣₐₙ are the same as for V⁽¹⁾sy, V⁽¹⁾ₛₜ, V⁽ⁿ⁾ᵣₐₙ, of which comparisons were made for several types of populations by W. G. Madow and L. H. Madow [6] and W. G. Cochran [1]. Some of the results will be reviewed in Section 3. The object of this note is to compare systematic sampling with s random starts, as described above, with systematic sampling employing only one random start but using a sample of the same size $ns$. To make this comparison we obviously have to assume that k is an integral multiple of s, say $k = ls$. The latter procedure then consists in choosing one element at random from the first l elements ₁, ⋯, yₗ\ and selecting every lth consecutive element. We denote the variances of the sample mean of the two procedures by V⁽ˢ⁾ₖ, V⁽¹⁾ₗ respectively, indicating by the subscript the size of the initial "counting interval." (In our notation V⁽ˢ⁾sy ≡ V⁽ˢ⁾ₖ.) We shall show in Section 4 that V⁽¹⁾ₗ = V⁽ˢ⁾ₖ in the case of a population "in random order," but V⁽¹⁾ₗ < V^(s)ₖ for a population with a linear trend or with a positive correlation between the elements which is a decreasing convex function of their distance apart. Some numerical results on the relative precision of the two procedures will be given in Section 5 for the case of a large population with an exponential correlogram.