Sampling Interval

When a probability sample is selected through use of a systematic random sampling design, a random start is chosen from a collection of consecutive integers [Page 792]that will ensure an adequate sample size is obtained. The length of the string of consecutive integers is commonly referred to as the sampling interval.

If the size of the population or universe is N and n is the size of the sample, then the integer that is at least as large as the number N/n is called the sampling interval (often denoted by k). Used in conjunction with systematic sampling, the sampling interval partitions the universe into n zones, or strata, each consisting of k units. In general, systematic sampling is operationalized by selecting a random start between 1 and the sampling interval. This random start, r, and every subsequent fcth integer would then be included in the sample (i.e. r, r + k, r + 2k, etc.), creating k possible cluster samples each containing n population units. The probability of selecting any one population unit and consequently, the probability of selecting any one of the k cluster samples is 1/k. The sampling interval and its role in the systematic sample selection process are illustrated in Figure 1.

For example, suppose that 100 households are to be selected for interviews within a neighborhood containing 1,000 households (labeled 1, 2, 3,…, 1,000 for reference). Then the sampling interval, k = 1,000/ 100 = 10, partitions the population of 1,000 households into 100 strata, each having k = 10 households. The random start 1 would then refer to the cluster sample of households {1, 11, 21, 31, 41,…, 971, 981, 991} under systematic random sampling.

In practice, the population size may not be an even integer multiple of the desired sample size, so the sampling interval will not be an integer. To determine an adequate sampling interval, one of the following adjustments may be useful.

1.
Allow the sample size to be either (n − 1) or n. The sampling interval, k, is then chosen so that (n − 1) × k is smaller than N and n × k is larger than N. Choosing a random start between 1 and k will imply a final sample size of either (n − 1) or n units. For example, if a sample of 15 houses is desired from a block containing 100, then N/n = 100/15 = 6.67. Choosing a sampling interval of k = 7 and allowing the sample size to be either 14 or 15 would then satisfy the requirement: (15 − 1) × 7 = 98<100 and 15 × 7 = 105 > 100. In this case, the sampling interval would be k − 7; random starts 1 and 2 would yield samples of size 15 while random starts 3 through 7 would yield samples of size 14.
2.
Allow circular references in the selection. In this case, the sampling interval is conveniently defined to be any integer no larger than N. A random start from 1, 2,…, N is chosen and that unit along with every successive fcth unit is selected—if the numbers being selected surpass N, simply continue counting from the beginning of the list as though the population identifications are arranged in a circular fashion. Continue selection until the desired sample size is reached. For example, suppose a sample of 5 households is to be selected from a block having 16 households; a sampling interval of 3 and a random start of 2 results in sampling households 2, 5, 8, 11, 14, 1 (identification number 17 exceeds the population size by 1, so the first element in the list is selected).
3.
Use fractional intervals. This approach combines the last approach with a modified computation of sampling interval. For example, suppose there were 200 high school students within a particular graduating [Page 793]class of which a sample of 16 was desired. The corresponding (fractional) sampling interval is k = 200/16 = 12.5.

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