Random and systematic sampling

Systematic sampling looks like a shortcut, but the probability arithmetic behind it is exact, and seeing it removes the guesswork. Write the sampling frame as a numbered list from 1 to N, and set k = N ÷ n. If k is a whole number, the frame splits into exactly k columns when you write it in rows of k: the first column holds people 1, k + 1, 2k + 1 and so on, while the last holds k, 2k, 3k and so on. Every possible random start, a whole number from 1 to k, selects exactly one column. Each person therefore sits in exactly one column, so exactly one start out of the k available starts can select them, and the probability of selection is 1/k, which equals n ÷ N because k = N ÷ n. That equivalence is the entire justification of the method. Two practical consequences follow. First, if N ÷ n is not a whole number — say N = 1,000 and n = 300, giving k ≈ 3.33 — the interval cannot be a count of people. The usual fixes are to round k to a whole number, to keep a fractional interval tracked on a running total, or to change the sample size; rounding silently changes the probability, so the choice must be stated in the report. Second, the periodicity danger is a consequence of the same arithmetic. If the frame repeats a cycle of length c and k is a multiple of c, the selected column always lands on the same position in the cycle, so the sample contains one kind of record only. The sample is then biased, not merely unlucky. Reordering the frame at random, or replacing systematic sampling with simple random sampling, breaks the alignment. The method is only as good as the order of the list you feed it.

The four methods are not interchangeable tools for the same job; each buys something and pays for it elsewhere. Simple random sampling is the reference standard: with a sampling frame in hand it gives every individual the same selection probability and every combination of n individuals a chance of occurring, and its precision is easy to describe. Stratified sampling buys precision. When the strata differ in the measured quantity, removing the differences between strata removes part of the total variation, so an estimate within each stratum is more precise than an estimate of the same size drawn from the mixed population. The gain is largest when the strata are internally alike and differ strongly from one another. Allocating the sample in proportion to stratum size — proportional allocation — keeps the overall sample representative; deliberately over-sampling a small stratum is also legitimate when that subgroup needs its own precise estimate, provided the analysis re-weights it and the report says so. Cluster sampling buys affordability. It is chosen when individuals cannot be listed but natural groups can, and when the population is scattered over a large area, because the field cost depends on the number of places visited and not only on the number of people interviewed. The price is statistical: people inside a cluster tend to resemble one another, so a cluster sample carries less information per person than a simple random sample of the same size. That inflation of uncertainty is called the design effect, and the practical rule is blunt — if you must cluster, use more clusters and fewer people per cluster, because information comes mainly from the number of independent groups, not from the number of interviews inside each one. A health survey that studies 20 villages of 50 children each is usually better than one that studies 4 villages of 250 children each.

Random selection sounds effortless until you have to do it for real, and the mechanics are where beginners introduce invisible preference. A random-number table works like this: number the frame from 1 to N, decide in advance how many digits you will read together — for a frame of 600, three digits cover 001 to 600 — choose a starting row and column by chance, and read in one fixed direction. Write down each number that falls in range, ignore numbers outside the range and numbers already kept, and continue until the required count is reached. Fixing the direction and the digit count before reading prevents the most common quiet cheat, which is to keep reading because the next number “feels” better. Computer-generated numbers follow the same logic but are produced by a formula, so they are pseudorandom: the sequence looks patternless yet is reproducible from a starting value called the seed. Reproducibility is a virtue in research, because a seed recorded in the methods section lets another analyst rebuild the identical sample. It becomes a weakness only if the generator is poor or if the seed is chosen to produce a convenient result. Sampling without replacement — never returning a selected number to the pool — is what makes the sample contain distinct individuals and what gives every member the same probability n ÷ N; sampling with replacement would allow the same patient to appear twice and would change the arithmetic. Finally, an honest report states the frame, the method, the tools, the interval or allocation, and the numbers actually drawn, so that a reader can see that chance, not preference, produced the sample. In clinical audit this is not pedantry: a supervisor who can rebuild your sample can verify that the records you reviewed were not simply the convenient ones.