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Write the ordered list
Eight labelled values alternate 10, 30, 10, 30, 10, 30, 10, 30. The finite population mean is 20. Select four units by choosing start 1 or 2 uniformly and taking every second position.
List the two systematic samples
| Start | Selected positions | Mean | Probability |
|---|---|---|---|
| 1 | 1,3,5,7 | 10 | 1/2 |
| 2 | 2,4,6,8 | 30 | 1/2 |
Calculate the actual design variance
The estimator expectation is 20. Its variance is 0.5 × (10 − 20)² + 0.5 × (30 − 20)² = 100. Every unit nevertheless has inclusion chance 1/2. For a size-four simple random sample without replacement, S² = 800/7 and mean variance is (1 − 4/8)(800/7)/4 = 100/7 ≈ 14.2857.
Avoid a universal ranking from this counterexample
The illustration shows how a particular periodic order harms this systematic design. A different ordering can produce different behavior. It does not prove that systematic sampling is always less precise. Inspect the actual ordered frame and allowed samples instead of using a method label as a guarantee.