S3.2 - Sampling

In a simple random sample of size n, every possible set of n distinct population units has the same chance of selection. A complete sampling frame is therefore needed so that every unit can be identified before random selection.

Number the N units consistently, for example 00–69 for 70 students. Read random digits in groups with enough places to represent the largest label. Accept a number only when it is in range and has not already been selected; ignore out-of-range values and repeats until n distinct units have been obtained.

For 70 students labelled 00–69, two-digit random numbers are required. If the stream begins 47, 83, 05, 47, 62, then select 47, ignore 83 because it is outside 00–69, select 05, ignore the repeated 47, and select 62.

The labels identify units but must not influence selection; the random-number source supplies the chance mechanism. Rejecting invalid and repeated labels preserves a sample without replacement while treating eligible labels symmetrically.

Choose a sampling method from the available frame, the population structure and the practical constraints. Stratified, systematic and quota sampling can all spread a sample across a population, but only the first two use a specified probability-based selection rule when carried out correctly.

For population size $N$ and stratum size $N_h$, allocate$$n_h\approx n\frac{N_h}{N},$$then adjust rounding so all allocations total $n$. With 800 employees split 430, 250 and 120 across three cities and sample size 100, proportional allocations are approximately 54, 31 and 15.

For an ordered list of $N=280$ students and sample size $n=40$, the interval is $k=N/n=7$. Choose a random start from 1 to 7, then select that unit and every seventh unit afterwards until 40 are chosen.