Sampling
Random sampling methods share one property: every sampling unit has a known, calculable chance of being chosen. That single property is what lets you trust — and put numbers on — how representative the resulting sample is.
What makes a method "random"?
A sampling method is random if every sampling unit in the population has a chance of selection that can be stated in advance — and that chance is non-zero for every unit. This is what makes probability theory applicable to the sample: because selection is genuinely governed by chance, the sample's behaviour can be predicted and its reliability quantified. There are three random sampling methods on this course: simple random, systematic, and stratified.
Simple random sampling
A simple random sample of size $n$ is one where every possible sample of size $n$ has an equal chance of being chosen. In practice, this means every individual sampling unit has an equal chance $\dfrac{n}{N}$ of appearing in the sample, out of a population of size $N$.
Two methods are used to actually generate a simple random sample from a numbered sampling frame:
Simple random sampling is free of bias and each unit has a known, equal chance of selection — but it needs a complete sampling frame, and for a large population it can be slow to number every unit and awkward to physically locate the selected ones.
Systematic sampling
Order the sampling frame (for example, alphabetically or by an existing reference number), then select units at a fixed interval $k$ after a randomly chosen starting point, where
The starting point is chosen at random from the whole numbers $1$ to $k$ inclusive; every subsequent unit in the sample is then found by repeatedly adding $k$.
If the first unit were simply chosen as, say, "the first person on the list," the method would no longer be random — that first unit's selection wouldn't be a matter of chance, and neither would any unit's position relative to it. A genuinely random starting point is what keeps the whole sample random, not just conveniently spread out.
Stratified sampling
Split the population into non-overlapping groups, called strata, based on a shared characteristic (such as gender, year group, or region). A simple random sample is then taken from within each stratum, with the number taken from each stratum kept proportional to that stratum's size in the population:
Where this calculation doesn't give a whole number, round to the nearest whole number of people.
Stratifying is worthwhile whenever the characteristic used to form the strata is likely linked to the thing being measured — because then simple random sampling alone could, purely by chance, under- or over-represent a subgroup with distinctly different behaviour. If the characteristic is unrelated to the outcome, stratifying adds complexity for no real statistical benefit.
Worked example 1
Always state what happens if a random number repeats ("discard and generate again") — omitting this is a common way to lose a mark, since without it the method could accidentally select fewer than 8 distinct members.
Worked example 2
Systematic sampling assumes there's no hidden pattern in the frame that lines up with $k$. If, for instance, items were produced in batches of 14 with a quality dip at the start of each batch, item 9 would always land in the same part of that cycle — this is a real limitation to be ready to discuss.
Worked example 3
The number found for each stratum is not the end of the method — members within each stratum must still be chosen by simple random sampling. Selecting them by convenience instead would turn this into quota sampling, which is not a random method.
Worked example 4
Both methods are equally "random" in the technical sense — the real question is which one controls for a source of variation that's actually relevant to what's being measured. Choosing a sampling method is as much about the context of the question as it is about the mechanics of the method.
Don't describe systematic sampling without a random starting point, and don't describe stratified sampling without a final random-selection step inside each stratum — both omissions are the most frequent reasons full marks are lost on these methods.
Check yourself
A call centre has 180 staff, listed 1 to 180. A systematic sample of 15 staff is required, with a random starting point of 6. List the first four register numbers that would be selected.
(Hint: find $k$ first, then add it repeatedly starting from 6.)