Introduction
In statistics, we almost never have access to data from every individual in the group we are interested in. Instead, we select a sample — a subset of that group — and use it to draw conclusions about the whole. This is the starting point for all of statistical inference, which occupies a large portion of the PS2 paper.
For a sample to lead to valid conclusions, it must be chosen in a way that is free from bias. The key tool for achieving this is randomness. Exam questions in this area typically ask you to explain why a particular sampling method is unsatisfactory, or to describe how random numbers could be used to select a sample. These questions are almost always worth 2–4 marks and reward precise, concise language.
Core Concept
Population and Sample
A population is the complete set of individuals (or items, measurements, outcomes) under study. A sample is any subset of that population that is actually observed or measured.
Why not just study the whole population? Studying every member of a population (a census) is often impractical — it may be too costly, too time-consuming, or even destructive (e.g. testing every battery until it fails). A well-chosen sample provides reliable information at a fraction of the cost.
The Necessity of Randomness
A sample is random if every member of the population has a known (and typically equal) chance of being selected, and selections are made independently. Randomness is essential because:
- It removes human bias — the sampler cannot (consciously or unconsciously) favour certain individuals.
- It prevents systematic bias — structural features of the population are not allowed to skew the sample.
- It validates statistical theory — the mathematical results used in hypothesis testing and confidence intervals (covered later in PS2) are derived under the assumption that sampling is random.
When a Sampling Method is Unsatisfactory
A sampling method is unsatisfactory when some members of the population have no chance, or a systematically different chance, of being selected. This produces a biased sample, meaning the sample is not representative of the population, and any estimates derived from it will be unreliable.
Common sources of bias to recognise and describe in the exam:
| Source of bias | Why the sample is unsatisfactory |
|---|---|
| Self-selection (volunteers only) | People who respond tend to share certain characteristics; non-respondents are excluded |
| Convenience sampling | Only accessible or nearby individuals are included; the rest have zero chance of selection |
| Systematic human choice | The sampler may unconsciously favour certain individuals, e.g. choosing "approachable-looking" people |
| Sampling from an incomplete list | If the sampling frame omits part of the population, those members can never be chosen |
| Periodic/structured sampling without randomness | A fixed-interval method can align with a periodic pattern in the population |
Using Random Numbers to Produce a Random Sample
A random number table or a random number generator (e.g. on a calculator or computer) produces digits (or integers) in which every value is equally likely and successive values are independent.
Procedure for selecting a simple random sample of size from a population of members:
- Number every member of the population from to (or to ).
- Read off numbers from a random number table (or generator), taking groups of digits of the appropriate length.
- Select the member corresponding to each number obtained, ignoring repeats and any numbers outside the range to .
- Continue until distinct members have been selected.
This procedure guarantees that every member has probability of being included, and every possible sample of size is equally likely.
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