Random Sampling and Bias

Populations and Samples

In statistics, the population is the complete collection of individuals or objects we want to draw conclusions about. In practice, populations are often too large, too costly, or too time-consuming to study entirely. We therefore select a sample — a manageable subset — and use it to draw inferences about the population.

Numbers that describe a population are called parameters. They are fixed (though often unknown). Numbers calculated from a sample are called statistics. Statistics vary from sample to sample and are used to estimate parameters. For example, the population proportion p is a parameter; the sample proportion p̂ is the corresponding statistic.

The goal of a good sampling method is to produce a sample that is representative of the population — one from which valid inferences can be made.

Simple Random Sampling

In a simple random sample (SRS) of size n, every possible subset of n individuals from the population is equally likely to be selected. This gives every member of the population an equal chance of inclusion and eliminates deliberate or unconscious selection bias.

In practice, an SRS can be taken by assigning every population member a number, then using a random number generator or table to select n numbers. The individuals corresponding to those numbers form the sample.

Systematic Sampling

In systematic sampling, the population of size N is listed in some order. We compute the sampling interval k = N/n, choose a random starting point between 1 and k, then select every k-th individual thereafter.

Potential weakness: if the list has a periodic pattern with period k, the sample may systematically include (or exclude) items sharing a particular characteristic, introducing bias.

Stratified Sampling

In stratified sampling, the population is divided into non-overlapping subgroups called strata (e.g. by year level, gender, or region). An SRS is taken from each stratum. The number sampled from each stratum is usually proportional to the stratum’s share of the population.

Advantage: all subgroups are represented, giving more precise estimates when subgroups differ substantially in the characteristic being measured.

Cluster Sampling

In cluster sampling, the population is divided into groups (clusters) — often by geography or natural grouping (e.g. schools, suburbs). A random selection of clusters is chosen, and every member of each chosen cluster is surveyed.

Advantage: practical and cost-effective when the population is spread over a large area. Disadvantage: members within a cluster often resemble each other more than the wider population, which can reduce the statistical precision compared with an SRS of the same total size.

Sampling Bias

Sampling bias occurs when the method of selecting the sample systematically favours some individuals over others, making the sample unrepresentative. Common sources include:

• Voluntary response bias: only individuals who feel strongly (often negatively) bother to respond
• Convenience sampling: selecting whoever is easiest to reach, rather than a random cross-section
• Undercoverage: some groups in the population have little or no chance of being selected
• Non-response bias: people who do not respond may differ systematically from those who do

A census surveys every member of the population, eliminating sampling bias entirely. However, a census is expensive, slow, and sometimes impossible (e.g. destructive testing). A well-designed random sample is usually preferred.