Probability sampling designs

What makes a design a probability design

A probability sample is one in which every unit has a known, non-zero chance of selection, and the selection is made by a chance mechanism — a random-number generator, not a person's judgement. That is what makes the sampling error calculable in Modules 8–13. Four designs are examined.

Simple random sample (SRS). Every possible subset of $n$ units is equally likely to be the sample. Note the wording: it is not enough that each unit is equally likely — a design that takes every unit with the same probability but can only ever produce certain subsets (systematic sampling, below) is not an SRS. Sampling is normally without replacement; for a large population the distinction barely matters, and the course's formulas assume it does not.

Stratified sample. Divide the population into groups (strata) that are homogeneous within and different between — branches, provinces, account sizes — then draw an SRS inside every stratum. Precision improves because the between-stratum variation is removed from the sampling error. Two allocation rules: proportional (each stratum's share of the sample equals its share of the population) and equal (the same $n$ in every stratum, used when each stratum must be estimated on its own).

Cluster sample. Divide the population into groups (clusters) that are each a miniature of the whole — the stores of a chain, the pages of a ledger — draw an SRS of clusters, and measure every unit in the chosen clusters. It saves travel and listing cost; it costs precision, because units within a cluster resemble each other.

Systematic sample. From a numbered list choose a random start between 1 and $k$, then take every $k$-th unit. Fast for a file of invoices; dangerous if the list has a cycle of length $k$ (every 12th entry a month-end batch).

Worked example — allocating an audit sample

An auditor will test 200 receivables accounts at a client with three branches holding 6,000, 4,200 and 1,800 accounts ($N=12,000$). Proportional allocation gives each branch $n\times N_h/N$:

$$n_A=200\times \frac{6,000}{12,000}=100,n_B=200\times \frac{4,200}{12,000}=70,n_C=200\times \frac{1,800}{12,000}=30$$

The three shares add to 200. Under equal allocation each branch would get $200/3\approx 67$ — more from the small branch than its share, which is the right call if the auditor must report on each branch separately and needs the small one estimated as precisely as the big ones.

A second case: a firm sends 240 internal auditors across four provinces and wants each province's control environment assessed on its own. Equal allocation: $240/4=60$ per province, regardless of how many locations each province has.

Telling them apart

The examined confusion is stratified versus cluster. Both start by grouping. In stratified sampling every group is used and units are sampled inside each; in cluster sampling some groups are sampled and everything inside them is used. Ask: "were all the groups touched?" — yes means strata, no means clusters.