Suppose you want to investigate the working efficiency of nationalised bank in…

Concept. Sampling designs are told apart by how the researcher reaches the ultimate units that will actually be observed. When a population is very large, spread over a wide territory and arranged in a natural hierarchy of levels — a whole system, then intermediate groupings inside it, then smaller groupings inside those, and only at the bottom the individual units — a probability design can draw the sample in a succession of stages: at every stage a fresh random selection is made only inside the units already selected at the stage above. Because only the units chosen at a given stage have to be listed, a complete frame of every ultimate unit is never required. A design of this shape is called multi-stage sampling.

Application. A nationalised bank in India is organised in exactly such a hierarchy — a head office above zonal offices, zonal offices above regional or district offices, and those above thousands of individual branches. "Working efficiency" is something that is observed at the level of the individual branch, and no single practicable list of every ultimate unit exists. The study is therefore built up stage by stage:

1. Stage 1 — treat the bank's zonal offices as the first-stage units and draw a random subset of them.

2. Stage 2 — inside each selected zone, draw a random subset of the regional or district offices under it.

3. Stage 3 — inside each selected circle, draw a random subset of branches; only these branches now have to be listed.

4. Final stage — at each selected branch, observe the efficiency measures themselves (turnaround times, transaction volumes, staff and customer records), sub-sampling them if needed.

Each stage keeps a known, computable selection probability, so an estimate for the bank as a whole can be built back up from the branches actually visited, while the listing work stays confined to the units that were selected.

Cross-check against the other designs.

• Area sampling takes geographical units — blocks, wards, villages, map segments — as the clusters and works from a map-based frame. It too can be carried out in successive stages, so the two designs share machinery; what separates them is the frame along which the descent runs: area sampling descends territorial levels, whereas the levels this study must descend are the institutional ones of zone, region and branch.

• Sequential sampling leaves the sample size open and draws units one at a time or in small successive groups, stopping the moment a decision rule is satisfied; it answers an accept/reject question rather than building a structured cross-section of an organisation.

• Quota sampling is non-probability: pre-set counts for each category are filled with whatever convenient units the investigator can reach, so selection probabilities are unknown and system-wide estimates cannot be defended.

Result. The investigation is carried out by multi-stage sampling.