Sampling

A rigorous guide to populations, frames, probability samples, stratification, clusters, and inferential scope.

Question → sample → inference. A population question leads to a sampling design, a data summary, and a conclusion with uncertainty. The return arrow requires revisiting earlier choices whenever verification exposes a design or interpretation problem.

Precise definition

A sample is a subset or collection of units observed to learn about a target population. In a probability sample, each unit has a known nonzero inclusion probability. Simple random sampling gives every sample of a fixed size equal probability; stratified sampling samples within defined groups; cluster sampling selects groups of units.

Notation and mathematical language

The target population is the group the research question concerns; the sampling frame is the operational list or mechanism used to reach it. A statistic estimates a population parameter. Sampling variability describes how the statistic changes over repeated samples under the design.

Conceptual picture

Random sampling spreads selection chance across the frame and supports design-based inference. Stratification can improve precision when groups are internally similar and all important strata are represented. Clustering can reduce cost but often increases variance because units within clusters resemble each other.

Fully worked example

Interpretation and application

Sampling designs underpin surveys, audits, ecology, and quality control. Precision calculations assume the actual design; treating a clustered convenience sample as a simple random sample usually understates uncertainty.