Sampling

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.

Conditions and key results

A good random draw from a poor frame does not cover omitted units. With finite populations, sampling without replacement creates dependence and standard errors may use a finite-population correction. Random sampling supports generalization; random assignment supports causal treatment comparisons.

A reliable strategy

1. Define the target population, unit, parameter, frame, and practical exclusions.
2. Choose a probability design suited to heterogeneity, geography, cost, and required subgroup estimates.
3. Draw according to the design and record inclusion probabilities, nonresponse, substitutions, and weights.
4. Estimate with design-aware uncertainty and limit generalization to populations actually supported by coverage and response.

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.

Common mistakes

Verification and reasonableness

• Audit the frame against population totals and calculate response by stratum.
• Reproduce selection probabilities and confirm no eligible unit had zero chance without disclosure.
• Compare weighted and unweighted estimates and use design-correct standard errors.

Practice

1. What does simple random sampling equalize?
2. Why stratify?
3. What primarily supports causal inference in an experiment?

Further deduction

For simple random sampling without replacement from population size $N$, the standard error of a mean includes $\sqrt{(N-n)/(N-1)}$, the finite-population correction. It matters when the sampling fraction $n/N$ is substantial and approaches zero as a census is reached. It does not correct coverage or nonresponse bias.

Systematic sampling chooses a random start and then every $k$th frame unit. It can approximate a spread-out probability sample when frame order is benign, but periodic ordering synchronized with $k$ can create serious bias or variance. Recording the random start, interval, and ordering is necessary to evaluate the design.

Stratified and cluster samples solve different design problems. Stratification samples units within every stratum, often improving precision when members within a stratum are similar. Cluster sampling selects whole groups or samples within selected groups, often reducing travel cost but increasing variance when members of a cluster resemble one another. Correct standard errors must reflect selection probabilities and clustering; analysing a complex sample as a simple random sample can substantially understate uncertainty.

Related topics

• Bias in Sampling
• Central Limit Theorem
• Data Collection