Must mixed sampling combine probability and nonprobability methods?

No. It can combine probability methods throughout.

Does using two stages always make a plan mixed?

No. The basic methods may remain the same.

Can equal-probability units still be dependent?

Yes. Clustered or systematic selections can link which units appear together.

AI Glossary / Statistics and probability / Intermediate

Mixed sampling

Definition

Mixed sampling is a sampling plan that combines two or more basic selection methods. For example, a survey may select branches by simple random sampling and then select customers systematically within each chosen branch. The term describes how the sample is selected. It does not, by itself, specify the estimator, guarantee representativeness, or mean that numerical and textual data are being combined.

What it means

Different parts of a population may require different practical ways of selecting observations. A complete list of branches may exist centrally, while customer records are available only after a branch is chosen. Combining selection methods can make a survey feasible without treating the final sample as if it came from one simple random draw.
Terminology is not completely uniform. The OECD glossary uses mixed sampling for combinations of basic methods and cautions against automatically applying the label merely because replacement rules differ across stages.

How it works

Define the target population and identify the sampling units at each step. Specify each selection method, its sampling frame, and any randomization. A multistage design might first choose geographic areas, then establishments, and finally records. It is mixed when the basic selection approaches differ across those steps.
For probability sampling, determine each final unit's inclusion probability under the complete design. In a simple two-stage setting with one route to selection, multiply its first-stage inclusion probability by its conditional second-stage probability. The reciprocal provides a base sampling weight.
Systematic sampling typically selects a random starting position and then units at a specified interval. Its properties depend on list ordering, interval rules, and population size. Periodic patterns aligned with the interval can cause highly variable samples, even when individual inclusion probabilities are equal.
Estimation must account for the actual design. Clustering, stratification, unequal probabilities, and systematic selection can affect variance. Knowing the final sample size and base weights is not always sufficient to obtain a valid standard error.
Retain stage identifiers, frame information, probabilities, and any response adjustments. If one component uses convenience selection, do not assume that a probability-based component makes the combined sample a probability sample. Inference then needs an explicit justification for the nonprobability part.
Assess coverage and nonresponse separately from selection. A carefully randomized draw from an incomplete list can still omit important population groups.

Three practical examples

Customer research: randomly choose stores, then systematically select customer records within each selected store.
Document audits: stratify accounts by risk category and apply different justified selection methods to the resulting groups.
AI evaluation: select service locations through a probability design, then sample interactions from each location using a documented systematic procedure.

Think of it this way

Think of planning a journey using different forms of transport. The complete route matters, not just the final vehicle. Similarly, a selected record carries the consequences of every sampling step that made it eligible to appear.

Worked example

A fictional company has 20 branches, each with exactly 100 eligible customer records. It selects four branches by simple random sampling without replacement. Within each selected branch, it chooses a random start from positions 1 through 10 and then takes every tenth record, producing ten records per branch.
A branch has inclusion probability 4/20 = 0.2. Conditional on its branch being selected, every record has probability 10/100 = 0.1 because exactly one of the ten equally likely starts selects it.
The overall record inclusion probability is 0.2 times 0.1 = 0.02. Each selected record therefore has base weight 1/0.02 = 50. The 40 selected records have total base weight 2,000, matching the known population size.
If 12 sampled records show a particular issue, the weighted estimated total is 12 times 50 = 600 and the estimated proportion is 600/2,000 = 30%.
These are point estimates under the stated equal-size, complete-response design. Their standard errors cannot be obtained by pretending the 40 records were independently sampled from all 2,000 records.

Illustrative case study

Fictional case: Northbank Services audits AI-generated account summaries across branches. A central team randomly chooses branches, and local staff use documented random starts to sample records systematically.
The analyst preserves branch identifiers and checks whether record ordering follows a repeating operational cycle. When reviewing results, managers receive design-aware uncertainty and a separate explanation of excluded records. The audit's credibility rests on the documented selection and analysis, not on the label mixed sampling alone.

Common misconceptions and mistakes

Mixed sampling is not the same as mixed-mode data collection, which might combine telephone and online responses. It is also distinct from mixed-frequency time-series modeling and from drawing observations from a mixture distribution.
Multistage sampling is not automatically mixed sampling: stages can use the same basic selection method. Conversely, a mixed plan need not be described solely through stages.
Equal weights do not establish independence or justify simple-random-sample variance formulas. Replacing unavailable sampled units with convenient substitutes can change the selection mechanism and invalidate the original weights.

Business use

Businesses can use mixed sampling to balance access, cost, and the precision required for a decision. The sampling plan should explain why each method is used and how the final estimates will be calculated. Preserve enough design metadata for analysts to reproduce selection probabilities and assess uncertainty. This is especially useful when audits combine central records with locally maintained information.

Check your understanding

Why multiply probabilities in the worked example?

Answer: A record must pass both branch selection and conditional selection within its branch.

Why retain branch identifiers?

Answer: They describe clustering relevant to analysis and uncertainty, even when base weights are equal.

Related terms

Multistage Sampling
Simple Random Sampling
Cluster Sampling
Nonresponse
systematic sampling
inclusion probability
sampling weight
sampling frame

Learn next

calculate inclusion probabilities for an unequal-size two-stage design, then examine how clustered selection changes variance relative to simple random sampling.

Simple Random Sampling

Sources

Sources: OECD Glossary of Statistical Terms, mixed sampling, printed page 336: https://www.oecd.org/content/dam/oecd/en/publications/reports/2008/09/oecd-glossary-of-statistical-terms_g1gh9ad7/9789264055087-en.pdf . Statistics Canada, Probability sampling: https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm . Relevant primary institutional text was checked. The design and calculations are original illustrations.

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