Are two samples always inspected?

No. A first-stage decision can end inspection; the second is used only when the plan requires it.

Is average sample size always halfway between the two possible totals?

No. It depends on the probability of continuation.

Does double sampling mean two-phase surveys?

In survey contexts it often does, but that usage differs from the industrial acceptance plan described here.

AI Glossary / Statistics and probability / Intermediate

Double sampling

Definition

Double sampling, in the OECD glossary's principal industrial-inspection sense, is an acceptance-sampling plan that permits a decision after a first sample or, when required, after a second sample using their combined results. The first inspection can lead to acceptance, rejection or continuation. The term also commonly denotes two-phase survey sampling, which is a different design. State the context explicitly; this entry focuses on lot-acceptance inspection while distinguishing the survey usage.

What it means

The purpose is to avoid inspecting the maximum number of items when the first sample already supports a decision under predetermined rules. Clearly good or poor lots may receive an early decision, while less decisive results trigger additional inspection. The plan must specify both stages and their thresholds in advance. It is not an invitation to keep sampling until a desired result appears. Its performance is described through acceptance probabilities and expected inspection effort at different quality levels, alongside the risks of wrong decisions.

How it works

Specify first and second sample sizes and the first-stage acceptance, rejection and continuation rules. Inspect the first sample and count the relevant nonconforming items. If the result falls in the continuation region, inspect the second sample and apply the final rule to the combined count. Under a suitable independent-item approximation, binomial probabilities can calculate the operating-characteristic and average-sample-number curves. For a second sample always inspected completely when needed, expected sample size equals n1 + n2 times the probability of continuation. Finite-lot sampling without replacement may require hypergeometric calculations. The defect definition, selection method and any stopping within the second sample must be documented.

Three practical examples

Incoming inspection: evaluate a lot using a prespecified two-sample plan when inspection is costly but a decision is required.
Quality planning: compare a double-sampling plan with alternatives using producer risk, consumer risk and expected inspection effort.
Survey methodology: use the separate two-phase meaning to collect inexpensive information broadly and detailed measurements on a subsample.

Think of it this way

Think of a checkpoint with three clearly marked exits: accept, reject or inspect more. A second checkpoint resolves the cases sent for further inspection using the combined evidence. The rules must exist before the first checkpoint is used; otherwise the route can be changed opportunistically after seeing the result.

Worked example

Consider an illustrative plan with ten items in each sample. Accept after the first sample if there are zero defects, reject if there are at least two, and continue if there is exactly one. After continuation, accept only if the combined defect count is at most one, which requires zero defects in sample two. Under independent items with defect probability 0.10, continuation probability is 10 x 0.10 x 0.90^9 = 0.387420489. Expected inspected items are 10 + 10 x 0.387420489 = 13.87420489. Acceptance probability is 0.90^10 + 0.387420489 x 0.90^10, approximately 0.4838. Thus about 48.38% acceptance is a modeled plan property at this defect rate, not assurance that accepted lots are defect-free. This toy plan is for arithmetic illustration and has not been designed to meet particular quality-risk requirements.

Illustrative case study

Fictional case: Juniper Components considers a double-sampling inspection plan for incoming parts. The quality team calculates acceptance and expected-sample-size curves over relevant defect rates, then compares them with agreed risk requirements and inspection costs. Staff receive clear instructions for continuation and final decisions, and records retain both sample results. This invented case illustrates plan evaluation and execution. It does not endorse the worked example's thresholds or claim that fewer average inspections always outweigh the administrative complexity of two-stage decisions.

Common misconceptions and mistakes

Do not simply add a second sample after disliking the first result. Do not forget first-stage defects when the plan calls for a combined decision. The maximum sample size differs from the average sample number, which varies with the true quality level and stopping rules. Acceptance sampling makes a lot decision; it does not certify every item or improve the production process by itself. Two-phase survey sampling and two-stage cluster sampling have different purposes and inferential structures despite occasional overlapping terminology. A repeated independent sample is not automatically a double-sampling acceptance plan.

Business use

Businesses use double sampling to balance inspection effort and decision risks. Compare candidate plans under the quality levels that matter, including the chance of accepting poor lots and rejecting good ones. Account for delays, sample handling, destructive tests and the cost of administering a continuation stage. AI-assisted inspection does not remove the need to assess measurement error: if defect detection is imperfect, observed counts may not match true defect counts. Review the plan and its measurement system together before relying on calculated operating characteristics.

Check your understanding

Why is the expected sample size about 13.87 rather than twenty in the example?

Answer: The first ten items are always inspected, but the next ten are inspected only with probability about 0.3874. Expected effort therefore adds ten times that continuation probability to the initial ten.

Related terms

Acceptance sampling
Multistage Sampling
operating-characteristic curve
average sample number
producer risk
consumer risk
nonconforming item
sequential sampling
two-phase sampling
inspection plan

Learn next

enumerate first-stage outcomes and identify which trigger continuation. Calculate acceptance probabilities and average effort at several defect rates. Then compare acceptance sampling with two-phase survey designs, ensuring that shared terminology does not obscure different objectives, selection rules and uncertainty calculations.

Acceptance sampling

Sources

OECD Glossary of Statistical Terms, Double sampling: https://www.oecd.org/content/dam/oecd/en/publications/reports/2008/09/oecd-glossary-of-statistical-terms_g1gh9ad7/9789264055087-en.pdf NIST/SEMATECH, What is Double Sampling?: https://www.itl.nist.gov/div898/handbook/pmc/section2/pmc24.htm Statistics Canada, Probability sampling, multiphase usage: https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm

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