ISO 28596:2022 – Sampling Procedures for Inspection by Attributes — Two-Stage Sampling Plans for Auditing and for Inspection Under Prior Information

Introduction to Two-Stage Sampling with Prior Information

ISO 28596:2022 provides a framework for two-stage attribute sampling plans that exploit prior information about the quality of submitted lots. This standard is particularly valuable in audit situations and for inspection of infrequent lots where historical data exists but is insufficient for full switching rules. The two-stage approach allows the inspector to examine a first sample and decide to accept, reject, or take a second sample, with the second sample size potentially reduced when prior information suggests high quality.

Two-Stage Plan Structure and Decision Rules

In the two-stage plan, the first sample (n₁) is drawn and inspected. If the number of nonconforming items in n₁ is ≤ Ac₁ (acceptance number for stage 1), the lot is accepted. If ≥ Re₁ (rejection number for stage 1), the lot is rejected. If between Ac₁ and Re₁, a second sample (n₂) is drawn. The lot is accepted if the total nonconforming in n₁+n₂ ≤ Ac₂. The prior information influences the selection of n₁ and n₂ — stronger prior evidence of quality allows smaller sample sizes.

Prior Information Quantification and Update Rules

ISO 28596 provides methods for quantifying prior information from three sources: historical inspection data, supplier certification status, and process capability evidence. The prior distribution is updated using Bayesian principles after each inspection result. The standard includes pretabulated plans for beta-binomial prior distributions with parameters α and β representing prior conforming and nonconforming counts. After each lot inspection, the prior is updated: α’ = α + (number of conforming units) and β’ = β + (number of nonconforming units).

Engineering Application in Auditing and Quality Systems

For internal auditing of quality system records, two-stage sampling with prior information is particularly effective. The auditor can use prior audit findings as the prior distribution, reducing sample sizes for areas with consistently good performance while maintaining thorough coverage. The standard recommends an initial prior with β/α equivalent to the historical nonconformance rate, with a total prior weight equivalent to 3-5 audit cycles.

Bayesian Foundation and Prior Distribution Selection

The two-stage sampling plans in ISO 28596 are built on a Bayesian statistical foundation that formally incorporates prior information through a beta-binomial prior distribution. The prior distribution is parameterized by two shape parameters, α (representing the number of equivalent prior conforming units) and β (representing the number of equivalent prior nonconforming units). The ratio β/(α+β) represents the prior estimate of the fraction nonconforming, while the sum α+β represents the strength of the prior evidence in terms of equivalent sample size. The standard provides guidance on selecting appropriate prior parameters from three sources: historical inspection data (where α+β equals the total number of previously inspected units), supplier quality certification data (where the equivalent sample size is adjusted based on the certification level), and process capability evidence (where α+β = 30 × (Cpk)² as a rule of thumb). When multiple independent prior sources are available, the standard recommends combining them conservatively using the minimum equivalent sample size to avoid overconfidence. For suppliers with no prior history, the standard specifies a non-informative prior (Jeffreys prior: α = 0.5, β = 0.5) that produces sampling plans equivalent to classical non-Bayesian plans, ensuring that the system defaults to traditional sampling when no prior information exists.