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ISO 28801:2011 — Double Sampling Plans by Attributes with Minimal Sample Sizes
Acceptance sampling indexed by producer’s risk quality (PRQ) and consumer’s risk quality (CRQ) | Quality engineering guide
Introduction to ISO 28801
ISO 28801:2011 provides double sampling plans by attributes with minimal sample sizes, indexed by producer’s risk quality (PRQ) and consumer’s risk quality (CRQ). Unlike traditional AQL-indexed plans (such as ISO 2859-1), this standard is designed for isolated lots or short series of lots where switching rules cannot operate effectively. Prepared by ISO/TC 69/SC 5, it addresses the modern manufacturing reality of shorter production runs and higher quality levels.
These plans use the smallest possible acceptance and rejection numbers — typically 0 and 2 for the first sample — making them ideal for high-reliability industries such as aerospace, medical devices, and automotive safety components.
Key Technical Concepts
Plan Structure and Operating Procedure
The double sampling plan operates as follows: a first random sample is taken from the lot. If zero nonconforming items are found, the lot is accepted. If two or more nonconforming items are found, the lot is rejected. If exactly one nonconforming item is found, a second (smaller) random sample is taken; acceptance requires zero nonconforming items in the second sample. For nonconformities, similar decision rules apply with different numerical limits.
| Risk Pair (α, β) | Number of CRQ Values | Number of PRQ Values | First Sample Pattern | Second Sample Pattern |
|---|---|---|---|---|
| (5%, 5%) | 17 | 17 | Ac=0, Re=2 | Ac=0, Re=1 |
| (5%, 10%) | 17 | 17 | Ac=0, Re=2 | Ac=0, Re=1 |
| (10%, 10%) | 17 | 17 | Ac=0, Re=2 | Ac=0, Re=1 |
These plans are designed for isolated lots only. For continuing series of lots, AQL-indexed plans from ISO 2859-1 provide smaller sample sizes because they benefit from switching rule protection.
Engineering Design Insights
The statistical optimization behind ISO 28801 minimizes the expected total sample size subject to the constraint that nominal producer’s and consumer’s risks are not exceeded. The second sample size is always smaller than the first — typically about 60-80% of the first sample size — because the conditional probability of needing a second sample is low (equal to the probability of finding exactly one nonconforming item in the first sample).
The average outgoing quality (AOQ) curve for these plans shows a characteristic peak at intermediate incoming quality levels. The AOQ limit (AOQL) provides an important benchmark: regardless of incoming quality, the long-term average outgoing quality will not exceed this limit when rejected lots are 100% inspected and nonconforming items are replaced.
For high-reliability applications, these plans offer a significant advantage: the probability of accepting a lot with critical quality levels can be precisely calculated using the hypergeometric distribution, providing both producer and consumer with transparent, quantifiable risk levels.
Operating Characteristics
The operating characteristic (OC) curve for ISO 28801 double sampling plans differs from single sampling plans in an important way: the probability of acceptance drops more steeply around the consumer’s risk quality level, providing better discrimination between good and bad lots. The average sample size (ASN) curve is also lower than equivalent single sampling plans across most quality ranges, making these plans more economical in practice.
Practical Implementation Examples
A typical application of ISO 28801 in the automotive industry involves sampling incoming batches of safety-critical fasteners. For a producer’s risk quality (PRQ) of 0.65% nonconforming and consumer’s risk quality (CRQ) of 3.15% nonconforming with (5%, 10%) risks, the standard specifies a first sample of 50 items with acceptance number 0 and rejection number 2, and a second sample of 30 items with acceptance number 0 and rejection number 1. This plan requires only 50 items in most cases (when quality is good or bad), rising to 80 items only in the narrow range where the first sample yields exactly one nonconforming item.
In the medical device industry, where lot sizes are often small (200-500 units) and the cost of inspection is high, ISO 28801 provides significant cost advantages over single sampling plans. For a similar PRQ/CRQ combination, the average sample size is approximately 30-40% lower than the equivalent single sampling plan, with only a marginal increase in sampling variability. The standard’s tables for nonconformities (rather than nonconforming items) are particularly useful for assessing cosmetic defects in pharmaceutical packaging, where multiple minor defects per unit are possible but each individual defect carries low risk.
Quality engineers should note that the AOQL (average outgoing quality limit) for ISO 28801 plans is typically 5-15% lower than for equivalent single sampling plans, providing better protection for the consumer. When rejected lots undergo 100% screening, the AOQL provides a ceiling on the worst-case average quality reaching the customer.
Another practical consideration is the selection of the appropriate risk pair. The (5%, 5%) pair provides the most protection for both parties but requires the largest sample sizes. The (10%, 10%) pair minimizes sampling but increases risk for both sides. The (5%, 10%) pair is a common compromise, providing strong producer protection while allowing slightly more consumer risk. The standard’s tables enable quick selection by entering with the desired PRQ and CRQ values and reading the sample sizes directly. For chains of lots from the same supplier, quality history should be considered — consistently good quality may justify more economical plans, while variable quality warrants the larger sample sizes of the more protective plans.
Frequently Asked Questions
Q: When should I use ISO 28801 instead of ISO 2859-1?
A: Use ISO 28801 for isolated lots, short production runs, or when switching rules cannot be applied. For ongoing production with lot-by-lot inspection, ISO 2859-1 is more economical due to reduced sample sizes from switching rule protection.
A: Use ISO 28801 for isolated lots, short production runs, or when switching rules cannot be applied. For ongoing production with lot-by-lot inspection, ISO 2859-1 is more economical due to reduced sample sizes from switching rule protection.
Q: How are PRQ and CRQ values determined?
A> PRQ (producer’s risk quality) is the quality level at which the probability of acceptance is 1-α (typically 95% or 90%). CRQ (consumer’s risk quality) is the quality level at which the probability of acceptance is β (typically 5% or 10%). These are agreed between producer and consumer based on process capability and quality requirements.
A> PRQ (producer’s risk quality) is the quality level at which the probability of acceptance is 1-α (typically 95% or 90%). CRQ (consumer’s risk quality) is the quality level at which the probability of acceptance is β (typically 5% or 10%). These are agreed between producer and consumer based on process capability and quality requirements.
Q: Can these plans be used for nonconformities per 100 units?
A: Yes, separate tables are provided for sampling by nonconformities, with the same risk pairs and sample size patterns.
A: Yes, separate tables are provided for sampling by nonconformities, with the same risk pairs and sample size patterns.
Q: What happens when the second sample is also inconclusive?
A> The plans are designed so that the second sample always reaches a decision — either accept (0 nonconforming) or reject (1 or more nonconforming). No further sampling is needed.
A> The plans are designed so that the second sample always reaches a decision — either accept (0 nonconforming) or reject (1 or more nonconforming). No further sampling is needed.
