Advancing QE Skills (32) | Statistical Principles of Sampling Plans: AQL, OC Curves, and Two Types of Risks

A car parts company received a quality agreement from a customer: batches with a nonconforming product rate exceeding 1.5% must not flow into the customer's production line. The quality engineer (QE) directly filled in the AQL as 1.5% in the Sampling Inspection Plan (SIP), and according to a batch size of 2400 pieces and general inspection level II, found the code K, setting the plan as n=125, Ac=5, Re=6, and wrote "meets the customer's 1.5% requirement" in the plan description.

During the customer's audit, they asked only one question: If the actual nonconforming product rate of this batch is exactly 1.5%, what is the acceptance probability of your plan? The answer, recalculated later, was 98.8%—a batch that should be rejected at the customer's red line would almost certainly be deemed acceptable and released. The issue is not with the arithmetic but with the direction: the customer's statement refers to the "quality level that must be blocked" (consumer risk point), while AQL describes the "process average level that can be accepted" (producer risk point). Mixing these two from the start makes the plan incorrect.

1. Three Quality Levels, Two Types of Risks: The Foundation of Sampling Plans

Any counting sampling plan is essentially determined by two parameters: sample size n and acceptance number Ac. Once these two numbers are fixed, the entire OC curve (the curve of acceptance probability Pa against batch quality p) is set, and so are the three key quality levels:

A crucial criterion to remember is: the usability of the plan is determined by the discriminatory power OR = CRQ / PRQ. Practical experience shows that plans with OR less than 3 have no practical value—they provide random conclusions for batches "just near the red line" rather than a clear judgment. The 1.5% in the customer agreement must fall under CRQ, while AQL should be set separately based on the supplier's actual process average level. These two should not be conflated.

The risk anchor points of 95% and 10% are historical conventions, not natural laws, and can be redefined based on business needs: for valuable or safety-critical parts, the acceptance probability for PRQ can be increased to 99% (α=1%), and the acceptance probability for CRQ can be reduced to 5% (β=5%), at the cost of a significantly larger sample size. Additionally, note that AQL and nonconforming product rate are not the same measure—AQL is calculated based on the number of defects per hundred units, and a single product can have multiple defects. Using the nonconforming product rate to infer AQL will systematically result in a looser standard.

2. Practical Step One: Select the Correct Distribution Model

Which distribution to use for Pa is determined by the ratio:

• n / N ≤ 0.1: Use the binomial distribution, Pa = Σ C(n,i)·p^i·(1-p)^(n-i) (i from 0 to Ac);
• n / N > 0.1: Must use the hypergeometric distribution, the binomial will overestimate the risk of missed detection;
• p ≤ 1% and n ≥ 100: The Poisson distribution can be used as an approximation, which is more convenient for manual calculations.

The differences are not trivial. For example, with N=5000, n=444, and Ac=3, the binomial distribution calculates Pa(0.25%)=97.37% and Pa(1.5%)=9.96%; the hypergeometric distribution gives 98.29% and 8.92%, respectively. The deviation is even more significant for small batches with a high sampling ratio: for N=1000, n=200, and Ac=2 (n / N=20%), the binomial distribution gives 5.93% for a batch nonconforming product rate of 3%, while the hypergeometric distribution gives only 4.19%, a difference of nearly 1.7 percentage points. With a small sample size, allowing the model to miscalculate the risk undermines the basis for discussion of the plan.

3. Practical Step Two: Derive n and Ac from the Two Types of Risks

The correct sequence is: first define the requirements, then derive the plan, rather than first looking up the table and then explaining.

First, write the requirements as four numbers. For example: PRQ=0.25%, α ≤ 5%; CRQ=1.5%, β ≤ 10%. The "1.5%" in the customer agreement should fall under CRQ, while the internal process capability target should fall under PRQ.

Second, solve by enumeration. Use the conditions Pa(PRQ) ≥ 0.95 and Pa(CRQ) ≤ 0.10 to start testing from the smallest n. The solution for the example is n=444, Ac=3 (Pa=97.37% / 9.96%), with a sampling rate of 8.9% for a batch size of 5000.

Third, understand the determining factors of the sample size. With the same risk requirements, the lower the quality level and the narrower the gap between the two levels, the more the required sample size increases:

Fourth, understand that Ac is the trade-off knob between α and β. With a fixed sample size of 444, changing only Ac immediately shifts the risks:

Each increment of Ac approximately doubles the probability of bad batches being accepted. Therefore, "increasing Ac for more insurance" is incorrect: you are suppressing the producer's risk while sacrificing the consumer's risk. To simultaneously reduce β without increasing α, the only method is to increase the sample size, which directly translates to more inspection hours.

4. Practical Step Three: Evaluate the Long-Term Consequences of the Plan Using AOQL and ATI

Looking only at α and β only gives insight into a single sampling inspection, not the plan's performance over a year. Two additional indicators are:

• ATI (Average Total Inspection) = n + (1 − Pa) × (N − n). For a plan with n=444, Ac=3, and N=5000: when the batch quality is 0.25%, ATI≈564 pieces; at 1.0%, ATI≈3400 pieces; at 1.5%, ATI≈4546 pieces, approaching the full inspection of 5000 pieces. This means that once the process deteriorates, the "time-saving" advantage of sampling inspection will automatically disappear, and inspection capacity must be reserved for the worst-case scenario.
• AOQL (Average Outgoing Quality Limit). Under the agreement that rejected batches are fully inspected and defective items removed, the AOQL for this plan is approximately 0.40%, occurring at a batch nonconforming product rate of about 0.66% (where Pa≈66%, ATI≈1980 pieces). This means that the long-term average quality of the output will not exceed approximately 0.40%. This number is more suitable for writing into quality commitments to customers, as it describes the actual quality of the delivered items, not the probability of a single batch being accepted.

The criterion is simple: the calculated AOQL must be lower than the customer's requirement for outgoing quality. If AOQL is higher than the requirement, it indicates that looking only at CRQ is insufficient; either the plan must be tightened, or key characteristics must be moved from sampling inspection to SPC or poka-yoke control.

Another common pitfall is the applicability boundary: counting-adjusted sampling standards (such as GB/T 2828.1 series) are based on continuous batches and adjust between normal, tightened, and reduced inspection levels using transfer rules. They protect the long-term process average, not a single batch. For isolated batches such as one-time purchases, trial batches, or return batches, which lack a process average to rely on, isolated batch plans designed according to LQ (such as GB/T 8051 series) should be used, or n and Ac should be derived by explicitly defining CRQ and β. Using tables for continuous batches to judge isolated batches is equivalent to removing the statistical foundation of the plan—while the table remains the same, the conclusions are no longer reliable.

5. Common Pitfalls

• Treating the customer's "red line" as AQL. In the example, filling in 1.5% as AQL results in an acceptance probability of 98.8% for a batch with a 1.5% nonconforming product rate, which is exactly opposite to the customer's intention. The customer's "must not exceed" corresponds to CRQ, while AQL should be determined based on the internal process average.
• Only providing n and Ac, without Pa and LQ. A plan without risk data cannot be verified, and any questions from the customer or auditor will easily expose its flaws. The plan card should at least include four numbers: PRQ, CRQ, α, and β.
• Unconditionally using the binomial distribution. When the sampling ratio exceeds 10%, the binomial distribution overestimates the risk of missed detection, with deviations of one or two percentage points in small batch scenarios, directly changing the decision on the plan.
• Assuming adjusting Ac can satisfy both parties. Ac is a zero-sum knob (as seen in the table), and reducing one risk will necessarily increase the other. To improve both, the only method is to increase the sample size and recalculate ATI and inspection resources.

6. Self-Check List

1. Are PRQ, CRQ, α, and β clearly stated on the plan card, and does the customer's requirement fall under CRQ?
2. Is the sampling ratio n / N over 10%? If so, has the hypergeometric distribution been used for verification?
3. Is the discriminatory power CRQ / PRQ less than 3? If so, has the plan been redesigned?
4. Has AOQL been calculated, and is it lower than the customer's requirement for outgoing quality?
5. Does ATI match the inspection capacity, and is there a full inspection contingency plan in case the process deteriorates?
6. Has each adjustment of n and Ac been re-read on the OC curve for α and β, and has the SIP and sampling ledger been updated accordingly?

First define the two types of risks, then derive n and Ac

Knowledge code: 6.2.2

Version: v20261016

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools for quality management practitioners, helping companies continuously improve their quality capabilities.