Why Are There Still Defects After Passing Sampling Inspection? —— A Five-Step Method to Quantify and Manage the Risk of Missed Defects

An electronics company's IQC follows the standard sampling inspection: batch size of 1000 units, general inspection level II, AQL 1.0, normal single sampling, sample size 80, Ac=2, Re=3. Over half a year, more than a hundred batches were all "sampling inspection passed" and stored in the warehouse, but the customer found about 3% defects in each batch on the assembly line, leading to a batch being returned and a claim placed directly on the quality manager's desk.

Upon review, it was found that the sampling inspection process was entirely compliant: random sampling, correct judgment, and complete records. The problem lay in a number that no one had ever calculated—the acceptance probability of batches with a 3% defect rate is 56.8%. In other words, more than half of the defective batches would be legally released.

1. What Does "Sampling Inspection Passed" Really Guarantee?

Sampling inspection never guarantees that "this batch is qualified." It only does one thing: use a sample size and acceptance number Ac to reduce the risk of accepting a defective batch to a level you can articulate.

AQL is not the "upper limit of allowable defect rate," nor is it a threshold for a single batch. It targets the process average of consecutive batches: when the actual defect rate of a batch is exactly equal to the AQL, the scheme accepts it with about a 95% probability, and the remaining 5% is the producer's risk α (good batch rejected). Correspondingly, the consumer's risk β (defective batch accepted) is characterized by the limiting quality LQ. Neither of these risks can be avoided; they must be explicitly written into the scheme—OC curve (operating characteristic curve) is the only tool that can present both risks simultaneously.

To put it another way: sampling inspection can only provide a probabilistic assessment for "this batch," never a definitive conclusion. Therefore, the rationality of the scheme must be clearly demonstrated with numbers before the inspection, rather than defending it with "we sampled according to the standard" after a customer complaint.

2. Calculate a Pa Table: Reveal the Scheme's True Nature

The OC curve is the curve of the acceptance probability Pa as a function of the batch defect rate p. Given a fixed sample size n and acceptance number Ac, the curve is determined using the binomial distribution:

Pa(p) = Σ〔C(n,i)·p^i·(1-p)^(n-i)〕, i from 0 to Ac

When the batch size is much larger than the sample size, the binomial distribution is sufficient. When the batch size is close to the sample size (such as in 100% inspection batches), the hypergeometric distribution should be used, which will yield slightly more optimistic results. Using the above scheme (n=80, Ac=2), the table is calculated as follows:

Three interpretations:

• AQL Point—The p corresponding to Pa ≈ 95% is the AQL (1.0% in this example), with α ≈ 5% being the risk borne by the producer.
• LQ Point—The p corresponding to Pa ≈ 10% is the limiting quality (about 6.5% in this example), with β ≈ 10% being the risk borne by the consumer.
• Discrimination—LQ/AQL ≈ 6.5, indicating that the scheme can only distinguish batches with obvious differences, such as "1% and 6.5%"; batches with defect rates between 2% and 4% are largely accepted or rejected by chance.

This table can also be read in reverse: if the customer explicitly requires that "batches with a defect rate of 2% or more must have an acceptance probability of no more than 10%," then the LQ must be around 2%, while the current scheme has an LQ of 6.5%, clearly not meeting the requirement—this translates a vague "customer requirement" into a calculable scheme indicator. Based on this, you can decide whether to tighten the AQL, increase the inspection level, or switch to 100% inspection and poka-yoke.

3. Five-Step Method: Calculate and Document the Scheme's Risk

Step 1: Define the Input. Determine the batch size, inspection level, and AQL. Generally, use general inspection level II; for destructive testing, high unit price, or high inspection costs, use level I or special levels; if the customer has special requirements (CSR), follow the customer's requirements and document the basis in the SIP.

Step 2: Look up n, Ac, Re, and Verify the Arrow. When encountering an arrow in the standard table, the sample size must be adjusted according to the arrow, not just the number. In this example: batch size 1000, level II → code J; AQL 1.0 → n=80, Ac=2, Re=3.

Step 3: Calculate Two Points: Pa(AQL) and LQ. One formula in Excel is enough: =BINOM.DIST(Ac, n, p, TRUE). A sampling scheme that only specifies n and Ac without Pa is equivalent to not conducting a risk assessment.

Step 4: Compare with the Cost of Escaped Defects to Decide on Sampling or 100% Inspection. Let the batch size be N, the inspection cost per unit be c, and the cost of escaped defects per unit be L, ignoring the inherent risk of 100% inspection:

100% Inspection Cost = N·c; Sampling Inspection Cost = n·c + (1-Pa)·N·p·L

In this example, N=1000, c=2 yuan, L=50 yuan, the cost of 100% inspection is 2000 yuan, and the break-even point is p ≈ 4.9%. When the batch defect rate is below 4.9%, sampling inspection is more cost-effective; when it is above 4.9%, 100% inspection is cheaper. Prior to this, IQC had been releasing batches with a 3% defect rate based on "sampling inspection passed," causing downstream processes to sort out several more defective units per batch.

Step 5: Document the Scheme and Its Risks. The ledger should at least include: batch size, inspection level, AQL, n/Ac/Re, Pa(AQL), LQ, current inspection status, and the judgment results of the last 10 batches. Transfer rules are not just formalities—consecutive 2 rejections trigger a stricter inspection, and 5 consecutive acceptances after tightening return to normal. This ensures that the scheme tightens when batch quality deteriorates.

4. Two Common Adjustments, Both with Costs

When the risk of missed defects is found to be high, the two common adjustments should be carefully evaluated for their costs, not just increased based on intuition:

The conclusion is straightforward: it is difficult to achieve "zero defects in the customer's hands" solely through sampling schemes. The improvement in discrimination is limited while inspection costs rise linearly. The real solution lies at the source—reducing process defect rates, adding poka-yoke, and controlling key characteristics through SPC or 100% automatic inspection. The role of sampling schemes is to quantify and control residual risks to an acceptable level, not to replace process control.

5. Four Common Misinterpretations

• Treating AQL as a Pass Rate. A "sampling inspection passed" by the customer only indicates that the number of defects in the sample did not exceed Ac; it does not mean that the batch defect rate is below AQL.
• Defending with "We Sampled According to the Standard." A scheme with AQL 1.0 still has a 56.8% acceptance probability at a 3% defect rate. If the actual tolerance on the customer's production line is much lower, the inspection level, AQL, or sampling/100% inspection strategy should be adjusted, not just repeatedly explaining compliance.
• Only Increasing Sample Size or Adjusting Ac. Both changes will alter the balance between α and β. Any adjustment should be recalculated on the OC curve and documented in the SIP and ledger.
• Evaluating Supplier Performance Based on Single Batch Judgments. A single rejection does not indicate that the supplier's process average is poor, and a single acceptance does not indicate that it is good. The judgment results should be accumulated over batches to form a process average for evaluation; otherwise, the triggers for tightening and relaxing will be inaccurate.

The quality of a sampling scheme is not written in the word "qualified," but in the moment you can articulate Pa and LQ.

Passing sampling inspection does not mean the batch is qualified: only by calculating Pa and LQ can you manage the risk of missed defects.

Knowledge code: 11.1.2

Version: v20261012

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.