Why 3 Defectives in a Sample of 80 Rejects the Whole Lot: The Math Behind GB/T 2828.1 AQL Sampling

A truckload of 1200 stainless steel balustrade posts arrives on site. Eighty are inspected. Three fail on wall thickness. The entire lot is rejected. The supplier objects: three out of twelve hundred is 0.25 percent, how can you reject the whole delivery? The buyer objects back: the contract says AQL 1.0 and the sampling plan is the sampling plan.

Both arguments miss the point. The real issue is this: a sampling plan never tells you how many defectives are in the lot. It tells you the probability that a lot with a given defect rate gets accepted. Those are very different statements, and the gap between them is where most site disputes are born. This article computes that gap explicitly using the binomial distribution, then answers three practical questions: what does the AQL figure actually commit anyone to, is it worth paying for a bigger sample, and what happens after a lot is rejected.

Standards and assumptions: sampling plans follow GB/T 2828.1 Sampling procedures for inspection by attributes, single sampling plans, general inspection level II, AQL 1.0. GB/T 2828.1 is the national adoption of ISO 2859-1, so the code letters, sample sizes and accept/reject numbers quoted here match the ISO tables. Acceptance probabilities are computed exactly from the binomial distribution rather than from the Poisson approximation. Chemical composition limits are taken from GB/T 20878; zinc coating thickness from GB/T 13912, the national adoption of ISO 1461; aluminium extrusion properties from the GB/T 5237 series. Steel density is taken as 7850 kg/m3 and zinc density as 7.14 g/cm3. Always verify against the current edition of the standard and against the inspection level agreed in the contract.

1. What AQL 1.0 Actually Promises (It Is Not a 1 Percent Cap)

AQL stands for Acceptable Quality Limit. The definition that matters is this: when the process average of submitted lots is equal to the AQL, the lot should be accepted with high probability. Read the subject of that sentence carefully. It is about the process average being accepted, not about the lot containing at most AQL worth of defectives.

Writing AQL 1.0 into a contract therefore means the buyer is saying: I tolerate your production process running at roughly one percent defective over the long run, and at that level I will still receive your goods. AQL describes a mutually tolerated process average, not a zero-defect promise on an individual lot.

The distinction matters because it determines what lever actually works. A buyer who genuinely wants zero defects and responds by tightening AQL from 1.0 to 0.65 or 0.1 does not get cleaner goods. What changes is the acceptance number Ac: it drops, good lots start getting rejected, lead times stretch, and eventually both sides work around the standard entirely. Section 6 covers the route that does work.

2. Looking Up the Plan: N = 1200 Gives n = 80, Ac = 2, Re = 3

GB/T 2828.1 is a three-step lookup, and each step has a defined input:

1. Lot size N to code letter. At general inspection level II, N = 1200 falls in the 501 to 1200 band, giving code letter J.
2. Code letter to sample size. Code letter J corresponds to n = 80.
3. Code letter plus AQL to the plan. Code letter J at AQL 1.0 under normal inspection gives Ac = 2, Re = 3.

Ac is the acceptance number and Re is the rejection number. The decision rule is one line: accept if the number of defectives d in the sample is at or below Ac, reject if d is at or above Re. Ac and Re differ by exactly one, so there is no middle ground:

That is the answer to the argument at the top. Three defectives lands exactly on Re = 3, so the lot must go. That number 3 was not derived from 3 divided by 1200. It came from the intersection of code letter J and AQL 1.0 in the master table. The supplier's 0.25 percent calculation is simply irrelevant to the plan.

One more rule that is worth stating explicitly: rejection applies to the whole lot, not to the three pieces that failed. That is the basic premise of sampling inspection - the sample represents the lot, and a failing sample means a failing lot. If you only want to pull out the pieces you happened to find, that is 100 percent screening, not sampling, and it is a completely different exercise in cost and in principle.

3. Computing the Acceptance Probability: the OC Curve Is a Binomial, Not a Feeling

Every (n, Ac) plan has an OC curve - an operating characteristic curve. The horizontal axis is the true fraction defective p in the lot; the vertical axis is the probability P_a that such a lot is accepted. This curve is not an empirical estimate. It can be computed directly.

Drawing n pieces at random from a large lot, the probability of finding exactly d defectives follows the binomial distribution:

The probability of acceptance is the sum from d = 0 up to Ac:

Work it through term by term for n = 80, Ac = 2, at a true defect rate of p = 4 percent:

That figure is the whole point of this article. A lot that genuinely contains four percent defectives still sails through this plan 37.5 percent of the time. Deliver three lots at that quality and on average one of them is accepted. Treating a passed sample as proof that the lot is clean means treating 37.5 percent as if it were zero.

4. Counter-intuitive Result 1: 2.5x the Sample Buys Half the Discrimination

The obvious response to weak discrimination is to inspect more. Move the lot size band up to 1201 to 3200 and code letter K gives n = 125, Ac = 3. Move further to 3201 to 10000 and code letter L gives n = 200, Ac = 5. Here is the full acceptance probability table for all of them at AQL 1.0, computed exactly from the binomial:

Look at the 4.0 percent row. Going from 80 to 200 pieces means inspecting 120 extra items, two and a half times the cost, and the pass probability only falls from 37.5 to 18.5 percent - roughly half the gap, for two and a half times the money. The 2.0 percent row is worse: across n = 80, 125 and 200 the acceptance probability runs 78.5, 75.8 and 78.5 percent. It does not improve at all, and at n = 200 it actually goes back up.

The reason is structural. The shape of the OC curve is governed by the product n x p, and Ac scales up in step with n - inspect more pieces and you naturally tolerate finding more defectives. Once Ac rises in proportion, the steepness of the curve barely moves. Buying discrimination with sample size is the least cost-effective move available in sampling inspection.

5. Counter-intuitive Result 2: Tightening Ac Is Far More Effective, and the Supplier Pays

What does move the curve quickly is the acceptance number. Tightened inspection under GB/T 2828.1 keeps the sample size unchanged and simply reduces Ac: code letter J at AQL 1.0 under tightened inspection is n = 80, Ac = 1, Re = 2. Here is how it compares:

The conclusion is blunt: tightened inspection inspects not one extra piece and still pushes the 4 percent pass rate down to 16.5 percent, stricter than the n = 200 plan at 18.5 percent. At 2 percent the gap is even wider - 52.3 percent against 78.5 percent. The same gain in discrimination, at zero inspection cost.

But the cost lands somewhere else. Look at the 1.0 percent row. That is the AQL point, the lot that by contract should sail through. Under tightened inspection its acceptance probability drops from 95.4 to 80.9 percent. The missing portion is the producer risk, alpha:

This is the real reason c = 0 plans - reject the lot if you find a single defective - struggle to survive in long-term supply agreements. Their discrimination is genuinely the best on the table: n = 50 alone drives a 4 percent lot down to 13.0 percent. But they also reject 39.5 percent of perfectly conforming lots. Two good deliveries out of five get sent back. No ongoing supply contract can be written that way.

Which explains something about the standard itself: GB/T 2828.1 defaults to Ac = 2, 3 and 5 rather than zero because the balance point is deliberately set on the producer's side. The standard assumes a continuing relationship between buyer and supplier, and prefers to let some moderately off lots through rather than risk wrongfully rejecting conforming ones. The practical consequences follow directly:

• If your objective is better incoming quality, the lever is tightened inspection (change Ac) and process control - not a bigger n.
• If your objective is fewer wrongful rejections and reliable lead times, reduced inspection or a larger n is the right direction, and you pay for it in discrimination.
• Chasing zero defects by lowering the AQL number alone does not work. Tightening AQL is just tightening Ac by another route: discrimination does improve, but alpha climbs with it until either the schedule breaks or both parties stop following the standard.

6. Switching Rules: When Ac Moves from 2 to 1

GB/T 2828.1 is not a single fixed plan. It defines switching rules between normal, tightened and reduced inspection, and the value of those rules is that inspection effort automatically flows towards suppliers that need it.

The rule most often ignored in practice is that the switching rules count lots, not defectives. One rejected lot counts once, whether three pieces failed or thirty. The rational supplier response is therefore to keep the process average of every lot inside the AQL, not to ship one bad lot and make up for it on the next - the latter walks straight into tightened inspection.

So what does work if you genuinely want critical defects to be near zero? Not the sampling plan. It is the combination of inspection strategies:

1. Reserve 100 percent inspection for critical characteristics. Wall thickness on load-bearing members, weld integrity, chemical composition - anything whose failure has serious consequences should not sit in the AQL pool at all. Inspect it fully or require certified third-party test reports per batch.
2. Let general characteristics be sampled. Appearance, packaging, non-structural dimensions: AQL 2.5 to 4.0 is entirely adequate, and it frees up inspection budget.
3. Move control upstream to the factory. Require first-article approval and factory release records so that defects are found on the production line rather than on site. A site rejection costs more than ten times an in-house rework.

7. A Rejected Lot Cannot Be Re-sampled - The Most Expensive Mistake on Site

Once a lot is rejected, the most common response on site is to draw another sample and see how it goes. That is wrong both mathematically and procedurally.

GB/T 2828.1 is explicit: the rejected lot is returned to the supplier, who must carry out 100 percent screening or rectification before resubmitting, and resubmitted lots are inspected under tightened inspection. There is no re-sample step in the standard.

Why is it forbidden? Because re-sampling after a rejection hands a failing lot a second draw. Take a genuine 4 percent lot: the first sample of 80 passes it 37.5 percent of the time. Even after a rejection (62.5 percent of cases), a second independent sample passes it with the same 37.5 percent. Stack the two chances and the overall pass probability becomes 1 - 0.625 x 0.625 = 60.9 percent, some 23 points above the 37.5 percent the plan was supposed to deliver. Every extra chance dilutes the guarantee the plan was designed to provide, and sampling inspection stops meaning anything.

There are three legitimate routes: (1) return the lot for 100 percent screening and resubmit under tightened inspection; (2) downgrade or accept under concession, which requires written design approval and a traceability record; (3) screen 100 percent on site at the responsible party's cost. For route 3, who pays must be settled in the contract, or it becomes the next dispute.

8. Three Material Checks That Decide Whether the Goods Are Real

The sampling plan answers how many to draw and how to decide. What actually determines whether the material can be used is what to measure and where to set the limit. With metal products the most commonly substituted item is the alloy itself, and it is also the item most often misjudged by field folklore. Each of the three materials has its own ruler.

8.1 Stainless steel: only measured Ni and Mo content counts

The most widespread field test is the magnet: if it sticks, it is 201; if it does not, it is 304. On fabricated stainless this method essentially does not work. Grade 304 is metastable austenite; cold bending, pressing and forming induce deformation martensite, so the worked zones pick up measurable magnetism. A genuine 304 post will attract a magnet at the bend and not along the straight. Judged by magnet, conforming material gets condemned - and genuinely low-nickel 201 in the solution-annealed condition, which is very weakly magnetic, gets passed.

Chemical spot tests for nickel answer only whether nickel is present, not how much, and they are strongly affected by the passive film. They are a rough screen, never an acceptance criterion. The only defensible acceptance method is portable optical emission spectrometry, a PMI gun, checked against GB/T 20878 (equivalent grades in EN and ASTM shown for reference):

The single most useful line in that table: decide 316L on molybdenum alone. The nickel ranges of 304 and 316L overlap between 10 and 11 percent, so nickel cannot separate them. Molybdenum belongs to 316L alone, and anything under 2.0 percent is an outright fail with no room for argument. For coastal projects, swimming pool plant rooms and wastewater treatment works - anywhere chloride is present - molybdenum is a mandatory test item, at a sampling ratio of at least 10 percent, and 100 percent on critical load-bearing components.

One further trap: do not compare materials across alloy families using a single PREN number. With PREN = Cr + 3.3 x Mo + 16 x N, grade 304 (Cr 19, N 0.05) works out at about 19.8 while 201 (Cr 17, N 0.20) comes to about 20.2 - the 201 scores marginally higher, yet it rusts first at the coast. The 16x coefficient for nitrogen was calibrated for solution-annealed austenite; in high-manganese steel much of the nitrogen precipitates as Cr2N and other nitrides, so the nitrogen that actually contributes to pitting resistance is far below the assay value. High manganese also brings MnS inclusions, which are preferred initiation sites for pitting. PREN is only comparable within a similar alloy family. Across families it is meaningless. Verify nickel and molybdenum instead.

8.2 Hot-dip galvanizing: local minimum and batch average are two parallel criteria

The classic galvanizing dispute is: the average is fine, so why is the lot rejected? The answer is in the standard. GB/T 13912, the national adoption of ISO 1461, sets both a local minimum thickness and a batch average minimum thickness, and both must be satisfied.

The conversion rests on the density of zinc, 7.14 g/cm3, so one micrometre of coating corresponds to 7.14 g/m2 (85 x 7.14 = 607 g/m2). When measuring on site with a magnetic gauge to GB/T 4956, where you place the probe matters more than what it reads:

• Keep away from corners and edges. Surface tension during withdrawal leaves corners measurably thicker. The reading flatters the article but says nothing about the corrosion protection of the main faces, and the standard does not take readings there.
• Keep away from threads and machined faces. Areas machined or tapped after galvanizing have the coating removed by design; they are not part of the assessment.
• Distribute readings over the surface, typically no fewer than five per article, spread across different zones. Five readings in one spot is one measurement, not five.
• Judge on the minimum, not the mean. If the batch averages 90 micrometres but one article has a local reading of 68 micrometres in the t >= 6 mm band, that article fails, and a healthy average does not release it.

8.3 Aluminium extrusions: film thickness, Webster hardness and base strength each cover a different thing

Aluminium needs three separate checks, and mixing them up is the easiest way to misjudge a delivery:

The Webster hardness test hides the deepest pit of the three. An anodic film reaches 300 to 500 HV, while 6063-T5 base metal sits around 65 HV. Test through the film and the indenter is measuring the oxide, giving a reading that can be double the true value and that says nothing about whether the extrusion was over-aged or under-aged. Grind the test point back to bright metal first, and avoid thin ribs and webs where the anvil effect distorts the result.

9. Five Ways This Goes Wrong

1. Reading AQL as a cap on the defect rate in the lot. Consequence: assuming an AQL 1.0 lot contains at most one percent defectives, when a four percent lot still clears the plan 37.5 percent of the time. Fix: treat AQL as the tolerated process average and read the actual per-lot protection off the OC curve.
2. Buying discrimination with sample size. Consequence: n goes from 80 to 200 at two and a half times the cost and the 4 percent pass rate only moves 37.5 to 18.5 percent; at 2 percent it does not move at all. Fix: switch to tightened inspection at Ac = 1, which reaches 16.5 percent for nothing.
3. Re-sampling a rejected lot. Consequence: two chances stack to give a 4 percent lot an overall 60.9 percent pass rate against the 37.5 percent the plan intended. Fix: return for 100 percent screening and resubmit under tightened inspection.
4. Identifying stainless with a magnet. Consequence: cold-worked 304 condemned as 201, and weakly magnetic 201 passed as conforming. Fix: measure with a PMI gun; nickel at or above 8 percent for 304, molybdenum at or above 2.0 percent for 316L.
5. Checking only the average coating thickness. Consequence: the lot passes on average while individual articles run thin locally, and those articles start rusting at the thinnest point with no per-point record to support a claim. Fix: apply average and local minimum together, spread the readings, and keep the individual records on file.

10. Summary and an Incoming Inspection Checklist

Back to the argument at the loading bay. Drawing 80, finding three and rejecting the lot is not a defective plan - it is a plan with expectations attached that neither side had stated. It does not guarantee that the lot is free of defectives. It guarantees that, over time, lots running near the process average the contract tolerates will usually pass. Writing that premise into the contract is worth more than any argument afterwards.

In one line: AQL 1.0 means a lot running at a one percent process average is still accepted 95.4 percent of the time, not that the lot holds at most one percent defectives. Against a four percent lot, n=80/Ac=2 passes 37.5 percent; a sample of 200 barely helps at 18.5 percent, while tightened inspection at Ac=1 reaches 16.5 percent without inspecting a single extra piece - and the price is that wrongful rejection of good lots rises from 4.6 to 19.1 percent, which is exactly why the standard sets the default balance on the producer's side.

Frequently Asked Questions

Does AQL 1.0 mean the lot contains at most one percent defectives?

No. AQL is the acceptable quality limit, defined by the property that a lot whose process average equals the AQL should be accepted with high probability. For n=80, Ac=2: at a true one percent defect rate the acceptance probability is 95.4 percent, and at four percent the lot still passes 37.5 percent of the time. AQL constrains the long-run process average, not the defect content of an individual lot. If you genuinely need individual lots to be defect free, take the critical characteristics out of the sampling pool and inspect them 100 percent or require certified third-party reports.

To improve incoming quality, should I increase the sample size or change the AQL?

Changing the AQL - which effectively changes the acceptance number - is far more efficient than increasing the sample. With n=80, Ac=2 a four percent lot passes 37.5 percent of the time; raising the sample to 200 pieces costs two and a half times as much and only reaches 18.5 percent, whereas holding at 80 pieces and cutting Ac to 1 under tightened inspection reaches 16.5 percent with no extra inspection work at all. The price is producer risk: acceptance at the AQL point falls from 95.4 to 80.9 percent, so wrongful rejection of conforming lots rises from 4.6 to 19.1 percent. Tightened inspection is therefore the right tool for a supplier whose quality has slipped, not something to apply across the board indefinitely.

Once a lot is rejected, can we draw another sample?

No. GB/T 2828.1 requires the rejected lot to be returned to the supplier for 100 percent screening or rectification before resubmission, and resubmitted lots are inspected under tightened inspection. Re-sampling is mathematically equivalent to giving a failing lot a second draw: a four percent lot passes at 37.5 percent per attempt, so allowing a re-attempt lifts the overall pass rate to 60.9 percent and systematically dilutes the protection the plan was meant to provide. If the material is genuinely needed on site, the only route is a concession with written design approval and a traceability record.