Acceptance Sampling: How a Small Sample Decides the Fate of an Entire Lot

Why sampling instead of inspecting everything

When a shipment of 5,000 fasteners arrives at incoming inspection, nobody has the budget or the patience to gauge every single part by hand. What actually happens runs on an old piece of probability theory: pull a sample of n units from the lot of size N, count the defectives in that sample, and compare the count against an acceptance number c. If the number of defectives found is at or below c, the whole lot passes. If it exceeds c, the lot is rejected, screened part by part, or sent back to the supplier. That's acceptance sampling, and the four numbers that define a plan — Lot size N, Sample size n, Acceptance number c, and the plan's sensitivity to the Actual defect rate p — are the only levers an inspector has to balance two very different kinds of risk.

The first risk belongs to the producer: a genuinely good lot, with a defect rate at or below the agreed acceptable quality level (AQL), gets rejected anyway because the sample happened to pull an unlucky cluster of bad parts. The second belongs to the consumer: a genuinely bad lot slips through because the sample, purely by chance, looked clean. Every acceptance sampling plan is a compromise between these risks, and the compromise is entirely determined by how n and c are set relative to N and the defect rate you actually expect to see.

The binomial machinery behind the OC curve

The probability of accepting a lot, Pa, at a given true defect rate p is a cumulative binomial (or, for large N and small p, a very good Poisson approximation). For a sample size n, acceptance number c, and defect rate p:

Pa(p) = sum_{k=0}^{c} C(n,k) * p^k * (1-p)^(n-k)

Poisson approximation (accurate when n is large and p is small):
mean defectives in sample:  np = n * p
Pa(p) ≈ sum_{k=0}^{c} e^(-np) * (np)^k / k!

Plot Pa against p and you get the operating characteristic (OC) curve — the single most important diagnostic for any sampling plan. A steep OC curve that drops sharply near the AQL is a discriminating plan: it accepts good lots reliably and rejects bad lots reliably, with little gray zone in between. A shallow OC curve means the plan can barely tell a 1% defect lot from a 5% defect lot, which is a much bigger problem than it sounds — it means your inspection program is essentially rolling dice regardless of supplier quality.

Two more outputs matter once a plan is running in production. The sampling fraction n/N tells you how much of the lot you're actually looking at — a useful sanity check, because a fixed sample size against a shrinking lot silently changes your risk profile even though n and c haven't moved. And the average outgoing quality (AOQ) answers a different question entirely: given that rejected lots get 100%-screened and their defectives replaced with good parts, what's the defect rate of everything that actually ships, averaged over all possible incoming quality levels? AOQ has a maximum somewhere in the middle of the p range — the AOQL, average outgoing quality limit — which is often the number a customer actually cares about contractually, more than the AQL the plan was nominally designed around.

Walking through a real plan

Suppose a lot of N = 5,000 injection-molded housings arrives, and the agreed plan calls for a sample size n = 200 with acceptance number c = 5. The supplier's process has been running at roughly p = 2% defective historically. What happens when this lot is inspected?

First, the expected number of defectives in the sample, mean defectives np, is:

np = n * p = 200 * 0.02 = 4.0

Using the Poisson approximation with mean 4.0, the probability of finding 5 or fewer defectives in the sample — the probability the lot is accepted — comes out to:

P(X=0) = 0.0183
P(X=1) = 0.0733
P(X=2) = 0.1465
P(X=3) = 0.1954
P(X=4) = 0.1954
P(X=5) = 0.1563
---------------------
Pa = sum = 0.7851  →  78.5%

So the probability of acceptance Pa is about 78.5%, which means the rejection probability sits at roughly 21.5%. That's a meaningfully high rejection rate for a supplier running at their historical 2% defect level — a sign that this particular plan (n=200, c=5) is fairly strict relative to that process capability, or that the AQL target is actually lower than 2% and the supplier needs to tighten up.

The sampling fraction n/N here is 200/5000 = 4%, a fairly light-touch inspection by volume — most of the parts in the lot are never touched. And the average outgoing quality AOQ, assuming rejected lots are fully screened and their defectives replaced with known-good parts, works out to:

AOQ = Pa * p * (N - n) / N
    = 0.7851 * 0.02 * (4800 / 5000)
    = 0.7851 * 0.02 * 0.96
    ≈ 1.51%

So even though the incoming lot ran at 2% defective, the AOQ mechanism (screening on rejection) pulls the outgoing quality down to about 1.5%. With Pa at 78.5%, the tool's lot verdict tendency output would read "likely accept," but the margin is thin enough that a slightly worse sample draw flips the outcome — which is exactly the kind of borderline case a plan designer needs to see before committing to n and c on paper.

Where these plans quietly fail

The most common mistake is choosing n and c to hit a round number — "let's sample 5% of every lot" — without ever plotting the OC curve for that choice. A 5% sample against a 200-unit lot behaves completely differently than a 5% sample against a 50,000-unit lot, because the OC curve's steepness depends on the absolute sample size n, not the fraction. Fixing the fraction and letting n float with lot size is a classic way to end up with wildly inconsistent producer and consumer risk from one lot to the next.

The second mistake is treating AQL as a promise rather than a statistical target. An AQL of 1.5% does not mean "lots with more than 1.5% defects will be rejected." It means the plan is designed so that lots at the AQL have a high (typically 95%) probability of acceptance — plenty of 2% or even 3% defective lots will still sail through on a bad sampling day. If a customer contract says "1.5% AQL" and expects zero lots above that rate to ever ship, the plan needs a much smaller acceptance number than intuition suggests, and the rejection probability for marginal suppliers will climb accordingly.

The third is ignoring lot homogeneity. The binomial model assumes defectives are randomly distributed through the lot. If a molding run has a bad patch — a tool wear issue that hit only the last 500 parts — a sample drawn from the front of the pallet will systematically underestimate p, and no amount of clever n/c selection fixes a sampling procedure that isn't actually random.

Try it yourself

The fastest way to build intuition for how Lot size N, Sample size n, and Acceptance number c interact is to move them one at a time and watch the OC curve, Pa, and AOQ respond in real time — a static formula sheet just doesn't convey how sharply a plan's discrimination changes as n grows even while n/N stays small. You can experiment with the full plan, including AOQ and sampling fraction, using the Acceptance Sampling Plan Simulator here. If your inspection program also tracks failure rates over a product's life rather than at receiving, the bathtub curve tool is a natural companion for reliability planning once parts are in the field.