Acceptance Sampling and OC Curves

Acceptance sampling is simply deciding whether to accept a whole shipment by inspecting a small sample from it. A plan is two numbers: $n$ (how many parts you check) and $c$ (how many bad ones you'll tolerate). The probability of accepting, $P_a = \sum_{k=0}^{c}\binom nk p^k(1 - p)^{n - k}$, is a binomial sum you've already done: the chance of finding $c$ or fewer bad parts. Graph it for every possible defect rate and you get the OC curve, the plan's report card.

Judging a lot you didn't make. A lot of 5,000 connectors arrives from a supplier. Testing every one is slow and expensive, and for a fuse or an airbag igniter the test destroys the part. So you test a random sample and decide for the whole lot. A single sampling plan $(n, c)$: test $n$ parts, accept the lot if at most $c$ are defective, reject it otherwise.

Why it's a binomial sum. If a fraction $p$ of the lot is defective and the lot is large compared with the sample (a common rule: $n$ at most 10% of the lot), each sampled part is defective with probability $p$, nearly independently. The number found is binomial, as in The Binomial Distribution, so $P_a(p) = P(X \le c)$ is the sum in the first formula. For small lots the exact model is hypergeometric (sampling without replacement). When $n$ is large and $p$ small, the Poisson approximation with mean $np$ is easier and nearly as accurate.

The OC curve. Plot $P_a$ against $p$ and you get the plan's operating characteristic curve. The ideal is a cliff: accept every lot better than some quality level and reject every lot worse. Only error-free 100% inspection does that. Real plans slope, and bigger samples make the slope steeper.

Two quality levels, two risks. Buyer and supplier agree on

• an acceptable quality level (AQL), a fraction defective good enough that such lots should almost always pass. The chance of rejecting one anyway is the producer's risk $\alpha = 1 - P_a(\text{AQL})$, often set near 5%;
• a lot tolerance percent defective (LTPD), quality so poor that such lots should almost always fail. The chance of accepting one anyway is the consumer's risk $\beta = P_a(\text{LTPD})$, often 10%.

These are the type I and type II errors of hypothesis testing, with "the lot is good" as the null hypothesis. Designing a plan means choosing $n$ and $c$ so the OC curve passes near $(\text{AQL}, 1 - \alpha)$ and $(\text{LTPD}, \beta)$.

Zero-acceptance plans. With $c = 0$, $P_a = (1 - p)^n$, which falls steeply from the start. It sounds strict ("no defects allowed"), but mostly it punishes the producer: with $n = 50$, a lot that's only 1% defective still fails 40% of the time, since $0.99^{50} = 0.605$.

Rectifying inspection and AOQ. Often a rejected lot isn't sent back but screened 100%, with every defective replaced. Accepted lots then ship with the defectives among their $N - n$ uninspected parts, about $p(N - n)$ of them, and rejected lots ship clean. Averaging over lots, the outgoing fraction defective is

AOQ is 0 when lots arrive perfect and close to 0 when they arrive terrible, since nearly all of those get screened. In between it peaks, and that peak, the average outgoing quality limit (AOQL), is the worst long-run quality the customer can receive, whatever the supplier ships.

A common mistake. It's tempting to treat a lot that passed as a good lot. It isn't necessarily: the $n = 50$, $c = 1$ plan still accepts 28% of lots that are 5% defective. Sampling "10% of every lot" gives wildly different protection to different lot sizes, because the OC curve depends on $n$, not on $n/N$. And sampling sorts quality after the fact; it can't make parts better. Modern supply chains rely on suppliers' control charts and capability data, and keep sampling as an audit.

Incoming inspection at assembly plants, the industrial sampling standards (tables of $(n, c)$ plans indexed by lot size and AQL), food-safety and drug-import inspections and batch release in pharmaceutical plants all rest on OC curves. The same calculation tells a software team how many test runs must pass before a release and tells an auditor how many invoices to check.