Accept the Shipment? Operating Characteristic Curves

You can't test every part in a shipment, so you test n of them and accept the lot if at most c are defective. The OC curve shows how often a lot passes at each true defect rate: a good plan passes good lots almost always and bad lots almost never, and n and c decide how sharply it separates them.

How it works

Acceptance sampling

In one sentence

For a single sampling plan $(n, c)$ applied to a large lot with defect fraction $p$ , the number of defectives in the sample is $X \sim \text{Binomial}(n, p)$ and $P(\text{accept}) = P(X \le c) = \sum_{k=0}^{c}\binom{n}{k}p^k(1-p)^{n-k}$ . The graph of $P_a$ against $p$ is the operating characteristic (OC) curve.

General formula

{\htmlData{sym=0}{P_a(p)}} = \sum_{{\htmlData{sym=4}{k}}=0}^{\htmlData{sym=3}{{c}}}{\htmlData{sym=5}{\binom{n}{k}}}{\htmlData{sym=1}{p}}^{\htmlData{sym=4}{{k}}}(1-{\htmlData{sym=1}{p}})^{{\htmlData{sym=2}{n}}-{\htmlData{sym=4}{k}}}

Sampling trades inspection cost against two risks: rejecting a good lot and accepting a bad one. The OC curve shows both risks at once for every possible quality level.

What the symbols mean

$P_a(p)$
Chance of acceptance
How often a lot with defect rate p passes the plan.
$p$
Defect rate
The fraction of the lot that's defective.
$n$
Sample size
Parts tested from each lot.
$c$
Acceptance number
The most defectives the sample can contain and still pass.
$k$
Defectives found
The number of defective parts in the sample.
$\binom{n}{k}$
Binomial coefficient
The number of ways to choose which k of the n sampled parts are defective.

If a fraction $p$ of the lot is defective and the lot is large, the number of defectives $X$ in a sample of $n$ is binomial with parameters $n$ and $p$ .
The lot is accepted when $X \le c$ , so the chance of acceptance is the binomial sum $P_a(p) = \sum_{k=0}^{c}\binom{n}{k}p^k(1-p)^{n-k}$ .
The producer's risk is $1 - P_a$ at the acceptable quality level (AQL, here 1% defective); the consumer's risk is $P_a$ at the lot tolerance percent defective (LTPD, here 8%).
Raising $n$ (with $c$ scaled to match) makes the curve drop more steeply, shrinking both risks.

Learn the math

Acceptance Sampling and OC CurvesEng Stats · Lesson 30, Statistical Process Control