Acceptance Sampling OC Curve Calculator

Understanding the acceptance sampling OC curve calculator — concept and principle

In acceptance sampling, you inspect a random sample of n items from a lot and accept the lot if the number of defectives found is at most c. The operating characteristic (OC) curve shows how likely a lot is to be accepted at each true defect rate. This calculator gives the probability of acceptance for a chosen plan and defect rate, and the average outgoing quality (AOQ).

Even a 2%-defective lot passes about three times in four — the OC curve shows this for every defect rate.

How it is calculated

Probability of acceptance (binomial model): Pa = Σ from k = 0 to c of C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, where p is the lot defect rate as a fraction and C(n, k) is the number of combinations.
AOQ: AOQ = Pa × p × (N − n) / N, shown in percent because p is entered in percent.

N is the lot size. AOQ assumes that rejected lots are 100% inspected and that every defective found, in the sample or in a rejected lot, is replaced by a good item.

The principle

Each sampled item is defective with probability p, so the count of defectives in the sample follows a binomial distribution. The lot is accepted whenever that count is 0, 1, …, c, and Pa is the sum of those probabilities. Under rectifying inspection, accepted lots go out with their remaining defectives (the N − n uninspected items), while rejected lots go out clean. Averaging over both outcomes gives the AOQ formula.

Worked example

Inputs: n = 50, c = 1, p = 2% = 0.02, N = 1,000.
P(0 defectives) = 0.98⁵⁰ = 0.36417.
P(1 defective) = 50 × 0.02 × 0.98⁴⁹ = 0.37160.
Pa = 0.36417 + 0.37160 = 0.73577, so 73.58%.
AOQ = 0.73577 × 2 × (1,000 − 50) / 1,000 = 1.397965649%.

So a lot that is 2% defective is accepted about 3 times in 4, and the outgoing stream averages about 1.4% defective.

Each point is the calculator re-run with only “Lot defect rate (%)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Lot defect rate (%)	Probability of acceptance
0.5	97.39%
1	91.06%
1.5	82.73%
2	73.58%
2.5	64.35%
3	55.53%
4	40.05%

How to read the result

For this plan, Pa is about 97% at p = 0.5%, 91% at 1%, 56% at 3% and 28% at 5%. A steep OC curve separates good from bad lots sharply. Two points describe a plan: the producer's risk α (chance of rejecting a good lot at the acceptable quality level) and the consumer's risk β (chance of accepting a bad lot). The maximum of the AOQ curve over all p is the AOQL; for this plan it is about 1.6%, near p ≈ 3.2%.

Common mistakes

Entering the defect rate as a fraction (0.02) when the field expects percent (2).
Reading Pa as the chance that a lot is good; it is the chance that a lot of this quality passes.
Assuming AOQ applies when rejected lots are scrapped or returned instead of screened.

Keep in mind: The binomial model assumes independent defects and a sample that is small relative to the lot. When n is more than about 10% of N, the hypergeometric distribution is more exact (73.60% here). Standard plans such as ISO 2859-1 should be taken from the standard, not designed ad hoc.

Frequently asked questions

Does a larger lot need a larger sample?

Not in proportion. Pa depends mainly on n and c, not N, though standard tables raise n with lot size.

What does c = 0 mean?

Accept only if no defectives are found. It is strict on paper but its OC curve falls gently, so it discriminates poorly near typical quality levels.

Sources: ISO 2859-1, Sampling procedures for inspection by attributes — Part 1: Sampling schemes indexed by acceptance quality limit (AQL) for lot-by-lot inspection · D. C. Montgomery, Introduction to Statistical Quality Control, Wiley

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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