Sampling Plan Decisions: OC Curves, Switching Rules, and Lot Risk

Reading a sampling table gets you a sample size and an accept number. A working inspector also has to know what that plan protects, and what it does not. How likely is a bad lot to slip through? When should inspection tighten? What do you do when a lot is mixed, too small, or comes back after a sort? These are the calls that separate someone who follows a plan from someone who can run one. Reminder: the plan and its AQL (acceptance quality limit) come from your control plan or customer. You do not choose them alone.

What You Will Be Able to Do

• Read an operating characteristic (OC) curve and state a plan's real risks in numbers.
• Compare two plans and say which gives better protection.
• Work out average outgoing quality when rejected lots are sorted.
• Keep a lot history and apply the switching rules.
• Handle mixed lots, small lots, resubmitted lots and defect classes correctly.

The OC Curve

An operating characteristic (OC) curve shows how likely a plan is to accept a lot, for each possible defect rate in that lot. You do not need to draw it. You need to read the numbers behind it, and quality software or a spreadsheet will give you those.

Worked example. Take the plan for a lot of 2,000 at General Level II, AQL 1.0, normal inspection: sample 125, accept on 3 or fewer defective, reject on 4 or more.

Three things stand out.

1. At the AQL, a good lot is still sometimes rejected. A 1 percent lot is rejected 3.7 percent of the time. That is the producer's risk at this quality level.
2. Between 2 and 4 percent, the plan is close to a coin flip. A 3 percent lot passes about half the time. One accepted lot tells you very little about a supplier.
3. Bad lots still get through. A lot at 5 percent, five times the AQL, passes about 1 time in 8. That is the consumer's risk at this quality level.

So when someone says "it passed AQL," the honest reply is, "at what quality?"

Comparing Plans

The OC numbers show why a bigger sample protects you better, and why the accept number matters as much as the sample size.

All the numbers are percent chance of acceptance. Look at the 5 percent row. The small plan accepts that lot more than twice as often as the large plan. Look at the 1 percent row. The small plan also rejects good lots more often. A small plan is worse on both sides. You pay for the larger sample in inspection time and you get a sharper line between good lots and bad lots.

This is why a small lot under the same AQL is less protected than a big lot. If a customer asks why you can't promise the same protection on a lot of 200, this is the answer.

Average Outgoing Quality

If every rejected lot gets a full sort, the plan also limits how bad the shipped average can get. Rejected lots come out clean. Accepted lots go out with whatever they held. The long-run average of what ships is the average outgoing quality (AOQ).

A handy approximation is:

AOQ = p x Pa x (N - n) / N

Here p is the incoming defect rate, Pa is the chance of acceptance, N is the lot size, and n is the sample size. This assumes bad parts found in the sample are pulled out and rejected lots are fully sorted.

Worked example. With N = 2,000 and n = 125, an incoming rate of 2 percent gives:

AOQ = 2 x 0.759 x (2,000 - 125) / 2,000 = 1.42 percent

The AOQ rises, peaks, and then falls. It falls because bad supply gets rejected and sorted more often. The peak is the average outgoing quality limit (AOQL), here a bit under 1.5 percent, near an incoming rate of about 2.3 percent.

Two warnings go with this. The AOQL is a long-run average over many lots, not a limit on any one lot. And it only holds if rejected lots are truly sorted. If they are accepted after a re-sample or shipped on a waiver, the math no longer applies.

Switching Rules in Practice

ANSI/ASQ Z1.4 is a scheme, not just a table. It is built to run over a series of lots from one source, and it tightens when the record goes bad. If you use it and skip the switching, you lose much of what it offers.

Original inspection means the first time a lot is inspected. A resubmitted lot does not count. The full conditions for reduced inspection, and the count that triggers stopping, can differ by edition, so read them in your copy. Your procedure also says who has the authority to approve the change.

Worked example. You keep this record for one supplier and part.

Lots 42 and 45 are both in the five-lot window from 41 to 45, so the second rejection triggers tightened inspection starting at lot 46. To go back to normal, you need five accepted lots in a row, 46 through 50. If lot 48 is rejected, the count starts again at lot 49.

Tightened plans use a stricter accept number or a larger sample. To see the effect, compare accept 3 with accept 2 at the same sample of 125. A 2 percent lot falls from a 75.9 percent chance of acceptance to 54.3 percent. A 3 percent lot falls from 48.1 percent to 27.3 percent. Those two numbers are only a comparison. Take the real plan from the tightened table in your shop's copy.

Inspection Levels

The inspection level sets how the sample size grows with lot size. General Level II is the usual default. Level I gives a smaller sample and less protection. Level III gives a larger sample and more protection. The special levels S-1 to S-4 give very small samples, for tests that are slow or that destroy the part.

For a lot of 2,000:

Do not change the level to save time on a busy day. A change of level is a change to the plan, so it goes through whoever owns the plan.

Forming a Fair Lot

Every number above assumes one lot made under one set of conditions. A lot put together from different sources breaks that assumption and hides the problem.

Worked example. Two machines each make 1,000 parts. Machine A runs at 0.2 percent defective, and machine B has a worn tool and runs at 6 percent. Someone packs them as one lot of 2,000.

• Combined, the lot is 3.1 percent defective. Under sample 125, accept 3, it passes about 45 percent of the time.
• Inspected on its own, machine B's lot of 1,000 uses sample 80, accept 2 (code letter J at AQL 1.0), and passes only about 13 percent of the time.

Mixing gave machine B's parts more than three times the chance of shipping. Keep lots separate by heat, machine, shift or supplier batch, as your plan requires. If you find a mixed lot, stop and ask before you sample.

Zero-Acceptance Plans Are Not Zero-Defect Plans

C=0 plans accept only when the sample has no defects. That can sound like a guarantee. It is not.

Worked example. A C=0 plan with a sample of 29 meets a lot that is 1 percent defective. The chance of finding no defects is 0.99 multiplied by itself 29 times, or about 74.7 percent. So three lots in four at 1 percent still pass. At 5 percent, 22.6 percent pass.

A C=0 plan gives a stronger reaction to a found defect, and it removes any idea that defects are acceptable. It still cannot see what it did not sample.

Other Ways to Sample

Variables sampling. ANSI/ASQ Z1.9 uses measured values instead of pass and fail. It usually needs a smaller sample for the same protection, because a number holds more information than a yes or no. It assumes the measurements follow a roughly normal (bell-shaped) distribution, so someone needs to check that first.

Skip-lot and process-based acceptance. When a supplier sends good capability data, a customer may reduce incoming inspection. That is a decision for your quality system, backed by records.

Calls You Will Have to Make

Common Mistakes

Check Your Understanding

1. A supplier's lot passes a plan where a 4 percent lot is accepted about 26 percent of the time. Your buyer says, "It passed, so it's under 1 percent." What is wrong?

A pass does not tell you the rate. Lots at several percent still pass some of the time. A trend over many lots says more than one result.

2. Rejections in the last six lots fall on lots 2 and 7 of a run from 2 to 7. Do you go to tightened inspection?

No, not on that record. Lots 2 and 7 are six lots apart, so they do not fall inside any window of five consecutive lots. Check your log for any other rejection in between.

3. Lots from two heats of steel arrive as one lot on one packing slip. Does that matter?

Yes. Different heats may differ, and mixing them can hide a bad one. Ask quality or receiving to split them, or to confirm how the plan treats them.

4. A C=0 plan with a sample of 50 shows no defects. Can you tell the customer the lot has no defects?

No. You can say it met the plan. A lot at 1 percent defective would still show no defects in a sample of 50 about 60 percent of the time.

5. Your plant sorts every rejected lot, and the AOQL is about 1.5 percent. A customer finds a lot at 4 percent. Did the AOQL fail?

Not necessarily. The AOQL is a long-run average over many lots, not a limit on one lot. But the lot should still be reported and looked into.

Key Takeaways

• An OC curve turns "it passed" into a real chance of acceptance at each quality level.
• A plan is sharper with a larger sample. Small plans are weaker for both the buyer and the seller.
• AOQL is a long-run average, and it depends on rejected lots really being sorted.
• Z1.4 is a scheme. Keep a lot history and follow the switching rules from your own copy.
• Keep lots separate, because a good source can hide a bad one.
• Zero acceptance reacts harder to a found defect. It is not a guarantee of zero defects.