The Question Behind the Question

How much to sample from an incoming lot is not a question about arithmetic. It is a question about two costs that move in opposite directions: the cost of inspecting and the cost of accepting a bad lot. Every sampling plan chooses a point between them, whether or not anyone did the calculation.

What a Sampling Plan Contains

A complete plan specifies five things: the lot size range, the sample size, the acceptance number, the rejection number and the inspection level or AQL that generated them. Sampling tables such as ANSI/ASQ Z1.4 and its international counterpart ISO 2859-1 provide these combinations for single, double and multiple sampling, under normal, tightened and reduced regimes.

Three terms describe how the plan behaves:

  • Probability of acceptance, the chance that a lot with a given true defect level passes the plan.
  • Producer risk, the chance that a good lot is rejected. It is conventionally set near five percent.
  • Consumer risk, the chance that a bad lot is accepted. It is conventionally set near ten percent.

The plot of probability of acceptance against true defect level is the operating characteristic curve, and it is the only honest way to describe what a plan does.

The Arithmetic of Acceptance

For a lot much larger than the sample, the probability of accepting can be approximated with the binomial expression: accept probability equals the sum, from zero to the acceptance number, of the combination of sample size and defect count, multiplied by the true defect proportion to that power and by one minus the proportion to the remaining power. In practice nobody computes this by hand for every lot; the tables already encode it. But understanding the expression explains why the acceptance number matters more than the sample size for small lots.

Starting From Zero Findings

A useful approximation for planning: if no defects are found in a sample of n units, the upper bound of the ninety-five percent confidence interval for the true defect rate is approximately three divided by n. A sample of thirty with no findings therefore does not prove a rate below one percent; it is consistent with a rate near ten percent. A sample of three hundred with no findings supports a rate below about one percent.

This single relationship resolves most arguments about sample size. If the requirement is to demonstrate a rate below one percent, the sample must be in the hundreds, not the tens, regardless of what the table level suggests.

The Break-Even Method

When the sampling table and the supplier history disagree, a break-even calculation settles it. Compare the cost of inspecting the whole lot against the expected cost of the defects that full inspection would remove. Full inspection is worth it when the defect probability multiplied by lot size multiplied by the cost per escaped defect exceeds the lot size multiplied by the inspection cost per unit. Cancelling the lot size gives a simpler rule: inspect everything when defect probability multiplied by defect cost exceeds inspection cost per unit.

The same logic sets the sample size. For a given plan, the expected total cost is the inspection cost of the sample plus the probability of acceptance multiplied by the expected defect cost remaining in the unsampled portion. Compute that for two or three candidate plans and choose the lowest. The result is often smaller than the table default for cheap, low-risk components and larger for a component whose failure stops the line.

Applying It to Zipper Components

Components differ enough that one sampling level for all of them is a mistake. A workable structure separates them by consequence:

  1. Tape and yarn: colour and width checked against the approved standard, with strength tested per lot. Appearance is judged against a physical standard under fixed lighting.
  2. Sliders and pulls: dimensions and function, with particular attention to the locking spring and the puller attachment.
  3. Teeth and wire: dimensional and material verification, usually supported by a supplier certificate reviewed against periodic testing.
  4. Plating chemicals and dyes: certificate of analysis checked, with periodic verification rather than routine full testing.
  5. Packaging and labels: count, artwork and barcode verification, where an error is cheap to find now and expensive at the customer.

Critical characteristics justify tightened inspection or full verification; cosmetic characteristics usually do not. Data source for all of this is the incoming inspection record plus the supplier's certificate of analysis, and the two should be reconciled rather than filed separately.

When to Reduce Inspection

Sampling should respond to supplier performance, not remain fixed forever. Reduced inspection is justified when a defined number of consecutive lots have passed, the supplier's process has been reviewed, and a change control agreement exists. It should be withdrawn automatically on a failure, not on a judgement call.

The main additional step is skip-lot inspection, where whole lots are accepted without sampling on a defined schedule. It should be tied to an audited process and a written agreement, with a scheduled verification lot rather than trust.

Common Errors

  • Sampling from the top layer of a carton, where the supplier also placed the best pieces.
  • Using one level for every component regardless of consequence.
  • Treating a passed sample as proof that the lot is defect-free.
  • Recording results without recording the plan, so the number cannot be interpreted later.
  • Re-sampling a rejected lot without a documented concession or supplier corrective action.

YCC ties the inspection level to a supplier rating that is reviewed quarterly, so that a supplier with a stable record earns a lighter plan and a supplier with a recent failure returns to tightened inspection automatically. The rule is written, so the decision does not depend on how busy the receiving bay is on a given morning.

The Honest Summary

There is no universally correct sample size. There is a defensible one, derived from the consequence of a defect, the cost of inspection and the supplier's demonstrated performance, written down in advance and applied consistently. A plan that can be explained in those terms will survive an audit and, more importantly, will survive a bad month.