God, grant me the serenity to accept the things I cannot change, The courage to change the things I can, And the wisdom to know the difference. (Serenity Prayer, Reinhold Niebuhr)


Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference (Wikipedia).

Cooperation during Contract Execution

The cooperation usually concentrates on the following activities: The chose of evaluation method depends on a variety of factors:

Action on non-conforming product

Communications to the supplier on nonconformance must include a precise description of the symptoms of the defects. Product acceptance involves the disposition of product based on its quality. This disposition involves several important decisions:

Inspection Planning

Inspection planning is the activity of (1) designating the “stations” at which inspection should take place and (2) providing those stations with the means for knowing what to do plus the facilities for doing it.

Locating the Inspection Stations

The most usual locations are:

Increasingly, inspection is built into the process rather than being placed at the end of the process.

Automated Inspection

Automated inspection and testing are widely used to reduce inspection cost, reduce error rates, alleviate personnel shortages, shorten inspection time, avoid inspector monotony, and provide other advantages.
With the emphasis on defect levels in the parts-per-million range, many industries are increasingly accepting on-machine automated 100% inspection and testing.

A dramatic example of automated inspection is the concept of “machine vision”. A critical requirement for all automated test equipment is precision measurement.

Inspection Accuracy

Inspection accuracy depends on (1) the completeness of the inspection planning, (2) the bias and precision of the instruments, and (3) the level of human error. Human errors in inspection arise from multiple causes, of which four are most important: technique errors, inadvertent errors, conscious errors, and communication errors.

Measure of Inspector Accuracy

Some companies carry out regular evaluations of inspector accuracy as part of the overall evaluation of inspector performance.

Errors of Measurement

The difference between true value and the measured value can be due to one or more of five sources of variation:

Any statement of bias and repeatability (precision) must be preceded by three conditions:

Affect of Measurement Error on Acceptance Decisions

The probability of accepting a nonconforming unit is a function of measurement error. Measurement error can be a serious problem.
Observations from an instrument used to measure a series of different units of product can be viewed as a composite of (1) the varation due to the measuring method and (2) the variation in the product itself. This value can be expressed as:

The components of measurement error often focus on repeatability and reproducibility (R&R). Repeatability concerns variation due to measurement gauges and equipment; reproducibility concerns variation due to human “appraisers” who use the gauges and equipment. Studies to estimate these components are often called “gauge R&R studies”.
The ANOVA method is preferred to analyze the averages and ranges. A common practice is to calculate 5.15s (+/- 2.575s) as the total spread of the measurements that will include 99% of the measurements. If 5.15s is equal to or less than 10% of the specification range for the quality characteristic, the measurement process is viewed as acceptable for that characteristic; if the result is greater than 10%, the measurement process is viewed as unacceptable.

How Much Inspection is Necessary?

The amount of inspection to decide the acceptability of a lot can vary from no inspection to a simple of 100% inspection. The decision is governed mainly by the amount of prior knowledge available as to quality, the homogeneity of the lot, and the allowable degree of risk.

Prior knowledge that is helpful in deciding on the amount of inspection includes:

Competition to reduce costs has resulted in pressures to reduce the amount of inspection. The concept of inspection by the producers (self-inspection) has added to the focus of reducing inspection.

Economics of Inspection

We have several alternatives for evaluating lots:

The break-even point for sampling sizes is:
Pb =         Inspection cost per item
                damage cost incurred if a defective units slips through
If the lot quality (p) is less than Pb, the total cost will be lowest with sampling inspection or no inspection. If p is greater than Pb, 100% inspection is best. This principle is often called the Deming kp rule.
For example, a microcomputer device costs $0.50 per unit to inspect. A damage cost of $10.00 is incurred if a defective device is installed in the larger system. Therefore,
Pb = 0.50 / 10.00 = 0.05 = 5.0% If the percentage defective is expected to be greater than 5%, then 100% inspection should be used. Otherwise, use sampling or no inspection.

Acceptance Sampling

Acceptance sampling is the process of evaluating a portion of the product in a lot for the purpose of accepting or rejecting the entire lot. The main advantage of sampling is economy. In addition to this main advantage, there are others:

The disadvantages are sampling risks, greater administrative costs, and less information about the product than provided by 100% inspection.
Acceptance sampling is used when (1) the cost of inspection is high in relation to the damage cost resulting from passing a defective product, (2) 100% inspection is monotonous and causes inspection errors, or (3) the inspection is destructive. The concept of prevention (using SPC or other techniques) is the foundation for meeting product requirements. Acceptance sampling procedures are important in a program of acceptance control.

Quality Indexes for Acceptance Sampling Plans

There are several published quality indexes:
  1. Acceptable quality level (AQL) – the maximum percent nonconforming (or the maximum number of nonconformities per hundred units) that, for purposes of sampling inspection, can be considered satisfactory as a process average.
  2. Limiting quality level (LQL) – defines unsatisfactory quality. Lot tolerance percentage defective (LTPD). Because an LQL is an unacceptable level, the probability of acceptance for an LQL lot should be low.
  3. Indifferenace quality level (IQL) – a quality level somewhere between the AQL and LQL. It is fequenced defined as the quality level that has a probability of acceptance of 0.5 for a given sampling plan.

These indexes apply primarily when the production occurs in a continuing series of lots.

Types of Sampling Plans

There are two types of sampling plans:
  1. Attribute plans – a random sample is taken form the lot, and each unit is classified as acceptable or defective. The number defective is then compared with the allowable number stated in the plan, and a decision is made to accept or reject the lot.
  2. Variables plan – a sample is taken and a measurement of a specified quality characteristic is made on each unit. These measurements are then summarized into a sample statistic (e.g., sample average) and the observed value is compared with an allowable value defined in the plan. A decision is then made to accept or reject the lot.

Single, Double and Multiple Sampling

In single-sampling plans, a random sample of n items is drawn from the lot. If the number of defectives is less than or equal to the acceptance number (c), the lot is accepted. Otherwise, the lot is rejected.
In double sampling plans, a smaller initial sample is usually drawn, and a decision to accept or reject is reached on the basis of this smaller first sample if the number of defectives is quite large or quite small. A second sample is taken if the results of the first are not decisive. Because it is necessary to draw and inspect the second sample only in borderline cases, the average number of pieces inspected per lot is generally smaller in double  sampling. In multiple sampling plans, one, two or several still smaller samples are taken, usually continuing as needed until a decision to accept or reject is reached.


ANSI/ASQC Z1.4 (1993) is an attributes sampling system. Its quality index is the acceptable quality level (AQL). The AQL is the maximum percentage nonconforming (or the max number of nonconformities per 100 units) that, for purposes of sampling inspection, can be considered satisfactory as a process average. The probability of accepting material of AQL quality is always high but not exactly the same for all plans.

Disposition of Nonconforming Product

If an inspector finds that a lot of product is nonconforming, the lot is marked “hold” and is often sent to a special holding area to reduce the risk of mixups.

Decision Not to Ship

The investigation may conclude that the lot should not be shipped as is.

Decision to Ship

This decision may come about in several ways: