Historical Articles

January, 1954 issue of Plating

 

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Acceptance Sampling of Electroplated Articles

J.M. Cameron and Fielding Ogburn, National Bureau of Standards, Washington 25, D.C.

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ABSTRACT
Acceptance sampling procedures have found widespread usage in governmental and industrial purchasing. The purpose of this note is to present some of the basic ideas behind acceptance sampling.

INTRODUCTION
When articles are being purchased, some understanding must exist between the purchaser and the supplier concerning the requirements that the items must meet, in order to be considered acceptable. Industry and government have recognized the need for definite and exact statements of such requirements and have devoted much energy to the writing of material specifications. These specifications give, in addition to the detailed requirements, a description of the test procedure by which conformance of an individual item is to be ascertained.

Ordinarily it is not feasible to accept or reject individual items from a lot. One hundred per cent inspection or screening, as it is called, is needed when the elimination of all nonconforming items is required.

When the testing is destructive or expensive, screening is out of the question and it is almost inevitable that some nonconforming items will be passed along from supplier to purchaser. The purchaser is therefore forced to make a decision as to the acceptability of an-entire lot or shipment on the basis of evidence supplied by a sample of relatively few items. This procedure, known as acceptance sampling, has the aim of providing some control against the acceptance of lots with an excessive proportion of defective items. It also serves to assist the producer in the control of his product.

When acceptance or rejection is to depend upon the incomplete information supplied by a sample, there is inevitably a risk of accepting “bad” material or rejecting “good” material. It is important, therefore, that the-items to be tested be selected in such a way that they reveal the maximum amount of information about the overall quality of the complete lot of material submitted for acceptance and that the acceptance criteria be chosen so as to fix the risks of wrong decision at a suitably small value.

The purpose of this note is to present some of the basic ideas involved in acceptance sampling with special reference to electroplated material.

THE INSPECTION OF ELECTROPLATED COATINGS
The inspection of electroplated coatings is, in practice, generally limited to examination of thickness, salt spray resistance, appearance, and adhesion. In some cases, the thickness of a coating can be determined without destroying the article or coating, e. g., with a magnetic thickness gage. For coatings such as chromium over nickel, the coating is usually destroyed by the thickness test. Almost always, however, a thickness determination yields a measured value of the actual thickness besides indicating whether or not the coating passes thickness requirements.

Salt-spray tests are destructive. While there are rating systems for indicating how badly an article’ failed the test or how well it stood up, in general one observes only whether or not the item passed this particular test.

Appearance may include observations of such things as brightness, coverage, pits, and stains. Usually this is not done on a quantitative basis, but the piece is either satisfactory or unsatisfactory. Thus inspection of appearance is nondestructive; and does not yield a quantitative measure of quality.

Adhesion tests are all destructive and those commonly used are not quantitative.

The results from tests for these four qualities of electroplated coatings rate the coatings as either passing or failing or as being defective or non defective. The following discussion is concerned primarily with acceptance sampling plans for this type of test, since they are the simpler and more commonly used procedures.

An acceptance sampling plan is a specification of the number of items to be drawn as a sample, the manner of selecting these items, and an acceptance criterion. For example, a specification calls for 0.001 inch of nickel on steel. The acceptance sampling plan specifies that the coating thickness on 10 specimens be measured and that if more than one specimen have less than 0.001 inch of nickel, the whole lot or shipment shall be rejected.

THE LOT FROM WHICH THE SAMPLE IS TO BE DRAWN
The efficiency of any acceptance sampling scheme depends on the make-up of the lot from which the sample is to be drawn. For example, consider a shipment of 1000 items that arrives in two parts, half arriving one day, the remainder the next day. If it was known that the first day’s delivery of 500 was free from defects and the second day’s delivery includes 200 defective items, this knowledge could be used to advantage in designing an acceptance sampling plan. A natural subdivision of a shipment into lots (based on date of delivery, date of production, plating tank used, etc.), when known, would suggest the drawing of a sample from each such lot of the shipment. If in the example above a sample of 20 were drawn from each day’s delivery, and a sampling plan were used calling for rejection if one or more unsatisfactory items were found in the sample, then the first day’s delivery would be accepted and the second day’s delivery (containing 40 per cent defective items) would almost certainly be rejected. In such an instance the accepted material would be entirely free of nonconforming items.

However, if the two deliveries were composited into one aggregate of 1000 items and a single sample of 20 were drawn from this aggregate (20 per cent of which are defective) the chance of accepting the entire delivery (including the 200 defective items) is calculated to be about 1 in 90 for the same rule of rejection as above. If accepted, the quality of material accepted would be the same as that submitted, 20 per cent defective. An acceptance sampling plan can be used to reduce the probability of accepting undesirable lots, but if uniformly bad material is submitted for acceptance the quality of accepted material will not be different from that submitted.

A fundamental property of efficient acceptance sampling schemes is the requirement that a sample be drawn from a homogeneous aggregate of material, called a lot. A homogeneous aggregate is one in which the per cent defective (or other measure of quality) is essentially the same from one part of the lot to the next. A careful definition of what shall constitute a lot involves requirements such as “manufactured under essentially the same conditions,” “made from the same batch of raw material.”

The method of packaging or shipping may make it impossible or impractical to isolate homogeneous lots for the purpose of sampling. In such a case some attention must be given to the method of selection of individual items so that the sample represents the entire shipment fairly.

SELECTION OF SAMPLES FROM A LOT
The manner in which samples are selected from a lot is important. The classic example of this is the box of strawberries at the grocery store. The purchaser normally inspects the strawberries which he can see, those on top. This sampling procedure has, at times, been known to be misleading. While the example of the box of strawberries may involve deliberate rearrangement, such situations can occur accidentally or in the normal course of events. For this reason one should not inspect, for example, just the first ten specimens off the plating line or take the five most accessible cases of chromium-plated stock in the warehouse.

In selecting items from a lot to constitute the sample, equal opportunity should be given for each item to appear in the sample regardless of whether the item is defective or non defective, i. e., the items should be a random sample of the lot. Ideally, some scheme of numbering of the items and the use of a table of random numbers’ for selecting the ‘individual items is the best approach for insuring the randomness of the selection. If the material comes in packages or groups, a random sample can be obtained by applying selection by random numbers at two stages—first selecting the packages, then the items from the package. If the method of generating the lot is known, some form of systematic sampling (involving, perhaps, a random starting point) is very likely to prove more efficient or economical.

THE OPERATING CHARACTERISTICS OF ACCEPTANCE SAMPLING PLANS
Despite the care that may be taken-to assure a proper sample, the test results from the sample represent only partial information on the nature of the lot from which the sample was drawn. There is always the possibility that, by chance, the sample gives an unduly optimistic picture of the lot and a “bad” lot may be wrongly accepted, or conversely a “good” lot wrongfully rejected. It is possible to calculate these risks and to select a plan which will reduce them to suitably small values.

The quality of a lot is measured in terms of the percentage (or fraction) of items in the lot which are defective. If it is impossible to screen out the defective items that remain in an accepted lot, the purchaser will unavoidably find some defective items showing up along with the good items. What fraction of defective items can be-tolerated’ varies with the product and the use to which it is put—e. g., a larger fraction defective can be tolerated in nails for a building project than in rivets for an aircraft. In most cases the purchaser can decide, on economic or engineering grounds, the fraction defective which, if present, would be upsetting to his operations. (If it is possible to screen out defective items, then in determining what fraction defective he can afford to receive and accept, this purchaser must take into account the cost of screening and its efficiency in reducing the fraction which is defective to such a level as will not be upsetting to his other operations.) It is important that the chance of accepting a lot having an excessive number of defective items be kept small. At the same time the purchaser does not wish to reject too many lots of material which have an acceptably low fraction defective.

The chance of accepting a lot obviously depends on the fraction of defective items turned out by the producer. The larger this fraction, the greater is the chance that defective items will turn up in the sample and hence the less the chance of acceptance under a specific sampling plan. A graph showing the probability of acceptance of a lot as a function of the fraction defective in the lot is called an operating characteristic curvet. The dashed curve of Fig. 1 shows the operating characteristic curve for a sampling plan calling for a sample of 10 items and acceptance of the lot from which the sample was drawn if one or zero defective items are found in the sample.

For a given sample size the effect of increasing the allowable number of defectives is to increase the chance of acceptance of lots of a given fraction defective. Fig. 1 shows the operating characteristic curves for three sampling plans, where the sample size is fixed at 10 units and the allowable number of defectives (acceptance number) is zero, one, and two, respectively.

If the allowable number of defectives is held constant, the effect of varying the sample size is as shown in Fig. 2. As the sample size is increased, the probability of acceptance of lots of given fraction defective is decreased. These are the operating characteristic curves for the sampling plans given in the Federal Specification for cadmium and zinc plating QQ-F16 and QQ-Z-325.

It should be noted that the operating characteristic curve gives the long run chance of acceptance of lots from material with a given fraction defective. Material with an excessively high fraction defective will almost always be rejected, but there is a calculable chance of acceptance though it may be quite small.
It is important to remember that if a producer supplies material’ which always has about the same percentage of defective items, the quality of the material accepted (and also the quality of the material rejected) will be the same as the quality of the material submitted. In such a case, an acceptance sampling procedure serves no other useful purpose than to assist the producer, by knowledge of the situation, in effecting better control of the quality of his material.

Fig. 1. Operating characteristic curves for acceptance sampling plans calling for a sample of 10 items with acceptance number c = 0, 1, 2

DESIGN OF ACCEPTANCE SAMPLING PLAN
In order that a sampling plan be mutually acceptable to the supplier and purchaser, the risks of rejecting “good” material and of accepting “bad” material must be made suitably small. For example, a purchaser may feel that material showing 25 per cent or more defective items will seriously affect his operations; at the same time the supplier may not be able to produce economically material with less- than 2 per cent of defective items. If both agree that the chance of wrongly rejecting “good’’ material or wrongly accepting “bad” material be not greater than 1 in 20, then a sapling plan calling for a sample of 17 items, allowing 1 defective item in the sample, is required. Methods of computing sampling plans given such requirements are described by Cameron2.

It is important to note that although the mathematical solution of the question of sample size is exact, the designation of limits for “good” and “bad” material, and the selection of the risk levels are generally far from exact.

In situations where practical considerations dictate the size of sample, one has only the choice of the acceptance criterion at his disposal. A choice of the plan to be used can be made by comparing the operating characteristics of the various possible plans.

Fig. 2. Operating characteristic curves for acceptance sampling plans with acceptance number zero-and sample size n

Small lots present difficulties since usually only a very few items can be tested. This means that it is not possible to design a sampling plan that’ will have much power of discrimination between good and bad lots (for example, see Deming3, p. 286).

PRACTICAL CONSIDERATIONS IN THE SELECTION OF A SAMPLING PLAN
Perhaps the most important practical consideration in the selection of a sampling plan is cost. The cost of drawing items from the lot, of testing the items, and in some cases the cost of the items themselves, must be balanced against the risks of wrong decision regarding the lot. This problem is largely economic in nature. For example, the failure of an inexpensive resistor in some ordnance device may result in the loss of thousands of dollars. It may be economic, therefore, to spend considerably more on acceptance testing than the cost of the entire lot of resistors. Cost considerations may dictate an upper limit to the number of items that is to be drawn as a sample for test. The problem of the appropriate sampling criterion can be decided upon after a study of the operating characteristic curves for various acceptance criteria for that given sample size.

An important consideration in selecting a sampling plan is the level of quality (fraction defective) that is turned-out by the producers whose products have a relatively low incidence of nonconforming items. Little material would ever be procured if the sampling plan was so stringent as to reject 99 times out of 100 the material of the producer with the lowest incidence of nonconforming items. It is desirable that the sampling plan have a high probability of acceptance associated with the fraction defective representative of the best industrial practices.

The above considerations relate to the selection of sampling plans from considerations based on their operating characteristics. Such studies cannot be avoided, but there are other factors that enter into the picture. For example:

(1) A larger number of items can be required for simple and inexpensive tests. Thus for electroplated materials the appearance and nondestructive thickness tests may be’ made on a larger number of items than can adhesion or salt-spray tests. Since the four tests are not completely independent (poor workmanship or poor plating practices will often show up in more than one type of test), one is exercising control, at least partially, against deficiencies on all characteristics by a more stringent acceptance sampling plan for the nondestructive tests.
(2) A knowledge of the production record of the supplier can be used to an advantage in reducing the number of tests required. For example, if a long history of satisfactory products has been recorded for a particular manufacturer, relatively small sample sizes are quite safe. Procedures exist for dropping from “normal” to “reduced” inspection when a sequence of adequately good lots has been the experience. Also it may be that the quality produced is known to fall into two classes, either good or extremely bad. For such a case, adequate protection can be obtained from rather small sample sizes.
(3) If costs or other considerations do not permit one to test a sufficient number of items to insure the desired quality, one can raise the specification requirements. This, however, may increase the cost of the product.

GENERAL SAMPLING PLANS FOR ELECTROPLATED COATINGS
In some general plating specifications, sampling tables have been given. The sampling plans in these tables for most instances represent a minimum inspection plan, and seldom an optimum inspection plan.

It would be very desirable to have a universal sampling table which could be recommended for electroplated coatings. Unfortunately, no such sampling table would be satisfactory in all situations. This is evident when one considers that the optimum plan is dependent on such things as the costs of testing, the costs of obtaining samples for test, the costs of the articles destroyed by testing, how critical the requirements are, the lot size, the costs of a bad decision as to the acceptability of a lot, the time required for testing, and urgency for delivery of the plated articles. Each factor differs from situation to situation, and hence an optimum plan for each particular situation must be developed in the light of considerations such as those mentioned in this paper.

REFERENCES
1. Statistical Research Group, Columbia University, Sampling Inspection, McGraw Hill Book Company, Inc., New York, 1948.
2. J M. Cameron, “Tables for Constructing and for Computing the Operating Characteristics of Single-Sampling Plans”, Industrial Quality Control, Vol. IX, No. 1, Part I (July, 1952), pp.
3. W. E. Deming, Some Theory of Sampling, John Wiley and Sons, Inc., New York, 1950.

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