ABSTRACT
The distinction between discrete and continuous variables involves good grammar as well as good statistics. We state how many we have of a discrete variable and how much when the variable is continuous. We may be interested in how many cans of soup were underweight by as much as a milligram; or how many rivets were off center by as much as 0.5 mm. These statements imply that continuous (measurement) variables can be subjected to an attributes (go no-go) type test simply by counting the number of items in a sample beyond some limit. Thus, attributes sampling plans could be applied in these two cases. Alternatively, if the shape of the underlying distribution of individual measurements were known,
acceptance sampling could be performed directly on the measurements themselves. Such procedures form the basis for variables sampling plans for proportion nonconforming and, when applicable, provide a considerable savings in sample size. The basic idea of variables sampling for proportion nonconforming is to show that the sample
results are sufficiently far within the specification limit(s) to assure the acceptability of the lot with reasonable probability. Variables plans involve comparing a statistic, such as the mean X, with an acceptance limit A in
much the same way that the number nonconforming, d, is compared to an acceptance number, c, in attributes plans. A comparison of the procedures involved in variables and attributes sampling plans is shown in Figure 10.1. Some of the advantages of variables sampling are as follows:
1. Same protection with smaller sample size than attributes
2. Feedback of data on process which produced the units
3. More data available in waiver situations
4. Extent of conformity of each unit given weight in application of the plan
5. Increased likelihood of errors in measurement being detected
Some of the disadvantages of variables sampling are as follows:
1. Dependence of results on correctness of assumption of shape of underlying distribution of measurements
2. Applicable to one (only) characteristic at a time
3. Higher inspection cost per unit
4. Higher clerical cost per unit
5. Possibility of no nonconforming unit found to show to producer after rejection
The principal advantage of variables plans over attributes is reduction in sample size. For example, in comparing average sample sizes for plans matched to the single-sampling attributes plan n¼ 50, c¼ 2 we have, for a single specification limit:
Single attributes 50 Double attributes 43 Multiple attributes 35 Sequential attributes 33.5 Variables (s unknown) 27 Variables (s known) 12
Specification limits can be of two types. A single specification limit implies only one boundary value for acceptability, either upper U or lower L. Thus, a measurement does not conform to the specification limit if
X > U
for an upper specification limit, or if
X < L
for a lower specification limit. Double specification limits place both upper and lower boundary values on the acceptability of a measurement. That is, the measurement X is acceptable if and only if
L X U
FIGURE 10.1: Comparison of attributes and variables sampling. (a) Attributes, single sampling; (b) Variables, single sampling (upper specification limit). (Reprinted from Schilling, E.G., Qual. Progr., 7(5), 16, 1974b. With permission.)
Probably the most important consideration in applying variables sampling plans is the requirement that the shape of the underlying distribution of measurements to which the plan is to be applied must be known and stable. This means that probability plots or statistical tests on past data must show that the distribution of measurements involved actually is that assumed by the plan. Control chart evidence also is desirable to indicate its stability. For a known distribution, it is the underlying distribution of measurements which relates the proportion of units outside the specification limit to a fixed position of the population mean of the measurements. Variables plans for process parameter may then be used to confirm or deny that the population mean is in the proper position. In a crude way, this is how variables sampling works. In fact, some plans are devised in just this way. It must be emphasized that the underlying distribution must be known to be that assumed by the plan for variables sampling to be properly applied. The basic theoretical nature of the variables acceptance sampling plans is illustrated in Figure
10.2, which involves an upper specification limit and assumes the underlying distribution of individual measurements to be normal. If the procedure of Figure 10.1 is applied, the mean X of a sample of n measurements is compared to an acceptance limit A and the lot accepted if X is not greater than A. Figure 10.2 shows that if the distribution of individual measurements is as shown, with s known, a proportion p of the product above the upper specification limit U implies the mean of the distribution must be fixed at the position indicated by m. Sample means of size n then would be distributed about m as shown and so the probability of obtaining an X not greater than A is as indicated by the shaded area in the diagram. Published plans for known standard deviation often are given in terms of sample size and k, the distance in units of the standard deviation, between the (upper) specification limit U and the acceptance limit A. From Figure 10.2 we see
k ¼ U A s
¼ zU zA
FIGURE 10.2: Distributions in variables sampling. (Reprinted from Schilling, E.G., Qual. Progr., 7(5), 16, 1974b. With permission.)
for the distribution of individual measurements positioned as shown where the z values are taken from the standard normal table. The situation is analogous, but reversed, for a lower specification limit. Using Figure 10.2 and normal probability theory, the probability of acceptance Pa can be
calculated for various possible values of p, the proportion nonconforming. Figure 10.3 shows the operating characteristic (OC) curve of the variables plan n¼ 7, k¼ 1.44 for known standard deviation compared to that of the attributes plan n¼ 20, c¼ 1. OC curves of variables plans are generally considered to be Type B. It can be seen that the two OC curves are well matched, that is they give about the same
protection. The variables plan, however, uses only about a third as large a sample size as the attributes plan. Thus, the variables plan appears superior. It must be remembered, however, that the superiority of the variables plan rests on assumption of the normality of the underlying distribution of the measurements. If this assumption cannot be justified, the variables plans may give unreliable results and recourse must be either to an attributes or to a mixed variablesattributes plan. The danger involved in using a variables plan which assumes normality when, in fact, the
underlying distribution of individual measurements is actually nonnormal is illustrated in Figure 10.4. This shows the proportion of the product beyond z standard deviation units from the mean to be heavily dependent on the shape of the distribution. A variety of distributions is represented by various shape parameters for the family of Weibull distributions. Note that the tail area beyond three standard deviations is over 2% for a Weibull distribution with shape parameter 0.5, while it is 0.13% for a normal distribution.
