Control Charts for Attributes

17.5 Control Charts for Attributes

As we indicated earlier in this chapter, many industrial applications of quality control require that the quality characteristic indicate no more than that the item “conforms.” In other words, there is no continuous measurement that is crucial to the performance of the item. An obvious illustration of this type of sampling, called sampling for attributes, is the performance of a light bulb, which either performs satisfactorily or does not. The item is either defective or not de- fective. Manufactured metal pieces may contain deformities. Containers from a production line may leak. In both of these cases, a defective item negates usage by the customer. The standard control chart for this situation is the p-chart, or chart for fraction defective. As we might expect, the probability distribution involved is the binomial distribution. The reader is referred to Chapter 5 for background on the binomial distribution.

698 Chapter 17 Statistical Quality Control

Table 17.2: Volume of Containers for 25 Samples in a Preliminary Sample (in cubic centimeters)

The p-Chart for Fraction Defective

Any manufactured item may have several characteristics that are important and should be examined by an inspector. However, the entire development here focuses on a single characteristic. Suppose that for all items the probability of a defective item is p, and that all items are being produced independently. Then, in a random sample of n items produced, allowing X to be the number of defective items, we have

As one might suspect, the mean and variance of the binomial random variable will play an important role in the development of the control chart. The reader should recall that

E(X) = np

and

Var(X) = np(1 − p).

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0 10 20 30 0 10 20 30 Sample Number

Sample Number

Figure 17.7: The ¯ X-chart with control limits es- Figure 17.8: The S-chart with control limits estab- tablished by the data of Example 17.2.

lished by the data of Example 17.2. An unbiased estimator of p is the fraction defective or the proportion de-

fective, ˆ p, where number of defectives in the sample of size n

p= ˆ

As in the case of the variables control charts, the distributional properties of p are important in the development of the control chart. We know that

p(1 − p)

E(ˆ p) = p,

Var(ˆ p) =

Here we apply the same 3σ principles that we use for the variables charts. Let us assume initially that p is known. The structure, then, of the control charts involves the use of 3σ limits with

" p(1 − p)

Thus, the limits are

n with the process considered in control when the ˆ p-values from the sample lie inside

the control limits. Generally, of course, the value of p is not known and must be estimated from

a base set of samples very much like the case of μ and σ in the variables charts. Assume that there are m preliminary samples of size n. For a given sample, each of the n observations is reported as either “defective” or “not defective.” The obvious unbiased estimator for p to use in the control chart is

p= ¯

m i=1

700 Chapter 17 Statistical Quality Control where ˆ p i is the proportion defective in the ith sample. As a result, the control

limits are LCL = ¯ p−3

p(1 − ¯ ¯ p)

centerline = ¯ p,

UCL = ¯ p+3

" p(1 − ¯ ¯ p)

Example 17.3: Consider the data shown in Table 17.3 on the number of defective electronic com-

ponents in samples of size 50. Twenty samples were taken in order to establish preliminary control chart values. The control charts determined by this preliminary period will have centerline ¯ p = 0.088 and control limits

= 0.2082. Table 17.3: Data for Example 17.3 to Establish Control Limits for p-Charts,

Samples of Size 50

Number of

Fraction Defective

Sample Defective Components

Obviously, with a computed value that is negative, the LCL will be set to zero. It is apparent from the values of the control limits that the process is in control during this preliminary period.

Choice of Sample Size for the p-Chart

The choice of sample size for the p-chart for attributes involves the same general types of considerations as that of the chart for variables. A sample size is required

17.5 Control Charts for Attributes 701 that is sufficiently large to have a high probability of detection of an out-of-control

condition when, in fact, a specified change in p has occurred. There is no best method for choice of sample size. However, one reasonable approach, suggested by Duncan (1986; see the Bibliography), is to choose n so that there is probability 0.5 that we detect a shift in p of a particular amount. The resulting solution for n is quite simple. Suppose that the normal approximation to the binomial distribution

applies. We wish, under the condition that p has shifted to, say, p 1 >p 0 , that

UCL − p 1

P (ˆ p ≥ UCL) = P Z≥

p 1 (1 − p 1 )/n

Since P (Z > 0) = 0.5, we set

UCL − p 1 = 0. p 1 (1 − p 1 )/n

Substituting

" p+3 p(1 − p) = UCL, n

we have

We can now solve for n, the size of each sample:

n=

p(1 − p),

where, of course, Δ is the “shift” in the value of p, and p is the probability of a defective on which the control limits are based. However, if the control charts are based on kσ limits, then

n=

p(1 − p).

Example 17.4: Suppose that an attribute quality control chart is being designed with a value of p = 0.01 for the in-control probability of a defective. What is the sample size per subgroup producing a probability of 0.5 that a process shift to p = p 1 = 0.05 will

be detected? The resulting p-chart will involve 3σ limits. Solution : Here we have Δ = 0.04. The appropriate sample size is

n=

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Control Charts for Defects (Use of the Poisson Model)

In the preceding development, we assumed that the item under consideration is one that is either defective (i.e., nonfunctional) or not defective. In the latter case, it is functional and thus acceptable to the consumer. In many situations, this “defective or not” approach is too simplistic. Units may contain defects or nonconformities but still function quite well for the consumer. Indeed, in this case, it may be important to exert control on the number of defects or number of nonconformities. This type of quality control effort finds application when the units are either not simplistic or large. For example, the number of defects may be quite useful as the object of control when the single item or unit is, say, a personal computer. Another example is a unit defined by 50 feet of manufactured pipeline, where the number of defective welds is the object of quality control; the number of defects in 50 feet of manufactured carpeting; or the number of “bubbles” in a large manufactured sheet of glass.

It is clear from what we describe here that the binomial distribution is not appropriate. The total number of nonconformities in a unit or the average number per unit can be used as the measure for the control chart. Often it is assumed that the number of nonconformities in a sample of items follows the Poisson distribution. This type of chart is often called a C-chart.

Suppose that the number of defects X in one unit of product follows the Poisson distribution with parameter λ. (Here t = 1 for the Poisson model.) Recall that for the Poisson distribution,

Here, the random variable X is the number of nonconformities. In Chapter 5, we learned that the mean and variance of the Poisson random variable are both λ. Thus, if the quality control chart were to be structured according to the usual 3σ limits, we could have, for λ known,

LCL = λ − 3 λ. As usual, λ often must come from an estimator from the data. An unbiased

UCL = λ + 3 λ,

centerline = λ,

estimate of λ is the average number of nonconformities per sample. Denote this estimate by ˆ λ. Thus, the control chart has the limits

Example 17.5: Table 17.4 represents the number of defects in 20 successive samples of sheet metal rolls each 100 feet long. A control chart is to be developed from these preliminary data for the purpose of controlling the number of defects in such samples. The estimate of the Poisson parameter λ is given by ˆ λ = 5.95. As a result, the control limits suggested by these preliminary data are

UCL = ˆ λ+3 ˆ λ = 13.2678 and LCL = ˆ λ−3 λ = −1.3678, ˆ with LCL being set to zero.

17.5 Control Charts for Attributes 703

Table 17.4: Data for Example 17.5; Control Involves Number of Defects in Sheet Metal Rolls Sample Number Number of Defects Sample Number Number of Defects

Figure 17.9 shows a plot of the preliminary data with the control limits revealed. Table 17.5 shows additional data taken from the production process. For each sample, the unit on which the chart was based, namely 100 feet of the metal, was inspected. The information on 20 samples is included. Figure 17.10 shows a plot of the additional production data. It is clear that the process is in control, at least through the period for which the data were taken.

Table 17.5: Additional Data from the Production Process of Example 17.5 Sample Number Number of Defects Sample Number Number of Defects

10 2 20 6 In Example 17.5, we have made very clear what the sampling or inspection unit

is, namely, 100 feet of metal. In many cases where the item is a specific one (e.g.,

a personal computer or a specific type of electronic device), the inspection unit may be a set of items. For example, the analyst may decide to use 10 computers in each subgroup and observe a count of the total number of defects found. Thus, the preliminary sample for construction of the control chart would involve several samples, each containing 10 computers. The choice of the sample size may depend on many factors. Often, we may want a sample size that will ensure an LCL that is positive.

The analyst may wish to use the average number of defects per sampling unit as the basic measure in the control chart. For example, for the case of the personal

704 Chapter 17 Statistical Quality Control

Number of Defects

Number of Defects

Sample Figure 17.9: Preliminary data plotted on the con- Figure 17.10: Additional production data for Ex-

Sample

trol chart for Example 17.5.

ample 17.5.

computer, let the random variable total number of defects

total number of defects U= n

be measured for each sample of, say, n = 10. We can use the method of moment- generating functions to show that U is a Poisson random variable (see Review Exercise 17.1) if we assume that the number of defects per sampling unit is Poisson with parameter λ. Thus, the control chart for this situation is characterized by the following:

UCL = ¯ U+3

centerline = ¯ U,

LCL = ¯ U−3 .

n Here, of course, ¯ U is the average of the U -values in the preliminary or base data

set. The term ¯ U /n is derived from the result that

E(U ) = λ,

Var(U ) = ,

and thus ¯ U is an unbiased estimate of E(U ) = λ and ¯ U /n is an unbiased estimate of Var(U ) = λ/n. This type of control chart is often called a U-chart.

In this section, we based our entire development of control charts on the Poisson probability model. This model has been used in combination with the 3σ concept. As we implied earlier in this chapter, the notion of 3σ limits has its roots in the normal approximation, although many users feel that the concept works well as a pragmatic tool even if normality is not even approximately correct. The difficulty, of course, is that in the absence of normality, we cannot control the probability of incorrect specification of an out-of-control state. In the case of the Poisson model, when λ is small the distribution is quite asymmetric, a condition that may produce undesirable results if we hold to the 3σ approach.

17.6 Cusum Control Charts 705