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Often it is desirable to classify a product as either defective or nondefective on the basis of comparison with a standard. This classi cation is usually done to achieve economy and simplicity in the inspection operation. For example, the diameter of a ball bearing may be checked by determining whether it will pass through a gauge consisting of circular holes cut in a template. This kind of measurement would be much simpler than directly measuring the diameter with a device such as a micrometer. Control charts for attributes are used in these situations. Attribute control charts often require a considerably larger sample size than do their variable measurements counterparts. In this section, we will discuss the fraction-defective control chart, or P chart. Sometimes the P chart is called the control chart for fraction nonconforming. Suppose D is the number of defective units in a random sample of size n. We assume that D is a binomial random variable with unknown parameter p. The fraction defective

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2 P

MD40...47(IV, M ) # h ( M ) ,

Therefore, a P chart for fraction defective could be constructed using p as the center line and control limits at UCL p 3 B p11 n p2 LCL p 3 B p11 n p2 (16-22)

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However, the true process fraction defective is almost always unknown and must be estimated using the data from preliminary samples. Suppose that m preliminary samples each of size n are available, and let Di be the number of defectives in the ith sample. The Pi Di n is the sample fraction defective in the ith sample. The average fraction defective is P 1 m m ia Pi 1 1 m mn ia Di 1 (16-23)

where h is the MD4 hash function, since there is a final transformation in MD4 that is not accounted for in M D ~ o . . . ~ ~XI)V This final transformation con( ., sists of adding the initial values to the output of the function and rearranging the order of the output words-see Table 5.1. The MD4 padding is also not considered in this attack. But any collision found by this attack will result in a collision for the full MD4 hash without any modification to M or M . Suppose ( Q j , Qj-1, Qj-2, Qj-3) = MD40...j (IV, M ) and Then define

Now P may be used as an estimator of p in the center line and control limit calculations. P Chart B B

The center line and upper and lower control limits for the P chart are UCL p 3 p11 n p2 CL p LCL p 3 p11 n p2 (16-24)

Our set G. along with concatenation operator '.*" and these relations, is also a group. Let us denote this group as

Table 16-4 Number of Defectives in Samples of 100 Ceramic Substrates Sample 1 2 3 4 5 6 7 8 9 10 No. of Defectives 44 48 32 50 29 31 46 52 44 48 Sample 11 12 13 14 15 16 17 18 19 20 No. of Defectives 36 52 35 41 42 30 46 38 26 30

These control limits are based on the normal approximation to the binomial distribution. When p is small, the normal approximation may not always be adequate. In such cases, we may use control limits obtained directly from a table of binomial probabilities. If p is small, the lower control limit obtained from the normal approximation may be a negative number. If this should occur, it is customary to consider zero as the lower control limit. EXAMPLE 16-4 Suppose we wish to construct a fraction-defective control chart for a ceramic substrate production line. We have 20 preliminary samples, each of size 100; the number of defectives in each sample is shown in Table 16-4. Assume that the samples are numbered in the sequence of production. Note that p (800 2000) 0.40; therefore, the trial parameters for the control chart are UCL LCL 0.40 0.40 3 3 10.40210.602 100 B 10.402 10.602 100 B 0.55 0.25 CL 0.40

The control chart is shown in Fig. 16-16. All samples are in control. If they were not, we would search for assignable causes of variation and revise the limits accordingly. This chart can be used for controlling future production. Although this process exhibits statistical control, its defective rate ( p 0.40) is very poor. We should take appropriate steps to investigate the process to determine why such a large number of defective units is being produced. Defective units should be analyzed to determine the speci c types of defects present. Once the defect types are known, process changes should be investigated to determine their impact on defect levels. Designed experiments may be useful in this regard.

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