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It is sometimes necessary to monitor the number of defects in a unit of product rather than the fraction defective. Suppose that in the production of cloth it is necessary to control the number of defects per yard or that in assembling an aircraft wing the number of missing rivets must be controlled. In these situations we may use the control chart for defects per unit, or the U chart. Many defects-per-unit situations can be modeled by the Poisson distribution. If each sample consists of n units and there are C total defects in the sample, U C n

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is the average number of defects per unit. A U chart may be constructed for such data. If the number of defects in a unit is a Poisson random variable with parameter , the mean and variance of this distribution are both . Each point on the chart is U, the average number of defects per unit from a sample of n units. Therefore, the mean of U is and the variance of U is n. UCL LCL 3 3 Bn Bn



Similarly, we call calF a right regular function in a domain D if for every closed hypersurface 8a in D,

If there are m preliminary samples, and the number of defects per unit in these samples are U1, U2, . . . , Um, the estimator of the average number of defects per unit is U 1 m m ia Ui 1 (16-26)


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These control limits are based on the normal approximation to the Poisson distribution. When is small, the normal approximation may not always be adequate. In such cases, we may use control limits obtained directly from a table of Poisson probabilities. If u 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 16-5 Printed circuit boards are assembled by a combination of manual assembly and automation. A ow solder machine is used to make the mechanical and electrical connections of the leaded components to the board. The boards are run through the ow solder process almost continuously, and every hour ve boards are selected and inspected for process-control purposes. The number of defects in each sample of ve boards is noted. Results for 20 samples are shown in Table 16-5. The center line for the U chart is 1 20 u 20 ia i 1 u Bn 32 20

1.6 1.6 B 5


1.6 1.6

In complex analysis, (C.31) leads to the Cauchy-Riemann equations. Here, left regularity leads to O:F = 0, which can be written in terms of components as

1.6 B 5

3.3 0

810 8xo \710

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The control chart is plotted in Fig. 16-17. Because LCL is negative, it is set to 0. From the control chart in Fig. 16-17, we see that the process is in control. However, eight defects per group of ve circuit boards are too many (about 8 5 1.6 defects/board), and the process needs improvement. An investigation needs to be made of the speci c types of defects found on the printed circuit boards. This will usually suggest potential avenues for process improvement.

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