1. Copyright © Cengage Learning. All rights reserved. 16 Quality Control Methods

2. Copyright © Cengage Learning. All rights reserved. 16.4 Control Charts for Attributes

3. 3 The term attribute data is used in the quality control literature to describe two situations: 1. Each item produced is either defective or nondefective (conforms to specifications or does not). 2. A single item may have one or more defects, and the number of defects is determined. In the former case, a control chart is based on the binomial distribution; in the latter case, the Poisson distribution is the basis for a chart.

4. 4 The p Chart for Fraction Defective

5. 5 Suppose that when a process is in control, the probability that any particular item is defective is p (equivalently, p is the long-run proportion of defective items for an in-control process) and that different items are independent of one another with respect to their conditions. Consider a sample of n items obtained at a particular time, and let X be the number of defectives and = X/n. Because X has a binomial distribution, E(X) = np and V(X) = np(1 – p), so

6. 6 The p Chart for Fraction Defective Also, if np  10 and n(1 – p)  10. has approximately a normal distribution. In the case of known p (or a chart based on target value), the control limits are If each sample consists of n items, the number of defective items in the ith sample is x i, and = x i /n, then are plotted on the control chart.

7. 7 The p Chart for Fraction Defective Usually the value of p must be estimated from the data. Suppose that k samples from what is believed to be an in-control process are available, and let The estimate p is then used in place of p in the aforementioned control limits.

8. 8 The p Chart for Fraction Defective The p chart for the fraction of defective items has its center line at height p and control limits If LCL is negative, it is replaced by 0.

9. 9 Example 6 A sample of 100 cups from a particular dinnerware pattern was selected on each of 25 successive days, and each was examined for defects. The resulting numbers of unacceptable cups and corresponding sample proportions are as follows:

10. 10 Example 6 Assuming that the process was in control during this period, let’s establish control limits and construct a p chart. Since, = 1.52, p = 1.52/25 =.0608 and cont’d

11. 11 Example 6 The LCL is therefore set at 0. The chart pictured in Figure 16.5 shows that all points are within the control limits. This is consistent with an in-control process. Figure 16.5 Control chart for fraction-defective data of Example 6 cont’d

12. 12 The c Chart for Number of Defectives

13. 13 The c Chart for Number of Defectives We now consider situations in which the observation at each time point is the number of defects in a unit of some sort. The unit may consist of a single item (e.g., one automobile) or a group of items (e.g., blemishes on a set of four tires). In the second case, the group size is assumed to be the same at each time point. The control chart for number of defectives is based on the Poisson probability distribution.

14. 14 The c Chart for Number of Defectives Recall that if Y is a Poisson random variable with parameter , then E(Y) =  V(Y) =   Y = Also, Y has approximately a normal distribution when  is large (   10 will suffice for most purposes). Furthermore, if Y 1, Y 1,..., Y n, are independent Poisson variables with parameters  1,  2,...,  n it can be shown that Y 1 +... + Y n has a Poisson distribution with parameter  1 +... +  n.

15. 15 The c Chart for Number of Defectives In particular, if  1 =... =  n =  (the distribution of the number of defects per item is the same for each item), then the Poisson parameter is n . Let  denote the Poisson parameter for the number of defects in a unit (it is the expected number of defects per unit). In the case of known  (or a chart based on a target value), LCL =  – 3 UCL =  + 3

16. 16 The c Chart for Number of Defectives With x i denoting the total number of defects in the ith unit (i = 1, 2, 3,...), then points at heights x 1, x 2, x 3,... are plotted on the chart. Usually the value of  must be estimated from the data. Since E(X i ) = , it is natural to use the estimate  = x (based on x 1, x 2,...,, x k ). The c chart for the number of defectives in a unit has center line at x and LCL = x – 3 UCL = x + 3 If LCL is negative, it is replaced by 0.

17. 17 Example 7 A company manufactures metal panels that are baked after first being coated with a slurry of powdered ceramic. Flaws sometimes appear in the finish of these panels, and the company wishes to establish a control chart for the number of flaws. The number of flaws in each of the 24 panels sampled at regular time intervals are as follows: 7 10 9 12 13 6 13 7 5 11 8 10 13 9 21 10 6 8 3 12 7 11 14 10

18. 18 Example 7 with  x i = 235 and = x = 235/24 = 9.79. The control limits are LCL = 9.79 – 3 =.40 UCL = 9.30 + 3 = 19.18 cont’d

19. 19 Example 7 The control chart is in Figure 16.6. The point corresponding to the fifteenth panel lies above the UCL. cont’d Figure 16.6 Control chart for number of flaws data of Example 7

20. 20 Example 7 Upon investigation, the slurry used on that panel was discovered to be of unusually low viscosity (an assignable cause). Eliminating that observation gives x = 214/23 = 9.30 and new control limits LCL = 9.30 – 3 =.15 UCL = 9.30 + 3 = 18.45 The remaining 23 observations all lie between these limits, indicating an in-control process. cont’d

21. 21 Control Charts Based on Transformed Data

22. 22 Control Charts Based on Transformed Data The use of 3-sigma control limits is presumed to result in P (statistic UCL) .0013 when the process is in control. However, when p is small, the normal approximation to the distribution of = X/n will often not be very accurate in the extreme tails.

23. 23 Control Charts Based on Transformed Data Table 16.3 gives evidence of this behavior for selected values of p and n (the value of p is used to calculate the control limits). Table 16.3 In-Control Probabilities for a p Chart

24. 24 Control Charts Based on Transformed Data Figure 16.6 Control chart for number of flaws data of Example 7

25. 25 Control Charts Based on Transformed Data In many cases, the probability that a single point falls outside the control limits is very different from the nominal probability of.0026. This problem can be remedied by applying a transformation to the data. Let h(X) denote a function applied to transform the binomial variable X. Then h(  ) should be chosen so that h(X) has approximately a normal distribution and this approximation is accurate in the tails.

26. 26 Control Charts Based on Transformed Data A recommended transformation is based on the arcsin (i.e., sin –1 ) function: Y = h(X ) = sin –1 Then Y is approximately normal with mean value sin –1 and variance 1/(4n); note that the variance is independent of p. Let y i = sin –1.

27. 27 Control Charts Based on Transformed Data Then points on the control chart are at heights y 1, y 2,.... For known n, the control limits are When p is not known, sin –1 is replaced by y. Similar comments apply to the Poisson distribution when  is Y = h(X) = 2, small. The suggested transformation is, which has mean value 2 and variance 1.

28. 28 Control Charts Based on Transformed Data Resulting control limits are 2  3 when  is known and y  3 otherwise. The book Statistical Methods for Quality Improvement listed in the chapter bibliography discusses these issues in greater detail.