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

2. Copyright © Cengage Learning. All rights reserved. 16.2 Control Charts for Process Location

3. 3 Suppose the quality characteristic of interest is associated with a variable whose observed values result from making measurements. For example, the characteristic might be resistance of electrical wire (ohms), internal diameter of molded rubber expansion joints (cm), or hardness of a certain alloy (Brinell units). One important use of control charts is to see whether some measure of location of the variable’s distribution remains stable over time. The most popular chart for this purpose is the chart.

4. 4 The X Chart Based on Known Parameter Values

5. 5 The Chart Based on Known Parameter Values Because there is uncertainty about the value of the variable for any particular item or specimen, we denote such a random variable (rv) by X. Assume that for an incontrol process, X has a normal distribution with mean value  and standard deviation . Then if denotes the sample mean for a random sample of size n selected at a particular time point, we know that 1. E( ) =  2. 3. has a normal distribution. X

6. 6 The Chart Based on Known Parameter Values It follows that where Z is a standard normal rv.  It is thus highly likely that for an in-control process, the sample mean will fall within 3 standard deviations of the process mean . X

7. 7 The Chart Based on Known Parameter Values Consider first the case in which the values of both  and  are known. Suppose that at each of the time points 1, 2, 3,..., a random sample of size n is available. Let denote the calculated values of the corresponding sample means. An chart results from plotting these over time—that is, plotting points and so on—and then drawing horizontal lines across the plot at LCL = lower control limit = UCL = upper control limit = X

8. 8 The Chart Based on Known Parameter Values Such a plot is often called a 3-sigma chart. Any point outside the control limits suggests that the process may have been out of control at that time, so a search for assignable causes should be initiated. X

9. 9 Example 1 Once each day, three specimens of motor oil are randomly selected from the production process, and each is analyzed to determine viscosity. The accompanying data (Table 16.1) is for a 25-day period. Table 16.1 Viscosity Data for Example 1

10. 10 Example 1 Extensive experience with this process suggests that when the process is in control, viscosity of a specimen is normally distributed with mean 10.5 and standard deviation.18. Thus so the 3 SD control limits are cont’d

11. 11 Example 1 All points on the control chart shown in Figure 16.2 are between the control limits, indicating stable behavior of the process mean over this time period (the standard deviation and range for each sample will be used in the next subsection). Figure 16. 2 chart for the viscosity data of Example 1 X cont’d

12. 12 Chart Based on Estimated Parameters

13. 13 Charts Based on Estimated Parameters In practice it frequently happens that values of  and  are unknown, so they must be estimated from sample data prior to determining the control limits. This is especially true when a process is first subjected to a quality control analysis. Denote the number of observations in each sample by n, and let k represent the number of samples available. Typical values of n are 3, 4, 5, or 6; it is recommended that k be at least 20. We assume that the k samples were gathered during a period when the process was believed to be in control. More will be said about this assumption shortly. X

14. 14 With denoting the k calculated sample means, the usual estimate of  is simply the average of these means: There are two different commonly used methods for estimating  : one based on the k sample standard deviations and the other on the k sample ranges (recall that the sample range is the difference between the largest and smallest sample observations). X Charts Based on Estimated Parameters

15. 15 Charts Based on Estimated Parameters Prior to the wide availability of good calculators and statistical computer software, ease of hand calculation was of paramount consideration, so the range method predominated. However, in the case of a normal population distribution, the unbiased estimator of  based on S is known to have smaller variance than that based on the sample range. Statisticians say that the former estimator is more efficient than the latter. The loss in efficiency for the estimator is slight when n is very small but becomes important for n > 4. X

16. 16 Charts Based on Estimated Parameters Recall that the sample standard deviation is not an unbiased estimator for . When X1, …., Xn is a random sample from a normal distribution, it can be shown that E(S) = a n   where and denotes the gamma function. A tabulation of a n for selected n follows: X

17. 17 Charts Based on Estimated Parameters Let Where S 1, S 2,..., S k are the sample standard deviations for the k samples. Then Thus So is an unbiased estimator of . X

18. 18 Charts Based on Estimated Parameters Control Limits Based on the Sample Standard Deviations where X

19. 19 Example 2 Referring to the viscosity data of Example 1, we had n = 3 and k = 25. The values of and s i (i = 1,..., 25) appear in Table 16.1, from which it follows that = 261.896/25 = 10.476 and s = 3.834/25 =.153. With a 3 =.886, we have These limits differ a bit from previous limits based on  = 10.5 and  =.18 because now = 10.476 and =.173.

20. 20 Example 2 Inspection of Table 16.1 shows that every is between these new limits, so again no out-of-control situation is evident. Table 16.1 Viscosity Data for Example 1 cont’d

21. 21 Charts Based on Estimated Parameters To obtain an estimate of  based on the sample range, note that if X 1,...,X n form a random sample from a normal distribution, then R = range(X 1,..., X n ) = max(X 1,..., X n ) – min(X 1,..., X n ) = max(X 1 – ,..., X n –  ) – min(X 1 – ,..., X n –  ) =   {max(Z 1,..., Z n ) – min(Z 1,..., Z n )} X

22. 22 Charts Based on Estimated Parameters =   {max(Z 1,..., Z n ) – min(Z 1,..., Z n )} Where Z 1,..., Z n are independent standard normal rv’s. Thus E(R) =   E(range of a standard normal sample) =   b n so that is an unbiased estimator of . X

23. 23 Now denote the ranges for the k samples in the quality control data set by r 1, r 2,...,r k. The argument just given implies that the estimate comes from an unbiased estimator for . Charts Based on Estimated Parameters X

24. 24 Charts Based on Estimated Parameters Selected values of b n appear in the accompanying table [their computation is based on using statistical theory and numerical integration to determine E(min(Z 1,..., Z n )) and E(max(Z 1,..., Z n )) X

25. 25 Charts Based on Estimated Parameters Control Limits Based on the Sample Ranges where and r 1,..., r k are the k individual sample ranges. X

26. 26 Recomputing Control Limits

27. 27 Recomputing Control Limits We have assumed that the sample data used for estimating  and  was obtained from an in-control process. Suppose, though, that one of the points on the resulting control chart falls outside the control limits. Then if an assignable cause for this out-of-control situation can be found and verified, it is recommended that new control limits be calculated after deleting the corresponding sample from the data set.

28. 28 Recomputing Control Limits Similarly, if more than one point falls outside the original limits, new limits should be determined after eliminating any such point for which an assignable cause can be identified and dealt with. It may even happen that one or more points fall outside the new limits, in which case the deletion/recomputation process must be repeated.

29. 29 Performance Characteristics of Control Charts

30. 30 Performance Characteristics of Control Charts Generally speaking, a control chart will be effective if it gives very few out-of- control signals when the process is in control, but shows a point outside the control limits almost as soon as the process goes out of control. One assessment of a chart’s effectiveness is based on the notion of “error probabilities.” Suppose the variable of interest is normally distributed with known  (the same value for an in-control or out-of-control process).

31. 31 Performance Characteristics of Control Charts In addition, consider a 3-sigma chart based on the target value  0, with  =  0 when the process is in control. One error probability is  = P(a single sample gives a point outside the control limits when  =  0 ) = P( >  0 + 3  / or X <  0 – 3  / when  =  0 ) X

32. 32 Performance Characteristics of Control Charts The standardized variable has a standard normal distribution when  =  0, so  = P(Z > 3 or Z < –3) =  (–3.00) + 1 –  (3.00) =.0026 If 3.09 rather than 3 had been used to determine the control limits (this is customary in Great Britain), then  = P(Z > 3.09 or Z < –3.09) =.0020 The use of 3-sigma limits makes it highly unlikely that an out-of-control signal will result from an in-control process.

33. 33 Performance Characteristics of Control Charts Now suppose the process goes out of control because  has shifted to  +  (  might be positive or negative);  is the number of standard deviations by which  has changed. A second error probability is.

34. 34 Performance Characteristics of Control Charts We now standardize by first subtracting  0 +  from each term inside the parentheses and then dividing by  = P(– 3 – < standard normal rv < 3 – ) =  (3 – ) –  (–3 – ) This error probability depends on , which determines the size of the shift, and on the sample size n.

35. 35 Performance Characteristics of Control Charts In particular, for fixed ,  will decrease as n increases (the larger the sample size, the more likely it is that an out-of-control signal will result), and for fixed n,  decreases as |  | increases (the larger the magnitude of a shift, the more likely it is that an out-of-control signal will result). The accompanying table gives  for selected values of  when n = 4. It is clear that a small shift is quite likely to go undetected in a single sample.

36. 36 Performance Characteristics of Control Charts If 3 is replaced by 3.09 in the control limits, then  decreases from.0026 to.002, but for any fixed n and ,  will increase. This is just a manifestation of the inverse relationship between the two types of error probabilities in hypothesis testing. For example, changing 3 to 2.5 will increase  and decrease . The error probabilities discussed thus far are computed under the assumption that the variable of interest is normally distributed.

37. 37 Performance Characteristics of Control Charts If the distribution is only slightly nonnormal, the Central Limit Theorem effect implies that will have approximately a normal distribution even when n is small, in which case the stated error probabilities will be approximately correct. This is, of course, no longer the case when the variable’s distribution deviates considerably from normality.

38. 38 Performance Characteristics of Control Charts A second performance assessment involves expected or average run length needed to observe an out-of-control signal. When the process is in control, we should expect to observe many samples before seeing one whose lies outside the control limits. On the other hand, if a process goes out of control, the expected number of samples necessary to detect this should be small.

39. 39 Performance Characteristics of Control Charts Let p denote the probability that a single sample yields an x value outside the control limits; that is, Consider first an in-control process, so that are all normally distributed with mean value  0 and standard deviation Define an rv Y by Y = the first i for which X i falls outside the control limits

40. 40 Performance Characteristics of Control Charts If we think of each sample number as a trial and an out-of-control sample as a success, then Y is the number of (independent) trials necessary to observe a success. This Y has a geometric distribution, and we seen earlier that E(Y) = 1/p. The acronym ARL (for average run length) is often used in place of E(Y). Because p =  for an in-control process, we have

41. 41 Performance Characteristics of Control Charts Replacing 3 in the control limits by 3.09 gives ARL = 1/.002 = 500. Now suppose that, at a particular time point, the process mean shifts to  =  0 + . If we define Y to be the first i subsequent to the shift for which a sample generates an out-of-control signal, it is again true that ARL = E(Y) = 1/p, but now p = 1 – . The accompanying table gives selected ARLs for a 3-sigma chart when n = 4.

42. 42 Performance Characteristics of Control Charts These results again show the chart’s effectiveness in detecting large shifts but also its inability to quickly identify small shifts. When sampling is done rather infrequently, a great many items are likely to be produced before a small shift in  is detected. The CUSUM procedures were developed to address this deficiency.

43. 43 Supplemental Rules for X Charts

44. 44 Supplemental Rules for Charts The inability of charts with 3-sigma limits to quickly detect small shifts in the process mean has prompted investigators to develop procedures that provide improved behavior in this respect. One approach involves introducing additional conditions that cause an out-of-control signal to be generated. X

45. 45 Supplemental Rules for Charts The following conditions were recommended by Western Electric (then a subsidiary of AT&T). An intervention to take corrective action is appropriate whenever one of these conditions is satisfied: 1. Two out of three successive points fall outside 2-sigma limits on the same side of the center line. 2. Four out of five successive points fall outside 1-sigma limits on the same side of the center line. 3. Eight successive points fall on the same side of the center line. A quality control text should be consulted for a discussion of these and other supplemental rules. X

46. 46 Robust Control Charts

47. 47 Robust Control Charts The presence of outliers in the sample data tends to reduce the sensitivity of control-charting procedures when parameters must be estimated. This is because the control limits are moved outward from the center line, making the identification of unusual points more difficult. We do not want the statistic whose values are plotted to be resistant to outliers, because that would mask any out-of-control signal.

48. 48 Robust Control Charts For example, plotting sample medians would be less effective than plotting as is done on an chart. The article “Robust Control Charts” by David M. Rocke (Technometrics, 1989:173–184) presents a study of procedures for which control limits are based on statistics resistant to the effects of outliers.

49. 49 Robust Control Charts Rocke recommends control limits calculated from the interquartile range (IQR), which is very similar to the fourth spread. In particular, For a random sample from a normal distribution, E(IQR) = k n  ; the values of k n are given in the accompanying table.

50. 50 Robust Control Charts The suggested control limits are The values of are plotted. Simulations reported in the article indicated that the performance of the chart with these limits is superior to that of the traditional chart.