2. [1] MA4104 Business Statistics Spring 2008, Lecture 06 Process Monitoring Using Statistical Control Charts [ Examples Class ]

3. [2] x Chart Structure UCL LCL Process mean when in control Center Line Time

4. [3] Control Limits for an x Chart Process Mean and Standard Deviation Known

5. [4] The Standard Error [ of the sample mean ] ASIDE: It is approximately true that

6. [5] Control Limits for an x Chart Process Mean and Standard Deviation Unknown where x = overall sample mean R = average range A 2 = a constant that depends on n; taken from “Factors for Control Charts” table = _

7. [6] Factors for x and R Control Charts n d 2 A 2 d 3 D 3 D 4 21.1281.8800.853 03.267 31.6931.0230.888 02.574 42.0590.7290.880 02.282 52.3260.5770.864 02.114 62.5340.4830.848 02.004 72.7040.4190.8330.0761.924 82.8470.3730.8200.1361.864 92.9700.3370.8080.1841.816 103.0780.3080.7970.2231.777 : : : : : :

8. [7] Interpretation of Control Charts The location and pattern of points in a control chart enable us to determine, with a small probability of error, whether a process is in statistical control. A primary indication that a process may be out of control is a data point outside the control limits. Certain patterns of points within the control limits can be warning signals of quality problems. – A large number of points on one side of the center line. – Six or seven points in a row that indicate either an increasing or decreasing trend. –... and other patterns.

9. [8] Clip gap measurements in twenty five samples of five measurements each

10. [9] Estimating  First, calculate the average range from stable data, –use the deletion principle then, convert to

11. [10] Applying the deletion principle identify the largest sample range value; calculate the average range from the remaining points; calculate a trial upper control limit and display the centre line and UCL on the line plot; check whether the largest and any other range values lie outside the limits: –if none are outside, recalculate average range including the most extreme point, –otherwise, delete the points outside the limits and repeat the whole process with the remaining points.

12. [11] Clip gap measurements in twenty five samples of five measurements each Sum of 25 Range values = 445. Average Range = 445/25 = 17.8 Exercise: Calculate deleted average range,, and UCL = 2.11 ×

13. [12] Clip gaps Range chart; trial limits with Sample 11 deleted

14. [13] Exclude Sample 11, delete Sample 21 Sum of 24 Range values = 405. R 21 = 30 Exercise: Calculate deleted average range,, and UCL = 2.11 ×

15. [14] Range chart with Sample 11 excluded; trial limits with Sample 21 deleted

16. [15] Convert to

17. [16] Convert to Exercise: A 2 = 0.58. Calculate. Did you get 7.26 ?

18. [17] Exercise Assuming a value of 7.3 mm for , use the Normal table to predict the proportion of clips whose gaps fail to meet the specification limits of 50 mm to 90 mm (i)when the process mean is 70 mm, Did you get 0.6 % ?

19. [18] Estimating  Identify stable data –a form of deletion principle Calculate

20. [19] X-bar chart for clip gaps with historical centre line, sample 11 excluded

21. [20] Clip gaps X-bar chart with with redrawn control limits

22. [21] Exercise, continued Assuming a value of 7.3 mm for , use the Normal table to predict the proportion of clips whose gaps fail to meet the specification limits of 50 mm to 90 mm (ii)when the process mean is 74 mm, (iii)when the process mean is 67 mm. (ii)Did you get 1.47 % ? (iii)Did you get 1.07 % ?

23. [22] Control Charts for COUNTS np Chart Used to monitor COUNTS, i.e., the number of defective items in a subgroup [ sample of size n ], with an overall proportion p defective.

24. [23] assumingnp > 5 and n (1-p) > 5 Note: If computed LCL is negative, set LCL = 0 Control Limits for an np Chart

25. [24] The Standard Error [ of the sample total defective ]

26. [25] Clerical error counts in weekly samples of 100 forms for a 30 week period. For this example, assume that p = 0.05, i.e., 5%. As n = 100 (forms), we have that np = 5

27. [26]

28. [27] NP chart for Error Count in 100 forms for a 30 week period

29. [28] Clerical error counts in weekly samples of 100 forms for a further 20 week period.

30. [29] NP chart for Error Count in 100 forms, extended to a 50 week period

31. [30] Post improvement : np chart with new limits