This chart was introduced by Page (1954) and is used primarily to maintain current control of a process. For more detailed information the reader is encouraged to see Burr (1976), Duncan (1986), Kemp (1962), Lucas (1982), Lucas and Crozier (1982), Page (1955), and Page (1954). Its advantage over the previous type of control chart (Shewhart chart) is that it may be equally effective at less expense. This occurs because the cusum chart can pick up a sudden and persistent change in the process average more rapidly , especially if the change is not large (see Figure 8.18).
Figure 8.18 shows a number of red beads obtained in successive drawings of 100 beads at random from a box containing 5% red beads. After 20 samples were drawn, a slight increase was made in the number of red beads in the box. The cusum chart on the bottom picks up this shift, whereas the Shewhart chart does not. In essence then, the cusum chart has as its objective to achieve desired discriminating power on minimum average run length (ARL).
Note | The "points" plotted on a cusum chart do not represent single observation. Starting from a given point, all subsequent plotted points contain information from all available observations. |
The data for working time (Table 8.4) is used to illustrate the method. Using a target value of 60 minutes, the difference between actual and target is first calculated. The cumulative sum of the differences (taking the sign ”negative or positive ”into account) is then obtained and plotted (Figure 8.19) chronologically. Figure 8.19a shows the total cumulative difference, and Figure 8.19b shows the same difference split between work time (minutes) and cusum (minutes).
Cusum Calculations | Target = 60 minutes | ||||||
---|---|---|---|---|---|---|---|
X | A | ˆ‘” | X | A | ˆ‘” | ||
1. | 65 | 5 | 5 | 35. | 68 | 8 | -95 |
2 | 63 | 3 | 8 | 36. | 65 | 5 | -90 |
3. | 63 | 3 | 11 | 37. | 67 | 7 | -83 |
4. | 62 | 2 | 13 | 38. | 62 | 2 | -81 |
5. | 65 | 5 | 18 | 39. | 61 | 1 | -80 |
6. | 61 | 1 | 19 | 40. | 62 | 2 | -78 |
7. | 63 | 3 | 22 | 41. | 63 | 3 | -75 |
8. | 67 | 7 | 29 | 42. | 61 | 1 | -74 |
9. | 62 | 2 | 31 | 43. | 59 | -1 | -75 |
10. | 59 | -1 | 30 | 44. | 57 | -3 | -78 |
11. | 57 | -3 | 27 | 45. | 60 |
| -78 |
12. | 55 | -5 | 22 | 46. | 60 |
| -78 |
13. | 52 | -8 | 14 | 47. | 64 | 4 | -74 |
14. | 53 | -7 | 0.7 | 48. | 62 | 2 | -72 |
15. | 51 | -9 | -2 | 49. | 60 |
| -72 |
16. | 45 | -15 | -17 | 50. | 62 | 2 | -70 |
17. | 44 | -16 | -33 | 51. | 64 | 4 | -66 |
18. | 48 | -12 | -45 | 52. | 64 | 4 | -62 |
19. | 59 | -1 | -46 | 53. | 75 | 15 | -47 |
20. | 55 | -5 | -51 | 54. | 59 | -1 | -48 |
21. | 54 | -6 | -57 | 55. | 55 | -5 | -53 |
22. | 57 | -3 | -60 | 56. | 52 | -8 | -61 |
23. | 55 | -5 | -65 | 57. | 50 | -10 | -71 |
24. | 59 | -1 | -66 | 58. | 57 | -3 | -74 |
25. | 57 | -3 | -69 | 59. | 55 | -5 | -79 |
26. | 53 | -7 | -76 | 60. | 72 | 12 | -67 |
27. | 50 | -10 | -86 | 61. | 74 | 14 | -53 |
28. | 47 | -13 | -99 | 62. | 72 | 12 | -41 |
29. | 55 | -5 | -99 | 63. | 62 | 2 | -39 |
30. | 54 | -6 | -110 | 64. | 64 | 4 | -35 |
31. | 61 | 1 | -109 | 65. | 70 | 10 | -25 |
32. | 61 | 1 | -108 | 66. | 66 | 6 | -10 |
33. | 59 | -1 | -109 | 67. | 69 | 9 | -10 |
34. | 66 | 6 | -103 | 68. | 59 | -1 | -11 |
From the method of calculation, it can be understood that the important feature of the cusum chart is the slope. When the plot is horizontal, the process is on target. An upward slope indicates a shift to a higher value. A downward slope indicates a shift to lower values. The greater the slope, the larger the shift.
Control limits can be drawn for the cusum chart in the form of a V mask. This is illustrated in Figure 8.20.
Statistical tables are available for constructing these V masks based on the extent of the process shift that one desires to detect, the confidence level, and the relative scales of the horizontal and vertical axes of the chart.
Cusum charts are not very convenient to use on a routine basis and operators often have difficulty understanding how the chart should be interpreted. This has limited their application.
The actual chart is shown in Figure 8.20. Recall that it is the slope of the cusum plot that indicates process level. Horizontal means on target or average. Downward means below, and upwards, above average. Control limits may thus be drawn as critical slopes.
For this example, Mask (a) will detect smaller deviations from target than Mask (b), but with a longer lead time. Data points within the V are in control. It is convenient to draw the V limits on transparent paper. Mask (c) is an illustration of detecting an out-of-control point.