ABSTRACT
If the control chart is used to record the history of the sum of the sample measurements, then the control limits are calculated as follows
DCL = nIL + 3~ (J CCL = nIL
LCL = nIL - 3~ (J Considering another hypothetical example, assume that the diameters of residen-
tial plumbing pipes are checked, in groups of five units each, at the end of the fabrication cycle. If the population of pipe diameter measurements is characterized from historical information to have a mean value of2.010 in. and a standard deviation of 0.010 in. then the control limits are calculated as
3 DCL = 2.010 + ff (0.010) = 2.023 CCL = 2.010
3 LCL = 2.010 - ff (0.010) = 1.997
Alternatively, using a sample sum control chart gives the following control limits
DCL = 5 x 2.010 + 3ff(0.010) = 10.117 CCL = 5 x 2.010 = 10.050
LCL = 5 x 2.010 - (3ff (0.010) = 9.933 Figure 4.2 shows a theoretical control chart for the pipe production process
where the successive sample averages are plotted with respect to time. From this chart it can be determined that the process is running reasonably well, although there have been periods in which the sample averages have been outside the control limits. The condition of being "out of statistical control" is indicated by a number of control chart characteristics. When one data point falls outside a plus or minus four sigma band or two or more successive points occurs outside of the three sigma control limits then a boundary limit has been exceeded which indicates that corrective action probably should be initiated. Similarly, when seven or more successive points occur on the same side of the mean value, a process shift is likely to have occurred while seven points that "go in the same direction" indicate a trend caused by some sort of process drift. Periodicity in the data also is abnormal from a theoretical point of view, and may indicate a need for a closer examination of the actual situation. Each of these phenomena indicates that something has changed within the manufacturing process and may be cause for concern. Of course, practical judgment must always be utilized in evaluating the severity of a problem. In general, one point outside the three sigma limits is cause for concern and experience may be required to estimate if this is the beginning
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of a problem. If the control limits are close to the process tolerances then these deviations can cause defective products. However, if the control limits are very small with respect to the tolerances then additional leeway exists. Another data pattern that should be recognized is the condition in which the data falls very close to the centerline as opposed to being distributed between the upper and lower control limits. Assuming the data is accurate, this condition indicates that the controllimits need to be recalculated so that a meaningful description of the process quality can be obtained. In Figure 4.2 none of the previously mentioned error conditions occur but there was an operating period, depicted by samples seven and nine, during which the level of process control appears to have been marginal.
