It has been observed that a person would seldom if ever be justified in concluding that a state of statistical control of a given repetitive operation or production process had been reached until he had obtained, under presumably the same essential conditions, a sequence of not less than twenty-five samples of four that satisfied Criterion I [i.e., fall within 3-sigma limits].
Walter A. Shewhart, 1939 The fact that an observed set of values for fraction defective indicates the product to have been controlled up to the present does not prove that we can predict the future course of this phenomenon. We always have to say that this can be done provided the same essential conditions are maintained, and, of course, we never know whether or not they are maintained unless we continue the experiment.
Walter A. Shewhart, 1931 Statistical control is ephemeral; there must be a running record for judging whether the state of statistical control still exists.
W. Edwards Deming, 1986b
Although it is possible (and practical) to establish control limits, at least on a trial basis, with relatively small amounts of data, you should be cautious about concluding that any process is in statistical control. States of statistical control are ideal states, and testing for them is much like testing for error-free software. You may be able to show that you have not reached the state, but you can never prove or know for certain when you have arrived. Nevertheless, the data you have, when plotted on control charts, are infinitely better than no data at all. Nothing can change the fact that you will have to bet on your predictions of the future performance of your processes. If your control charts signal out-of-control histories, you have little basis for extrapolating historical performance to the future. But as you identify and permanently eliminate assignable causes of unusual variation, your willingness to rely on extrapolation increases. Reliable predictions of process performance and the setting of achievable goals then become reasonable possibilities.
One important corollary to your caution in concluding that a process is stable is that you should almost never stop control charting any process, especially one that has had a history of going out of control. How else can you be assured that, once stabilized and made predictable, the process has not fallen back into its old ways?
Measured Rounded Observation X mR X mR 1 1.08 — 1.1 — 2 1.09 .01 1.1 0 3 1.15 .06 1.2 0.1 4 1.07 .08 1.0 0.2 5 1.03 .04 1.0 0 6 1.08 .05 1.1 0.1 7 1.1 .02 1.1 0 8 1.04 .06 1.0 0.1 9 1.07 .03 1.1 0.1 10 1.1 .03 1.1 0 11 1.12 .02 1.1 0 12 1.09 .03 1.1 0 13 1.03 .06 1.0 0.1 14 1.03 .0 1.0 0 15 1.09 .06 1.1 0.1 16 1.13 .04 1.1 0 17 1.02 .11 1.0 0.1 18 1.04 .02 1.0 0 19 1.03 .01 1.0 0 20 1.04 .01 1.0 0 21 1.14 .1 1.1 0.1 22 1.07 .07 1.1 0 23 1.08 .01 1.1 0 24 1.13 .05 1.2 0.1 25 1.08 .05 1.1 0.1 26 1.03 .05 1.0 0.1 27 1.02 .01 1.0 0 28 1.04 .02 1.0 0 29 1.03 .01 1.0 0 30 1.06 .03 1.1 0.1 31 1.02 .04 1.0 0.1 32 1.01 .01 1.0 0
Figure 7-2: Measured Values as Recorded and Subsequently Rounded
7.5
The Problem of Insufficient Granularity in Recorded Values
When measured values of continuous variables have insufficient granularity (i.e., are coarse and imprecise), the discreteness that results can mask the underlying process variation. Computations for X and sigma can then be
affected, and individual values that are rounded or truncated in the direction of the nearest control limit can easily give false out-of-control signals.
There are four main causes of coarse data: inadequate measurement instruments, imprecise reading of the instruments, rounding, and taking measurements at intervals that are too short to permit detectable variation to occur. When measurements are not obtained and recorded with sufficient precision to describe the underlying variability, digits that contain useful information will be lost. If the truncation or rounding reduces the precision in recorded results to only one or two digits that change, the running record of measured values will show only a few levels of possible outcomes. Fortunately, when this problem occurs, it is easy to identify. Figure 7-2 shows two sets of values for X (the measured process performance) and mR (the moving range of X). The left-most set of values lists 32 observations as they were recorded; the right-most set lists the observations after rounding (or as they might have been recorded if the measurements were insufficiently precise). The XmR charts produced from the two sets of values are shown in Figures 7-3 and 7-4. Notice that the charts do not appear to describe the same process. The out-of- control points in Figure 7-4 appear solely because the data do not correctly reflect the underlying process variation.
0 5 10 15 20 25 30 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 LCL = 0.965 CL = 1.07 UCL = 1.17
Moving Range
Individuals
Observation Number
0 5 10 15 20 25 30 0.00 0.02 0.04 0.06 0.08 0.10 0.12 CL = 0.0384 UCL = 0.125Figure 7-3: XmR Charts Constructed from Values as Originally Recorded
0.00 0.05 0.10 0.15 0.20 0 5 10 15 20 25 30 LCL = 0.931 CL = 1.06 UCL = 1.19
Moving Range
Individuals
Observation Number
0 5 10 15 20 25 30 0.95 1.00 1.05 1.10 1.15 1.20 CL = 0.0484 UCL = 0.158Figure 7-4: XmR Charts Constructed from Rounded Values
The solution to the problem of insufficient granularity is to ensure that the data used for control charts have a resolution that is smaller than the process standard deviation. A good rule of thumb for achieving this is to ensure that the set of points that are plotted always take on values that range over at least five possible levels of discreteness (the charts in Figure 7- 4, for example, have stepped appearances because the values in them have only three possible levels). Increasing the number of levels can be accomplished by measuring more
precisely, by decreasing the frequency of measurement to allow for variation to occur between measured values, or by increasing the size of subgroups to allow for more variation within a subgroup.
Never round data to the point where the values that result span less than five attainable levels. If this rule must be violated, the data can be plotted in a running record, but they should not be used to calculate control limits.
Additional examples and guidelines related to this subject can be found under the topic of “Inadequate Measurement Units” in Wheeler’s books [Wheeler 89, 92, 95].
7.6
Rational Sampling and Rational Subgrouping
…it is important to divide all data into rational subgroups in the sense that the data belonging to a group are supposed to have come from a constant system of chance causes.
Walter A. Shewhart, 1931 …emphasis…must be laid upon breaking up the original sequence into subgroups of comparatively small size. If this is not done, the presence of assignable causes will very often be overlooked.
Walter A. Shewhart, 1931 Obviously, the ultimate object is not only to detect trouble but also to find it, and such discovery naturally involves classification. The engineer who is successful in dividing his data initially into rational subgroups based upon rational hypotheses is therefore inherently better off in the long run than the one who is not thus successful.
Walter A. Shewhart, 1931
Control charts are founded on the concepts of rational sampling and rational subgrouping. These concepts deal with collecting and organizing data so that the questions at issue can be answered as reliably and economically as possible.