3.2 What Do We Gain from the mr Chart?

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1 Guide to Data Analysis While it is true that the X Chart tells the story, and that we will undoubtedly find the X Chart to be the most interesting part of the XmR Chart, there are times when the mr Chart can clarify and extend what we see on the X Chart. 3.2 What Do We Gain from the mr Chart? It has been suggested that the Moving Range Chart adds so little to the Chart for Individual Values that you should not bother to show it Simply show the X Chart and forget the mr Chart. The basis for this recommendation seems to be the documented fact that the combined XmR Chart does not have any appreciably greater ability to detect signals than does the X Chart alone. However, this mathematical analysis overlooks the interpretative benefits to be gained by including the mr Chart. Consider the XmR Chart for the Data from Bead Board No. 7 in Figure 3.3. The six moving ranges that exceed their upper limit identify six points where the funnel of the bead board was moved (see the dashed lines on the running record for the X values). Two of these six shifts do not correspond to points outside the limits on the X Chart. Thus, this Moving Range Chart provides additional information and reinforces the message of the X Chart X mr Figure 3.3: The XmR Chart for the Data from Bead Board No. 7 44

2 3 / Process Behavior Charts 11 1 X mr Figure 3.4: The XmR Chart for the Batch Weight Data When computing the limits for the Batch Weight Data in Figure 3.4, I only used the first 59 moving ranges to compute the Average Moving Range. Inspection of the mr Chart in Figure 3.4 will show why I did this there is an increase in the process variation after the first sixty values. Since the objective is to compute limits that characterize what the process is capable of doing, the first 59 moving ranges do this better than the rest. The XmR Chart in Figure 3.5 has limits based on the Median Moving Range. The six spikes on the mr Chart allow us to break the original data into seven segments and use a separate central line for each segment, resulting in an X Chart that shows the changes in level and also reveals the presence of other types of nonhomogeneity. Thus, while the mr Chart may not add much to the likelihood of detecting a change, it can be very helpful in interpreting changes that occur. The second reason that I cannot agree to the suppression of the mr Chart is that there are many people, and many software packages, that actually compute three standard deviation limits rather than three-sigma limits. If you are shown a naked X Chart you will have no way of knowing if the limits have been computed correctly. However, if you are shown an XmR Chart, you will immediately have a higher level of confidence that the limits have been computed correctly. Moreover, by using the central line of the mr Chart, you can quickly check to see if the limits are indeed cor-

3 Guide to Data Analysis Yield Values Moving Ranges Figure 3.5: The XmR Chart for the Creel Yield Data rectly computed. Thus, the mr Chart is the secret handshake of those who know the correct way of computing limits for an X Chart. Omit it and your readers cannot be sure that you are a member of the club. Finally, the mr Chart allows you and your audience to check for a problem that will be explained in Section 3.7, the problem of chunky data. Thus, while there may be little mathematical justification for showing the mr Chart, there are three practical reasons to do so, any one of which is sufficient to justify the inclusion of the mr Chart with your X Chart. 3.3 What Makes the XmR Chart Work? There are two basic ideas or principles that need to be respected when creating an XmR Chart. The first is that successive values need to be logically comparable. The second is that the moving ranges need to isolate and capture the local, shortterm, routine variation that is inherent in the data. While the use of the time-order sequence of the data will usually be sufficient to satisfy these two requirements of the XmR Chart, there are times when a careful con- 46

4 3 / Process Behavior Charts sideration of the structure in the data will require a different organization. As a case in point consider the Camshaft Bearing Diameters shown in Table 2.4. The strict time-order for these data is: Camshaft No. 1; Bearing 1, Bearing 2, Bearing 3, Camshaft No. 2; Bearing 1, Bearing 2, Bearing 3, etc. When the data are arranged in this order the Average Moving Range is 3.1. With an Average of we obtain Natural Process Limits of 41.6 to These limits are shown in Figure Piece Figure 3.6: An X Chart for the Camshaft Bearing Diameter Data in Time-Order With this arrangement the moving ranges represent the differences between the three bearings. Since the three bearings were produced by three separate parallel processes there could be systematic differences between the three bearings that have nothing to do with routine variation. Even though the three bearings are supposed to be the same, the fact that they are produced by parallel operations makes it a bit naive to assume that they are indeed the same. The idea of checking your data for homogeneity is based on a skeptical view of homogeneity, not a gullible one. Therefore, rather than using the strict time order shown in Figure 3.6, a more rational approach is to organize these data according to bearing number, and then to use the time order within each bearing to create moving ranges. When this is done the moving ranges will represent the natural, short-term, routine variation within each production process rather than the differences, if any, between the three processes. Now the Average Moving Range is 1.91, giving the limits of 44.7 to 54.9 shown in Figure 3.7. This running record and these limits allow us to see the differences between the bearings with greater clarity. 47

5 Guide to Data Analysis Bearing One Bearing Two Bearing Three Figure 3.7: The Second X Chart for the Camshaft Bearing Diameter Data Thus, when there is a structure within your data, it is imperative that you consider that structure when organizing the data for an XmR Chart. If there are logical partitions or subsets in your data, isolate those subsets from each other so that successive values will be logically comparable and the moving ranges can characterize the routine variation rather than the differences between the subsets. 55 Bearing One Bearing Two Bearing Three Figure 3.8: The Third X Chart for the Camshaft Bearing Diameter Data In fact, for the Camshaft Bearing Diameters we could take the next step and compute a separate set of limits for each bearing. This X Chart is in Figure 3.8. Not only are the three bearings different, but each bearing shows evidence of nonhomogeneity in the production processes. With the Hot Metal Delivery Times the changes occur so often that it is virtually impossible to obtain an Average Moving Range that will characterize the routine variation. In cases like this it is customary to shift to using the Median Moving Range. Since median values are less severely inflated by extreme values than averages, the use of the Median Moving Range allows us to extract an estimate of routine 48

6 3 / Process Behavior Charts Figure 3.9: The XmR Chart for the Hot Metal Delivery Times variation even in the face of an abundance of signals of change. Here the Median is and the Median Moving Range is 2, giving the limits shown in Figure 3.9. No Lower Natural Process Limit is shown in Figure 3.9 because the computed value was less than zero while the delivery times cannot be less than zero. Whenever a boundary or barrier value falls within the computed Natural Process Limits, that boundary value will take precedence over the computed limit, and you end up with a one-sided chart. While the Median Moving Range provided tighter limits and a more sensitive chart in Figure 3.9, you could argue that the Median Moving Range and the limits have still been inflated by the sheer multiplicity of large ranges. While this may be true, it is useful to recall that we are not trying to estimate the parameters of a probability model, but we are instead examining the data for evidence of nonhomogeneity. Having found this evidence, we know that these data did not come from one universe, but from several, so the whole question of estimation is moot. The limits are good enough to do their job. We need to discover why the process is operating with multiple personalities, and do something about this problem. Estimation will not help here; action is required. Once you have organized the data in a rational manner, there are many ways to use the limits to tell the story that is contained within the data. The objective is understanding and insight rather than computing a particular value. There is an ele- 49