Chapter 11 Screened I Chart – eliminating freak moving ranges before calculating limits
As discussed in earlier chapters, control limits are designed to represent natural (common cause) variation and thereby enclose almost all data points from a stable process. Estimation of this variation typically relies on within-subgroup variation. However, when subgroups consist of single observations – as in I charts – there is no within-subgroup variation to calculate. Instead, we use moving ranges, defined as the absolute differences between consecutive data points. In effect, pairs of consecutive observations are used to estimate the within-subgroup variation.
This approach has an important implication. If a shift occurs between two consecutive observations, the moving range at that point will be inflated and may produce a signal on the moving range chart.
To illustrate, consider the following 24 values:
18 16 8 9 10 11 26 14 15 14 18 19 18 11 28 20 16 17 12 13 24 16 15 11The average moving range (AMR) is 5. From Table 6.1, the natural upper limit for moving ranges is 3.267 x AMR (= 16.3). The moving range between observations 14 and 15 is |28 − 11| = 17, which exceeds this limit and is therefore unusually large, as shown in Figure 11.1.
Figure 11.1: Moving range chart with one range above the control limit
Some authors recommend removing such extreme moving ranges before calculating the control limits for the corresponding I chart, as this may improve sensitivity to meaningful changes (Nelson 1982). By default, qicharts2 applies this approach, as illustrated in Figure 11.2.
# default is to screen moving ranges before calculating control limits
qic(y,
chart = 'i',
title = 'I chart with screened MRs')
Figure 11.2: I chart with control limits calculated after removing freak moving ranges
The procedure is as follows. First, calculate the AMR from the original data (= 5). Next, remove any moving ranges greater than 3.267 × AMR (= 16.335) and recalculate the AMR (= 4.45). Finally, use this screened AMR to calculate the control limits for the I chart: (= 15.8 \(\pm\) 2.66 x 4.45).
To suppress this screening step in qicharts2, the qic.screenedmr option can be set to FALSE as in Figure 11.3:
# suppress screening of moving ranges
options(qic.screenedmr = FALSE)
qic(y,
chart = 'i',
title = 'I chart without screened MRs')
Figure 11.3: I chart with control limits calculated without removing extreme ranges
As expected, the control limits in the unscreened chart are slightly wider than those in the screened chart – in this example, just wide enough to suppress a signal that would otherwise have been detected.
One might argue that screening moving ranges is unnecessary if the moving range chart is presented alongside the I chart. However, in our experience, moving range charts rarely add value for non-technical users and may even introduce confusion. For this reason, we prefer to increase the sensitivity of I charts by excluding extreme moving ranges before calculating control limits, while reserving moving range charts for more technical analysis.