Chapter 12 Screened I Chart – eliminating freak moving ranges before calculating limits
As detailed in earlier chapters, control limits are designed to represent the natural, common cause variation and thereby enclose (almost) all data points from a stable process. To estimate the common cause variation we use the within subgroup variation. However, if the subgroups consist of single data elements, as is the case with I charts, there is no within subgroup variation to be calculated. Instead, we use the moving ranges, that is the absolute differences between consecutive data points. Thus, two subgroups is used to define the “within” subgroup variation.
Consequently, if a shift occurs between two subgroups, this will affect the moving range at this point and possible produce a signal on the moving range chart.
For example, take these 24 “random” numbers:
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 we find that the natural upper AMR limit is 3.267 x AMR (= 16.3).
Consequently, the range between observations numbers 14 and 15 (|28 – 11| = 17) is unusually large as shown by the moving range chart in Figure 12.1.
Figure 12.1: Moving range chart with one range above the control limit
Some experts recommend removing extreme moving ranges before calculating the control limits for the corresponding I chart (Nelson 1982), as this can slightly improve the chart’s sensitivity. qicharts2 does this by default as shown in Figure 12.2.
# default is to screen moving ranges before calculating control limits
qic(y,
chart = 'i',
title = 'I chart with screened MRs')
Figure 12.2: I chart with control limits calculated after removing freak moving ranges
To remove extreme moving ranges, first calculate AMR on the original data (= 5), then remove any MRs greater than 3.267 x AMR (= 16.335) and recalculate AMR (= 4.45). Finally, use the screened AMR to calculate the control limits of the I chart (= 15.8 \(\pm\) 2.66 x 4.45).
To suppress the screening of moving ranges before calculating control limits for I charts (and prime charts) in qicharts2, we can set the qic.screenedmr option to FALSE before plotting as in Figure 12.3:
# suppress screening of moving ranges
options(qic.screenedmr = FALSE)
qic(y,
chart = 'i',
title = 'I chart without screened MRs')
Figure 12.3: I chart with control limits calculated without removing extreme ranges
As expected, the control limits on the unscreened chart is sightly wider than on the screened chart. In this example, just enough to suppress a signal that would otherwise have been discovered.
One may argue that screening moving ranges is unnecessary as long as the moving range chart is shown alongside the I chart. However, in our experience, presenting moving range charts to non-technical SPC users rarely adds meaningful insight and can sometimes cause confusion. As a result, we prefer to enhance the sensitivity of I charts by excluding extreme moving ranges before calculating control limits while leaving the use and interpretation of moving range charts to data analyst.