by Stanley N. Deming
AL
Statistics in the Laboratory: Control Charts, Part 2
In the previous column (January/February 2016), we looked at the basic ideas behind control charts and saw how to construct a common X-bar and R chart. In this column we’ll look at two rules that can be used to detect out-of-control situations, and we’ll see why more is not better when it comes to such rules. In the next column we’ll differentiate between being out of control and out of specification—two entirely different concepts.
Figure 1 shows a set of three charts associated with a process. Let’s call it a measurement process, perhaps chromatography, though it could be any
quantitative analytical method. We’ll assume that the process has been running consistently for a long time, and the control limits are based on 20 or 30 initial sets of subgroups of data, not on the data shown in Figure 1. The measurement numbers and subgroup numbers all start at “1” in the figure, but these labels are relative to the figure—that is, mea- surement number 1 in the figure might actually represent the 3169th measurement since the chromatographic method was transferred from the development laboratory to the applications laboratory.
The upper chart in Figure 1 is called a run chart or a trend chart. It sim- ply plots the process output (in our case, the integrated peak area for a reference material) as a function of run number—that is, as a function of the sequence order. Because of accuracy requirements for this chromato- graphic method, the result must fall between the upper specification limit (USL) and the lower specification limit (LSL), shown in red. These might be called method acceptance limits for the chromatographic method to be considered still fit for use. All of the data in Figure 1 lies between these specification limits, so there is no reason to believe the chromatographic method isn’t producing valid results. We’re good to go.
The lowest chart in Figure 1 is a control chart, the familiar R chart in which subgroup ranges are plotted. In this example, the subgroup size is four. These subgroup ranges have been calculated from the data in the run chart at the top, taking the run data four at a time to produce each sub- group range plotted in this R chart. The process appears to be in statistical control as far as the ranges are concerned. Some ranges are larger than others, some ranges are smaller than others, but all of the ranges lie be- tween the control limits. Nothing unusual here.
The middle chart in Figure 1 is the familiar X-bar chart in which subgroup means are plotted. These subgroup means have also been calculated from the data in the run chart at the top, taking the run data four at a time to produce each subgroup mean plotted in this X-bar chart. The process appears to be in statistical control…until we reach subgroup 20, where the subgroup mean plots outside the statistical process control limits. This outlier is unusual!
Figure 1 – A three-sigma rule violation. AMERICAN LABORATORY 43
As we saw in the previous column, the probability that this will happen if the process is behaving now as it has behaved in the past is only 0.00270. So one of two things has happened here: either (a) this is just one of those events that’s expected to happen every now and then, about 3 times in
APRIL 2016
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