One risk with these charts is that the other points, which are outside control limits, can confuse the interpretation of the plot by suggesting that there are other key events which should be considered. This becomes a serious concern if the process is not stable and under statistical control, as is the case here.

A weakness of this approach using a chart for individuals is that it lacks the statistical power to identify a small shift in the average. Visual examination of a CUSUM plot may have identified a change point but the chart for individuals with its control limits might only provide the same indication somewhat later, limiting its value as a statistical test. Other rules from SPC, such as a run of points, can be applied if the change is not so pronounced, but the only thing such rules will confirm is that the average has changed, not when it changed. A chart of moving average, like the one in Figure 2, is a more sensitive detector of a small change if appropriate control limits are added but it also has the limitation of not identifying when the change occurred.

It is possible to apply control rules directly to a CUSUM chart but the methods are not as simple or intuitive as with conventional SPC. For the occasional user who is trying to solve a problem, they present a steep learning curve; for a statistician or experienced practitioner in SPC, they are worth pursuing. This article does not cover those control rules.

Attribute Data

CUSUM charts work well with counted data, such as the number of particles in a sample, or the number of rejects per day. If the batch size or lot size varies, then the step size on the horizontal scale should mimic this so that a percentage reject rate from a large batch has a greater visual weight than the same percentage from a small batch. There are different ways to do this in Excel, but an X–Y plot on which the X-axis has a running total from all the batches is straightforward to implement.

CUSUM Chart in a Report

Interpretation of the charts is not intuitive. People often look at the vertical scale and try to find the significance in positive or negative values. Only the slope is important, however, so this should be emphasized if a CUSUM chart is included in a report. It is best to superimpose a plot for raw data with a CUSUM, as in Figure 5, along with an explanation if the intended readers of the report are not familiar with the technique.

Statistical Process Control

The equivalents of control limits can be set up for CUSUM charts and are valuable if a process average can drift, needs tight control, or cannot be evaluated using large subgroup sizes. This can occur for process variables, such as concentration or batch weight, where repeated measurements are not possible or will give the same result. However, it is not easy to run the charts and interpreting them is not intuitive for people who are only accustomed to conventional trend charts.

SUMMARY

The case study example shows how CUSUM was used to eliminate one potential cause of a problem and thereby save time and a lot of unnecessary work.

CUSUM techniques are an effective method for identifying a change point and are especially valuable in problem solving when a trend chart shows that an average has shifted. They are simple to implement in Excel and will often give a precise answer. One risk with CUSUM charts is that a casual reader may not understand that the slope of the lines is important and consequently make errors in interpretation. A second risk is that small variations might be thought to be important because visually they appear to be significant. However, these problems are minor in comparison to the benefits that the cumulative sum method brings when it is applied in the appropriate situations.

George R. Bandurek, PhD, is the director of GRB Solutions, Ltd., West Sussex, UK, +44 1903 215175, george@grb.co.uk