Cumulative Sum Charts for Problem Solving

A retrospective analysis for problem solving using cumulative sum charts.
May 01, 2008
Volume 21, Issue 5


Cumulative sum (CUSUM) charts are a valuable tool in problem solving because they can reveal when a change occurred. This information then can be used to confirm or eliminate potential causes by showing that the problem appeared at the same or a different time. This article focuses on retrospective analysis for problem solving and not the use of CUSUM for controlling a process. The methods for converting a trend chart into a CUSUM chart in Excel are explained in depth. Because interpreting the charts is not entirely intuitive, a guide is presented, which shows how the chart should be read. One case study is used to provide a running example through the article.

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A manufacturer of excipients had process problems which arose during the course of a year. The major problem was that assay measurements of one characteristic of the product, the concentration of a contaminant, had increased over several months. As always with problem solving, many theories were cited about the likely cause. A favored one was that the assay standard had changed and the production measurements were consequently distorted but the material itself was still in specification.

Figure 1. Contaminant concentration in production batches. The trend chart shows how the detected quantities increased and eventually exceeded the limit of 10 ppm.
A trend chart (Figure 1) shows how the measurements deteriorated and eventually exceeded the limit of 10 ppm but does not show clearly when the change occurred.

Figure 2. Contaminant concentration in production batches with moving average. It is still not certain when the increase occurred.
In this situation, it is common to plot a moving average, especially because it is easy to do so using the trend line option in Excel and selecting "Moving Average." The second chart (Figure 2) shows the moving average, but even after examining it, one cannot be certain when the increase occurred. Interpreting this chart is difficult because the moving average is slow to respond to a change and always lags behind the raw data. For example, this chart shows that the line for the moving average was displaced to the right from the data points when the average rose during September.

Figure 3. The chart for the assay standard shows an increase in the second half of the year.
Another chart for the assay standard (Figure 3) shows a similar pattern where an increase in the second half of the year could be taken as evidence that the production measurements were mirroring the assay standard.

Figure 4. Ratio of production measurement to assay reference
A simple check to see if the production and reference results are related, is to calculate and plot the ratio of production value to reference value for each batch. It is easy to do this in Excel: if the ratio is constant, then the cause of the problem is probably in the assay reference, whereas if the ratio changes in step with the production results, then the cause is somewhere in the production process. The chart for ratio (Figure 4) is unfortunately inconclusive and shows some increase at about the right time but not enough to account for the change in production results.

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