Cumulative Sum Charts for Problem Solving - A retrospective analysis for problem solving using cumulative sum charts. - BioPharm International


Cumulative Sum Charts for Problem Solving
A retrospective analysis for problem solving using cumulative sum charts.

BioPharm International
Volume 21, Issue 5


As mentioned above, a line which slopes downwards indicates that the results are below the average or the target value that was selected. The steeper the line, the bigger the difference. A line that slopes upwards indicates that the selected values are above average.

Changes in Slope

The human eye is a wonderful tool for seeing patterns and it is easy to imagine that every slight variation in a line is significant. In this example, the extreme change in slope at the end of August is an important event but the others are debatable. For example, at the end of July, the slope of the line is upwards, which indicates that the raw data values are above the overall average, but not as much above average here as in September and October. This change may be significant or it could be a random variation, it is difficult to tell from the data alone. If the date of change coincides with a known event, such as a change of material or a process adjustment, then the data would support an assertion that something had affected the output of the process. Below we describe some statistical significance tests that give an objective method for assessing a change in the average.

Gaps in the Data

Gaps in the data, such as the times during March and April when there was no production, put a horizontal step into the CUSUM plot. It is usually possible to ignore such gaps by eye and make a visual estimate of the slope of the line to either side. If there are many gaps and the plot becomes confusing, it is simple in Excel to change the horizontal axis so that it shows batch number or successive measurements in sequence. A related problem occurs when a batch exists but there is no measurement for it. In this situation, Excel interprets a blank cell as zero and the CUSUM score can be sent very high or low, creating a vertical step. The solution is to either manually adjust the formulae so that they ignore the empty cell or to put in an IF function to do it automatically.

Changes in Vertical Height

The actual vertical position of the CUSUM plot is not relevant and can be changed very easily by adjusting the value of the first point in the sequence. In this example, the first point in the CUSUM is equal to the first measured value but it could be set to zero, or the average, or some arbitrary value, which makes plotting convenient.

Individual extreme values put a step into the line but the slope remains the same to either side. If there are only a few of these values, they can be ignored when looking at the charts. Alternately, the Excel formulae can be changed to exclude those values.

Average Value or Target Value

For problem solving, it is simple and convenient to use the overall average as the fixed, offset value in the CUSUM calculations. If a process target value is used instead, then the chart gives a clear indication if the process is running high or low. If it continues to run high or low, then the line will eventually fall outside the scale on the chart and a reset will need to be put in.

Find the Change Point

Figure 6. An example of how a CUSUM plot can be used to estimate the change point.
A graphical approach for finding the time when the average shifted is simple to use but does require a little judgment. Figure 6 shows the CUSUM for concentration of contaminant and two straight lines have been drawn through the points near the change point. The judgment is in deciding how many of the data points to use for the straight lines. Sometimes, it is helpful to mask the points nearest to the change time so that the lines are not distracted by a few points. Where the lines cross over gives the time when the average changed and it is usually much clearer than an estimate from a simple trend chart. In this example, the average went up starting on August 29.

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