Tabular Cusum Chart for the Average and Hawkins Statistic for Variability

The BIS.Net Team BIS.Net Team

The Cusum Chart for Averages

The Cusum chart is arguably the most sensitive classical control chart for detecting shifts in the process mean. Whereas a Shewhart X Bar chart can take a one standard error change a Cusum chart only takes 9 points on average.

The Cusum chart has been successfully used in applications requiring close control to a specified target. The Cusum chart is not appropriate to identify intermittent, non-persistent assignable causes. The Shewhart X-bar chart is more appropriate for these applications.

There are two types of Cusum Control Charts. One uses a V mask, the other a tabular (algorithmic) approach making it possible to draw a chart looking like a Shewhart Chart in the sense that points or bars are plotted with superimposed control limits. BIS.NET Analyst uses the latter as it is easier to use and understand. These charts are commonly called Cusum Status Charts.

The tabular Cusum chart separately accumulates deviations above and below the target as follows

Ci+ = max(0,xi-(target+k)+Ci-1) (above target)

Ci- = max(0,(target-k)-xi +Ci-1) (below target)


Ci+ is the cumulative value above the target

Ci-1+ is the previous cumulative value above the target

Ci- is the cumulative value below the target

Ci-1i is the previous cumulative value below the target

K is called the reference value, or allowance, or slack value. It is usually chosen to be half way between the target and the out of control value of the mean we wish to rapidly detect.

For example, if the process standard deviation is 1.5 and we wish to detect a one standard deviation change then k= (1.5/1)/2.

The procedure for calculating control limits are beyond this brief overview of Cusum charts. However, it is important to know how BISNET Analyst calculates process stand deviation if not known. For individual’s data without rational subgroups BIS.NET Analyst uses the average of the moving range to convert to standard deviation by dividing the result by a d2 factor, as used for Shewhart X-bar / Range charts. If rational subgroups are used BISNET uses the weighted average subgroup standard deviation divided by the c4 factor as used for Shewhart X-bar / Range charts.

For applications requiring a process correction to maintain the process on target the cusums can be reset to zero, once the change has been made. Additionally the cusum provides information on how much to adjust the process by.

The Cusum Chart for Variability

Variability is often measured with a Range or Sd chart.

It is however possible to construct a cusum control chart for monitoring variability. A procedure developed by Hawkins D.M. "A CUSUM FOR A SCALE PARAMETER", Journal of Quality Technology, Volume 13(4) pp 228-235 is an alternative used by BIS.NET Analyst.

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