c-Charts

Computing Limits for the XmR Chart

We know that the values for the centerlines for each chart were computed by simply taking the average of the values displayed (i.e., by adding up the values for each month and then dividing by the number of months/values to compute the average). How were the upper and lower limits calculated for the charts shown above? We can calculate the limits for both the X (Individual Values) chart and the Average Moving Range (mR) chart as follows :

• For the mR (moving range) chart. The upper range (or upper control limit, or upper natural limit) is computed by multiplying the average moving range (the centerline of the mR chart) by 3.27.

• For the X chart (individual values chart). The upper range for the X chart is computed by multiplying the average moving range of the associated chart by 2.66 and then adding the value for the centerline of the X chart. The lower range for the X chart is computed by multiplying the average moving range by 2.66 and then subtracting the value for the centerline of the X chart.

Notice that values for both representations (individual values and average moving range values) must be gathered and computed. The upper and lower limits for the individual values chart (X chart) depend on the average variations calculated for the centerline of the average moving range chart. Therefore, these charts are interdependent and can be used to show relationships between the two types of charts and the two types of data.

When you present these charts to management, they may decide that the variation looks a little too big for their comfort . So, they may suggest moving the limits to plus or minus some value. No! The only way to change the limits is to change the process itself. Remember: the charts show the "voice of the process." So, if management desires to improve the limits, they must improve the process. The numbers shown in the charts reflect the results of the process. Our job is to find what caused these results. When analyzing the causes, and adjusting the process, remember that the inputs and outputs of the process may be affected, as well as any other processes that depend on the original process under review. So, focusing on one or two causes will have downstream ramifications and will probably affect parts of your process, as well as other processes.

We have also seen the limits for the XmR charts calculated using median ranges instead of average ranges. The median moving range is often more sensitive to assigned causes when the values used contain some very high range values that inflate the average. Remember that the median range is that range of numbers that hover around the middle of a list sequenced in ascending or descending order: thus, the median range chart will automatically "throw out" the very high- or low-end values. Use of the median moving range approach is valid; however, the formulas (constants) change. The constant value 2.66 changes to 3.14, and the constant value 3.27 changes to 3.87. If you decide to use both methods on your data, you will find some slight differences in the limits, but not much. Try it and see.

Do not panic over computing these numbers. Most of the automated tools out there will do it for you, and draw the charts. While we make no recommendations, the tool we used to generate the charts in this chapter is an add-on to Microsoft Excel.