## spctoolkit

by Donald J. Wheeler

### When Do I Recalculate My Limits?

Correct limits allow the user to separate
probable noise from potential signals.

Of all the questions about Shewhart's charts, this is perhaps the most frequently asked question. While there is no simple answer, there are some useful guidelines.

The first guideline for computing limits for Shewhart's charts is: You get no credit for computing the right number-only for taking the right action.

Without the follow-through of taking the right action, the computation of the right number is meaningless. Now, this is contrary to everyone's experience with arithmetic. Early on we are trained to "find the right number." Thus, when people are introduced to Shewhart's charts, this natural anxiety will surface in the form of questions about how to get the "right limits."

While there are definite rules for computing limits, and right and wrong ways of computing such limits, the real power of Shewhart's charts lies in the organization's ability to use them to understand and improve their processes. This use of Shewhart's charts-as an aid for making decisions-is the true focal point of the charts. But it is so easy to miss and so hard to teach.

The second guideline for computing limits for Shewhart's charts is: The purpose of the limits is to adequately reflect the voice of the process.

As long as the limits are computed in the correct way and reflect the voice of the process, then they are "correct limits." (Notice that the definite article is missing-they are just "correct limits," not "the correct limits.") Correct limits allow the user to separate probable noise from potential signals. Shewhart's charts are a tool for filtering out the probable noise. They have been proven to work in more than 70 years of practice.

Shewhart deliberately chose three-sigma limits. He wanted limits wide enough to filter out the bulk of the probable noise so that people wouldn't waste time interpreting noise as signals. He also wanted limits narrow enough to detect the probable signals so that people wouldn't miss signals of economic importance. In years of practice he found that three-sigma limits provided a satisfactory balance between these two mistakes.

Therefore, in the spirit of striking a balance between the two mistakes above, the time to recompute the limits for Shewhart's charts comes when, in your best judgment, they no longer adequately reflect the voice of the process.

The third guideline for computing limits for Shewhart's charts is: Use the proper formulas for the computations. The proper formulas for the limits are well-known and widely published. Nevertheless, novices continually think that they know better and invent shortcuts that are wrong.

The proper formulas for average and range charts will always use an average or median dispersion statistic in the computations. No formula that uses a single measure of dispersion is correct. The proper formula for X-charts (charts for individual values) will always use an average moving range or a median moving range.

Within these three guidelines lies considerable latitude for computing limits. As Shewhart said, it is mostly a matter of "human judgment" about the way the process behaves, about the way the data are collected and about the chart's purpose. Computations and revisions of limits that heed these three guidelines will work. Calculations that ignore these guidelines won't.

So, in considering the recalculation of limits, ask yourself:
Do the limits need to be revised in order for you to take the proper action on the process?
Do the limits need to be revised to adequately reflect the voice of the process?
Were the current limits computed using the proper formulas?

So, if the process shifts to a new location and you don't think there will be a change in dispersion, then you could use the former measure of dispersion, in conjunction with the new measure of location, to obtain limits in a timely manner. It is all a matter of judgment.

Remember, Shewhart's charts are intended as aids for making decisions, and as long as the limits appropriately reflect what the process can do, or can be made to do, then they are the right limits. This principle is seen in the questions used by Perry Regier of Dow Chemical Co.:
Do the data display a distinctly different kind of behavior than in the past?
Is the reason for this change in behavior known?
Is the new process behavior desirable?
Is it intended and expected that the new behavior will continue?

If the answer to all four questions is yes, then it is appropriate to revise the limits based on data collected since the change in the process.

If the answer to question 1 is no, then there should be no need for new limits.

If the answer to question 2 is no, then you should look for the assignable cause instead of tinkering with the limits.

If the answer to question 3 is no, then why aren't you working to remove the detrimental assignable cause instead of tinkering with the limits?

If the answer to question 4 is no, then you should again be looking for the assignable cause instead of tinkering with the limits. The objective is to discover what the process can do, or can be made to do.

Finally, how many data are needed to compute limits? Useful limits may be computed with small amounts of data. Shewhart suggested that as little as two subgroups of size four would be sufficient to start computing limits. The limits begin to solidify when 15 to 20 individual values are used in the computation. When fewer data are available, the limits should be considered "temporary limits." Such limits would be subject to revision as additional data become available. When more than 50 data are used in computing limits, there will be little point in further revisions of the limits.

So stop worrying about the details of computing limits for Shewhart's charts and get busy using them to understand and improve your processes.