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Using Other than Three Sigma Limits on a Control Chart

Authors: Marilyn & Robert Hart

Q: Most control charts use three sigma limits. When should I use other than three sigma limits?

A: Three sigma limits are used because in practice they have been shown to minimize the cost of the two mistakes: tampering with the process when it should have been left along (overcorrecting) and failing to take action on a process that needs corrective action. The probability of the first mistake is called the alpha risk or the false alarm risk. When an Xbar & R control chart is made with three sigma limits, it is typically for 25 subgroups of a small size (usually 3 to 5). In this case, the alpha risk is around 0.07 for the Xbar chart alone. So when fewer than 25 or more than 25 subgroups are used, it is desirable to "adjust" the number of "sigma limits" accordingly to keep the alpha risk similar to the 0.07 value. Ellis Ott approached thi problem in his 1975 book "Process Quality Control" using a method he called "Analysis of Means." His approach depended both on the subgroup size and the number of subgroups. A simplified method that we developed is to use the number of sigma limits (T) given in Table 1.

Table 1. Process Evaluation for Special-cause Variation: Recommended Values for T for T-sigma Limits.*

# of subgroups T
2 1.5
3-4 2.0
5-9 2.5
10-34 3.0
35-199 3.5
200-1500 4.0

*The tabular values of T may be used for the usual case of "no standard given" with all attribute and variables charts.

It should be noted that an Xbar & s chart is preferred to an Xbar & R chart since the s chart works with large subgroups as well as small subgroups whereas the R chart is only good for small subgroups. Also:

  • At least 25 subgroups of time-ordered data should be used to predict future performance
  • If fewer than 10 subgroups are used, the subgroup sizes should be 25 or larger.