The Use of Three-Sigma for Control Limits

Robert F. Hart, Ph.D.
Marilyn K. Hart, Ph.D.

Three-sigma (σ) limits are generally used for control limits on a Shewhart control chart. Why? Well, Shewhart [1931] and Deming [1975] would tell you that they have been shown to work well in practice, that they minimize the total cost from both overcorrecting and under-correcting. But you have to remember that control charts traditionally have had 25 subgroups. The standard American Society for Quality (ASQ) paper that people used when we were still doing control charts by hand were set up for 25 subgroups. For normal data with a known mean and standard deviation (the case of standard given), the probability of getting one or more of the 25 points outside the control limits by pure chance is around 0.065. That’s because the probability of a given point being inside the 3σ control limit would be 0.9973. The probability of all 25 points being in the limits is 0.997325 = 0.9346. Hence, the probability that they are not all in is 1 – 0.9346 = 0.0654. We call this probability the false-alarm risk, the α-risk, or the probability of a type 1 error. This is the probability that the process really is in control, i.e. stable (and in our case normally distributed), and we still get an indication of lack of control by one or more points outside the control limits. Note that this is close to the typical value of α= 0.05 that is used in hypothesis testing. When the control limits and the corresponding mean and standard deviation are estimated from the data (the case of no standard given), the calculations become even more complicated.

When control charts are maintained over an extended period of time, the number of subgroups grows. With more points, the probability of points outside the control limits by pure chance grows, i.e., the α- risk grows. This problem has been acknowledged in the American National Standards Institute (ANSI) Standard Z1.3 [1958, 1975, p. 18]. It states that

it is usually not safe to conclude that a state of control exists unless the plotted points for at least 25 successive subgroups fall within the 3-sigma control limits. In addition, if not more than 1 out of 35 successive points, or not more than 2 out of 100, fall outside the 3-sigma control limits, a state of control may ordinarily be assumed to exist.

No mathematical basis is given and no suggestions are given for the evaluation of more than 100 subgroups. Similarly, the number of subgroups may be smaller than 25. For instance, you may be comparing the results of three shifts. This is Shewhart’s concept of rational subgroups.

One method for keeping a satisfactory a-risk (close to 0.05) is to use control limits other than 3s limits, which may vary with the number of subgroups and possibly with the subgroup size. The objective is to maintain the highest possible power in detecting special-cause variation without getting an a-risk that is too high. One approach developed by Ellis Ott [Ott, 1975; Schilling, 1973] is called Analysis of Means. It is a more sophisticated method of selecting the control limits to be used for a given alpha risk depending on both subgroup size and the number of subgroups. It requires extensive tables and is seldom used in practice. Studies done by Hart and Hart [1994, March 1996, April 1996, 1997] have simplified the methodology by the use of "T-sigma" limits, which keep the total a-risk to less than 0.09. The results for no standard given are provided in Table 1. [Hart, 2002]

Table 1. Recommended Values of T for T-Sigma Limits, No Standard Given

# of plotted points T
2 1.5
3 - 4 2.0
5 - 9 2.5
10 - 34 3.0
35 - 199 3.5
200 - 1500 4.3

Note that 3σ limits are consistent with the usual use of 25 subgroups. For ongoing process improvement, 3σ limits are often used to project into the future for use as standard given to continuously monitor the process and seek out special-cause variation. However, with rational subgroups (which may use very few subgroups) to compare only a few groups, control limits tighter than 3σ limits are used. And to simply judge whether a process is in control where you have many plotted points, use control limits that are wider than 3σ limits or you are too likely to falsely indicate lack of control.


ANSI (American National Standards Institute) Standards Z1.1, Z1.2, Z1.3 (American Society for Quality Control Publications B1 and B2). "Guide for Quality Control; Control Chart Method for Analyzing Data; Control Chart Method for Controlling Quality during Production." New York: American National Standards Institute, 1958, 1975.
Deming, W. Edwards. “On Some Statistical Aids toward Economic Production.” Interfaces, vol. 5 (4), pp. 1 – 15, August 1975.
Hart, Marilyn and Robert Hart. Statistical Process Control in Health Care. Duxbury, Pacific Grove, California, 2002.
Hart, Marilyn. "C Control Chart Limits for a Given Number of Subgroups." Proceedings of the Midwest Business Administration Association, Chicago, pp. 71 – 74, March 13 – 15, 1996.
Hart, Marilyn. "R and s Control Chart Limits for a Given Number of Subgroups." Proceedings of the Western Decision Sciences Institute, Seattle, WA, pp. 335 – 339, April 2 – 6, 1996.
Hart, Marilyn. "NP Control Chart Limits for a Given Number of Subgroups." Proceedings of the Decision Sciences Institute, San Diego, vol. 2, pp. 994 – 996, November 22 – 25, 1997.
Hart, Marilyn, and Robert Hart. "X-Bar Control Limits for an Arbitrary Number of Subgroups." Proceedings of the Western Decision Sciences Institute. Maui, Hawaii, pp. 637 – 639, March 29 – April 2, 1994.
Ott, Ellis. Process Quality Control. New York: McGraw-Hill, 1975.
Schilling, Edward G. "A Systematic Approach to the Analysis of Means." Journal of Quality Technology, pt. 1, vol. 5(3), pp. 93 – 108, July 1973, and pts. 2 and 3, vol. 5(4), pp. 147 – 159, October 1973.
Shewhart, Walter A. Economic Control of Quality of Manufactured Product. Princeton, NJ: Van Nostrand Reinhold Co., 1931. (Republished in 1981 by the American Society for Quality Control, Milwaukee, WI).

For more information, contact Drs. Robert and Marilyn Hart at or (541)412-0425.

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