One-Two-Three Sigma Control Limits - Calculating Control Limits for X-bar Charts



Hi there,

I'm writing an online course on Quality Assurance for purchasing professionals. I have some prior background in SPC but I'm finding some advice that's contradictory to what I learned and I want to be sure I can explain different schools of thought...and even better, say which one is right when they contradict. This site has been great at explaining these details.

This issue concerns calculating control limits for X-bar charts. It's pretty clear that they should be calculated using factor "A2" from published tables and that ASQ in their "control chart template" calls them "3 sigma" limits. (Template is at

Now I'm trying to explain the Western Electric Zone concept which requires knowing not just three sigma limits but one and two sigma limits to see if the process is under control. The explanations I've seen seem to advocate using the sigma-bar value and multiplying it by 1,2, or 3.

Going back to the ASQ template, it calculates 1,2 and 3 sigma control limits that are not proportional to each other. The 1 sigma upper limit is much more than a third of the three sigma upper limit.

This is a convoluted way of getting around to asking how ASQ and others calculate those lower sigma limits. I haven't been able to get into their spreadsheet and look or find an explanation on line.

Thanks much

Dick Locke
Elsmar Forum Sponsor


Hi -

I think you have 2 questions here. To answer your calculation question first, I looked at the ASQ template and while it's a protected file, the math is the same, so here's what they've done:

To get the standard 3 sigma limits, they have used the subgroup size based constants (A2, etc). I chose a subgroup size of 5 and looked at the results for the X-bar/R chart. With a mean of 1.424, the UCL is 15.66. The template shows a +1 sigma limit of 6.171.

They arrived at this by taking the span between the mean and the UCL and splitting it into thirds: (15.66-1.424)/3 = 4.74.

Then 1.424 + 4.74 = 6.17 (I didn't round mid calc and got 6.169333)

For a +2 sigma limit, take the mean and add 2*4.74, for 10.92 (rounded).

Now the reason for doing this is really your second question:

The control limits help us determine whether the observed result is due to random or non-random behavior of the process. If the point lands outside thé 3 sigma limit, there is less than a 0.0027% chance of that being due to normal, random process behavior, so we call it "special cause" or whatever label your instructor put on it.

The Western Electric rules (and other similar sets) all pretty much follow that guide: look for things that are less likely than what we expect due to random chance at a 3 sigms threshold. So you see rules looking at number of points more than 2 sigma from the mean, or number of points within 1 sigma from the mean, and so on. Each of these is a test, which flags when you exceed the number of observations matching the (approximate) 3 sigma probability. Graphically, having 1 and 2 sigma limits on the chart helps people see this, but most stats packages (Minitab, etc) flag the rules for you.

Hope this helps.



Bev D

Heretical Statistician
Staff member
Super Moderator
I don't know what ASQ does or why.
I do know that the western electric rules as developed by western electric involve calculating the control limits using A2 (X bar, R) or A3 (X bar, S) which are based on the within subgroup variation (Rbar or Sbar). The zones are created by dividing the distance between the average and the control limit by 3. The zones are of equal size.


Replying to myself here...I figured it out. The 1 and 2 sigma limits are proportional to the 3-sigma limits in their distance from the x-double-bar line, not to the zero line.

I'm now back in a state of control.
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