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In the last article we discussed the XR Charts or rather how to start off a Control Chart. In this section we are going to discuss how we decide where to set the control limits?


Well, after all our data falls within the specification limits we’ll use the following formulas for the averages section.

for the upper control limit;

for the lower control limit.


Then the following formulas for the ranges section

for the upper control limit;

for the lower control limit.

The new variables seen in these formulas are factors developed for control limits calculations (I’m sure there is a story behind them but, I don’t know it). A partial table showing the factors involved follows.

n A2 D3 D4

3 1.023 0 2.574
4 0.729 0 2.282
5 0.577 0 2.115


Some of the factors, D3 in this case although not very obvious, are directly proportional to the number of samples (n) while others, like A2 and D4, are inversely proportional to them.

So let’s take some real data that I took several years ago from the manufacturing of a gas valve. That kind of valves have a screw that regulates the gas flow, and whose depth has to be controlled and kept between 0.157” and 0.250” of an inch. Substituting the values in the formula we get

UCLx= Total Average + A2 x Average Range
= 0.2049 + (0.577) (0.02365) = 0.219

LCLx = Total Average - A2 x Average Range

= 0.2049 - (0.577) (0.02365) = 0.191
Thus we have our two control limits. Our original specification limits were from 0.157 to 0.250 inches, so notice that the control limits narrowed such tolerance between 0.191 and 0.219 making the quality of the product inherently better than acceptable.

Although for the ranges section there’s no specification, its limits must be calculated too, so we can have a visual aid of our process’ behavior. Substituting the values in the formula we get

UCLr = D3 x Average Range
<!--[if !supportLists]-->(0) (0.02365) = 0<!--[endif]-->

LCLr = D4 x Average Range
(2.115) (0.02365) = 0.050

As it was explained in the last article the Ranges section must show a line looking as much as possible like a straight line. So even when the averages in the process maintain themselves within the control limits, but our rages show a tendency going up or down , it should be interpreted as a warning. Something is potentially wrong with our process and something must be done to keep the ranges within its new limits. Why are ranges so important if in average our process seems to be Ok?

Let’s remember that a range is the difference between a maximum and a minimum values. In one subgroup it may be possible to have a measurement very close to upper control limit and another too close to the lower limit, so thinking about it, why should a critical operation have such a swinging differences? Would you ride with a friend that suddenly and frenetically steers from one side of the road to the other with oncoming traffic? Ok, that was an extreme example but it was appropriate because SPC is applied to control critical operations like the one above: a gas flow regulating screw.

Now we have a visual of our process, what’s next? Well, depending on the outcome of your XR Chart you can either keep monitoring, or keep improving, or look for solutions. In the last part of this article we’ll talk about some common problems and some other techniques to improve our process.
Sent By : - Ivan