X bar R Control Chart

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    Sonu S Andrews

    I have a small thought on the application of X bar R chart which I would like to share. Let us take a characteristic whose specification is 6.75/6.45. Now can we set a control limit by working back from the Cpk required as given below (in this example I have aimed for Cpk of 2)

    Cpk, Cp = 2 = USL-Xbar/(3 X Sigma)
    VSL = 6.75
    Xbar = 6.60 (mean of 6.75 and 6.45)
    Therefore 2 = 6.75-6.60 / (3 x sigma) and sigma = 0.025
    Rbar = sigma X d2 (from the formulae sigma =R bar/d2)
    = 0.025 X 2.33 (taking d2 = 2.33 for subgroup size of 5)
    = 0.06
    R Chart
    UCL = D4 x R bar
    = 2.11 X 0.06
    = 0.13
    LCLr = D3 x R bar
    = 0

    X bar Chart
    UCL = Xbar + (A2 x R bar)
    = 6.60 + 0.58 X 0.06
    = 6.60 + 0.03 = 6.63
    LCLBa = Xbar – (A2 x R bar)
    = 6.60 – 0.58 X 0.06
    = 5.57

    Hence would it be correct if I ask our production personnel to control the process with 6.63/6.57 as control limit for X bar-Chart and 0.13/0 as control limits for R chart to achieve a Cpk of 2.Can this principle be used to set the control limits based on the capability required.


    Neil Polhemus

    The basic principle of controlling to a standard is certainly sound, and solving backwards for the required sigma is commonly done. You might also check out the discussion of Acceptance Control Charts in Doug Montgomery’s book “Introduction to Statistical Quality Control”, where he works the specs into the establishment of the control limits from a different angle.


    Gary Cone

    This will only work if the inherent capability (Cp) is at least as good as your targeted capability. also make sure you have the ability to adjust to target.

    The setting of x bar to target is something I advise everyone to do (assuming we have the ability to adjust to target, sometimes we don’t). Why center your process at some arbitrary average when you know the desired target?


    Grant Blair

    You will first need to control chart your measurement system and determine its capability. Unless the limits for your measurement system are, at most, 1/2 of the limits you have calculated, you may have a problem.
    This will also depend upon your subgroup size.



    I don’t think back calculation is a good idea at all. First of all, as pointed out earlier, you can’t do better than setting the process target at other than the required target. If you do, then it makes no sense in dictating the limits within which you may control the process since it is an INHERENT property of the process and you cannot control as you wish!


    Shree Phadnis

    What you have shown is mathematically possible,but I would like to tell you that you are defeating the very purpose of the control chart that is whether the process has special causes or is only a process of commoncauses.Such a use of control charts will derail the purpose of control charts.While using control charts process capability is calulated on only the process that is stable.I request you to study Dr Demings pioneering book Out of the Crisis to understand control charts.


    Dr Phil Whateley

    In general the use of “tolerance based” control charts is not best practice, although it can be done. Control limits should be based on the actual statistical performance of the process. (The use of tolerance based limits would probably be a major audit non-conformance for QS9000 for example)

    Montgomery’s book (see other replies) is good in general, but I would recommend Bissell’s book “Statistical Methods for SPC and TQM”, precisely because it’s methods are Process rather than Tolerance based.


    Jeff Ayland

    I probably (99.99% sure) wouldn’t do it.

    1’stly get control, as a result of the process exhbiting only common cause variability ….once you have gained a “STABLE” process, then compute capability.

    (Oh No, not another big debate on what a “stable process” is !..)


    If capability isn’t at the desired level 1.66, 2.0, 3.0, 5.0 … or what ever you choose. Improve the process, the control chart will then “tell you” if you have made “an improvement” on an ongoing basis.

    Getting the guys in production to run to what are, effectively, compressed control limits, could mean they would interpret the signals given from the control chart “incorrectly”.

    Could those actions have a “cost” in the wallet ?

    I would just keep on “M – A – I” until i had the desired capability. Then run to a control chart in real time, and use the signals to deduce if “things” are “changing”.

    That’s me.



    Eoin Barry

    Setting specification limits based on proccess capability considerations underpins the six sigma DFM approach and is to be appaluded.

    However control limits are a function of the process itself and have no relation to the specification.

    If the fruits of six sigma are to be realised then SPC, which is expensive to implement and maintain, on the output specificaiton needs to be engineered out of the process.
    A process which is modeled correctly, is centred and has a has a process capability of 2.0 (n > 200 – CI 10%) is a candidate for removing SPC.

    SPC is really part of the cost of poor quality.


    Jose R. Deliz

    Control limits for X-bar R charts are the result of the inherent variability of your process (Man, machine, materials, methods, environment). The actual values indicate whether your process variability and mean are stable with respect to time. Thus Cpk will indicate whether inherent variability, process performance, meets customer requirement of Cpk = 2.

    When capability index is less than desired you must improve process to reduce variability and properly centralize mean within specs.

    Computing control limits from desired Cpk could be considered wishful thinking, not statistical thinking.


    Gary Cone

    All of these replies are precisely what the SPC books say.

    What if we have prior knowledge of the process? Isn’t that the basis of DFSS? If I build 6 sigma capability into my process/product design, wouldn’t I want to have a control chart that reflected these assumptions on day one before the first piece was ever produced?

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