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Sampling size X-R Chart

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  • #30604

    YH
    Member

    Hi,
       In production process, QC will draw certain sampling size per hour to measure specific parameter, such as dimension measurement.
       My questions are:
             1) How to determine the sampling size? Any formula? 
             2) According to this sampling size, X-R chart will be used to control the process.  What do the X-chart and R-chart stand for respectively?   If a point in R-chart is over control limit but the point in X-chart is within control limit,  what does this issue mean in the process?
          Look forward to your kind help. Thanks.
         YH. 

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    #79876

    Hemanth
    Participant

    Hi
    There is no specified formula for sample size in control charts, it depends on the cost and effort involved in measurement / inspection. I would suggest a book which will clear your doubts its Statistical Proces Control by Wheeler. Its a good book and will be helpful.
    Hemanth

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    #79881

    Jagdish R
    Participant

    All,
    A good min number of samples for a control chart is atleast 5. There is a significance of the samples size in control chart. The interpretation and the control becomes better with more samples but a min of 5 is recommended.

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    #79885

    RSloop
    Member

    Sample size is critical to the success of control charts.
     
    Sensitivity increases as sample size (n) increases.  Two types of error can result from making decisions based on control charts and these errors are dependent on each other and on sample size.
     
    Type I Error:   Alpha(a)
     
    a risk is the probability of concluding a process control element “out of control” when it was actually “in control”.  Results in over-adjustments and increases process variation.
     
    Type II Error:  Beta(b)
     
    Beta (b) risk is the probability of concluding a process control element “in control” when in fact it is “out of control”.  Results in missed process changes that allow the process to “drift”.
                                                   
    The two errors are dependent on each other.  As the risk of over-controlling (alpha (a) risk) increases, the risk of missing process changes (beta (b) risk) decreases.  The risks are related to sample size with the sensitivity (alpha (a) risk) increasing as the sample size increases.
     
    To determine a sample size (n) where the shift (D) is known:
     
    Sample size (n) = [(Z value of alpha/2 + Z value of beta)2     multipied by the sigma2] / D2
     
    Note:  Z value is taken from a standard Z table.  D is the shift that you wish to detect.
     
    If you have a sample size and wish to know what shift it will detect you can use the following formula:
     
    D = the square root of [(Z value of alpha/2 + Z value of beta)2     2     multipied by the sigma2] / n
     
    It’s hard to enter a formula here, but if you’d like, post an email and I’ll be glad to send better copy.
     
    Rick
     
    SPC Solutions

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    #79886

    Jagdish R
    Participant

    That’a an excellent and good explaination. Along with the Alpha and Beta risk going a step further will be confidence intervals.
    But good explanation
    Jagdish

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    #79889

    RSloop
    Member

    Good point Jagdish….
    Rick

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    #79920

    YH
    Member

    Thanks a lot for these useful information. I noted that there is a calculator linked in this website to calculate sampling size by inputing Alpha, Bete & others.  Thanks again if you could send me some bettere copy by e-mail. My e-mail address:  [email protected]
    Regards,
    YH
     

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    #80290

    Yee
    Member

    Where I could find this formula?  Are there any books that give formulas for different types of control charts?  Thanks for your help.
    Yee

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    #80292

    Gabilan
    Participant

    Hey, Rick this is a very good explanation.
    Could you email me the formula ?
    [email protected]
    Thanks.

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    #80293

    RSloop
    Member

    Yee,
    If you don’t mind posting your email I’ll be glad to send you what I have. 
    Rick

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    #80307

    singh
    Member

    Hello Rick,
    Its a very good eaxplanation.  Please mail me the formula at [email protected]
     
    Thanks.
    vaibhav

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    #80318

    Lin
    Participant

    Hi Rick,
    good explanation, i’d like to learn more. can you send me the formular for reference? thanks.
    Lin

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    #80319

    RSloop
    Member

    Lin,.
    I can’t get the formula listed here correctly.  If you leave your email I’ll be glad to send it to you or if there is a way to post it here for everyone I’ll be happy to do that.
    Rick
     

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    #80320

    Ron Bertani
    Member

    Hi Rick,
    If you would’nt mind please forward the formula to [email protected]
    Thanks,
    Ron
     

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    #80321

    Dave Strouse
    Participant

    Rick –
    The formula you are proposeing is seriously in error. It is simply the sample size formula for a two sample Z test. However, SPC does not normally use sample sizes consistent with this analysis.
    A two sample t-test is a close approximation to the correct method. Please see Wheeler, Montgomery or any other author on construction of  OC curves and ARL for the most precise ways to do this. Your method and the accepted method do appear to give better covergence with larger shifts, however the differences are still significant.
    Note that according to your formula, detecting a one sigma shift with alpha and beta at 95% would require (1.64 +1.64) squared or 10.75 samples in the group. Call it eleven. Using the simple 2 sample t-tets power calculator in MINITAB for this same problem shows groups of 27 required. You will find this number in close agreement to those published in references above.
    Bottom line is that you will have a low probability of detecting a one sigma shift using detection rule one alone on the first sample using the eleven samples per group given by your method.
    Wheeler”Advanced Topics in Statistical Process Conbtrol” is an excellant reference for further study.
     

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    #80323

    RSloop
    Member

    Dave,
    You made a few good points, but I think that not being able to post the full formula created some confusion.  In the example you present you have both alpha and beta at 95%, actually they are dependent.  As one increases the other has to decrease.  If beta is 95% then alpha is only 5%.  You also have to use only 1/2 of the alpha in this equation.  I have used ARL and OC curves, but in my experience the OC curve was used for acceptance sampling and not determining sample size for control charts.  I was also wondering what sigma you used for your example.  27 is a very large subgroup for a control chart and seems more in line with acceptance sampling of a lot.  It would seem that almost any chart with a subgroup of 27 would be overcontrolled. 
    If you have any info on using OC for control chart sample size I’d be interested in seeing it and if you would like the formula for this method leave your email and I’ll be glad to send it to you.
    By the way, it nice to have some debate on these things!  It help me to keep an open mind and learn some new methods.
    Rick
     

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    #80324

    Vallejo
    Participant

    Please, Rsloop
    My mail is [email protected] and I would like to receive THE formula… If you can…
    This subject is a problem when we need to estimate sampling size with a PPM error, and not with a percent… OCC has this limitation 
    Thanks
     

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    #80327

    Dve Strouse
    Participant

    Rick –
    I did miss the alphas halves initially, but it does not change the analysis.
    What makes you think alpha and beta cannot be the same? The constraint is that (1-beta) , the power must be greater than the significance, alpha. As alpha increases, beta does decrease but only if sample size is held constant. When sample size varies, this is not true.
    Yes, 27 is a stupidly large subgroup. I only used it as a very extreme example. Normally, no one will want to detect a 1 sigma shift on the next sample anyway. It’s too small to be concerned overly with. However, common practice when evaluating a control chart sensativity is to key on a 1.5 sigma shift.
    References to OC curves and ARL to evaluate sensativity of SPC are
    “Process Quality Control”Ott, Schilling,Neubauer pg 74 of the third edition. Start with this one.
    “Principles off Quality Control” Banks ,1989, pg 202.
    Banks prints a figure from Douglas Montgomery “Statistical Quality Control,1985”. Don’t have that one handy but since the figure is in Bank’s book, he must discuss it.
    “Advanced Topics in SPC” D. Wheeler, 1995. Chapter 9 is about power functions i.e. OC curves. Read it carefully. Dr, Don has a slighthly different but equivalent methodology. He uses the s.e. rather than s.d to show shifts. It will likely throw you on first reading.
    Pretty sure Grant and Leavenworth also cover this extensively. I’m sure I can find more, but none other handy at the moment.
    Common practice in my mind for sample size evaluation in a Xbar chart is –
    1) know what shift you wnat to detect with what proibability
    2) evaluate the standard charts of 4 and 5 piece subgroups for those measures.
    3) if they don’t get it done, consider more frequent sampling to lower the ARL or increase the sample size. Don’t neglect the comcepts of rational subgrouping.
    Can you provide any reference to the use of the sample size calculatoir for the two sample Z test you have shown here in evaluating control chart sensativity and subgroup size selection?
    Regards –
    Dave
     
     

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    #80333

    RSloop
    Member

    Dave,
    I’ll definitely re-read chapter 9 of “Advanced Topics …SPC” by Wheeler.  It’s one of the best SPC books around and even though I’ve been through it repeatedly, I’m sure that I could miss something.
    As far as references for the formula you can look up Sample Size for a Two-Tailed Hypothesis Test about a Population Mean in about any statistical text.  There is a great description in the “CQE Primer” from the Indiana Quality Council (section XI – 9), the class text that I sometimes teach from, “Statistics”, by Anderson, Sweeney and Williams explains it on page 325 of the third edition and Pyzdek gives a pretty good description in his “CQE Handbook”, page 287.  It is a pretty standard test.
    Maybe part of the misunderstanding is that you are thinking about a shift in the sigma.  The shift in this formula is the shift from the nominal to the actual average.  Sigma is taken into account, but the shift is the difference in the average itself.  You also have to be careful not to confuse Acceptance sampling with control chart sampling.
    By the way, I agree with your 3 steps.  They present the most practical way and having a rational subgroup can be much more important than the sample size.  Good point.
    Best Regards,
    Rick

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    #80340

    RSloop
    Member

    Dave,
     
    I did read over chapter 9 of “Advanced Topics in SPC”, Wheeler, again.  The method that Wheeler uses takes into account the number of subgroup measured before the shift is detected.  His method would result in higher sample quantities for a k=1 (1 subgroup) detection, but similar sample quantities as the formula I listed at higher k levels.  The formula that I presented is a basic stat formula used for sample size in normal charting applications.  The sigma used in the formula is usually based on a preliminary process capability study which includes more than one subgroup of data.  I’m sure that Wheeler’s method would give better results in a critical situation.
     
    Note:  after reading over some of my postings in this string, I didn’t like the way I explained the procedure and would like to make it a little clearer.  I mentioned that the risks are dependent and that as one increases the other decreases.  This is true, but the formula takes this into account.  As an example;  in most cases you would use an alpha (probability of concluding the process out of control when it is in) value of 0.0027 (for a normal distribution 99.73% within 6 sigma and 0.27 expected outside of 6 sigma) and thus with the 0.0027/2 or 0.00135, a Z of 3.  The alpha won’t change.  The risk change occurs as you set the beta value.  If you wish to be 90% certain that you detect a shift you would use a beta of 1 – .90 = 0.10.  You would be estimating a 10% probability that you would not conclude the process out of control when it actually is.  Therefore the Z for 0.10 would be equal to approximately 0.255.  You also have to use a Z table for Normal Curve areas.  As you change the beta the numerator changes, remember that you are simply adding the Z values, which adjusts the sample size.  As beta increases the sample size increases. 
     
    It’s hard for me to explain something without being able to draw on a white board, but I hope that this helps.
     
    Thanks for the tip on the Wheeler book.  I had not read that chapter in much detail and I did enjoy it.  I have also enjoyed this discussion and the feedback.
     
    Best Regards,
     
    Rick

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    #80514

    Hectir Jaramillo Macías
    Participant

    Hi Rick
     
     

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    #80515

    Hectir Jaramillo Macías
    Participant

    Hi Rick
     
    Your explanation about sample size on X-R charts is very interesting, could you please send me more infoprmation about the formula, my e-mail is [email protected]
     
    Thanks before hand

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    #80527

    Tedy
    Member

    Hi Rick,
    Could you please send me more information about the formula, my e-mail is [email protected]
    Tks & Regards,
    Tedy

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    #80533

    Ron
    Member

    Your very simple question has evoked a plethora of answers mostly incorrect.
    1) Process samplng is very different from population sampling which is what most people replied to your with. For an Xbar and R chart your sample should be the most you can take between 5 and 20 samples.
    2) The key to moving range charts is to watch the data in the range charts to determine process stability. If it is often oscillating out of control your process is out of control and needs to be stabilized prior to doing anything else.
    3) The XBar portion of the chart tells you your between group variaition. The Range protion of the chart tells you your within group variation.
    Great book by D Wheeler Understanding Statistical Process Control is amust read.

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    #80558

    YH
    Member

     Hi Ron,
          I still have a question about the sampling size in Xbar-R Chart:
               As mentioned in your reply of item 1), normally, the sampling size is 5-20.   Assume the process is stable with higher process capability such as CPK = 1.5, I believe it is OK if drawing 5 samples from 200pcs products per hour;  but, how about the sampling size for 20,000pcs per hour, still 5pcs or increasing to 20 or above?  Pls advise.
    Regards,
    YH.

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    #80561

    RSloop
    Member

    Ron,
     
    The formula that I posted is a standard method of estimating the sensitivity of a X bar R chart at different sample quantities.  It’s used in most control chart training courses and in the majority of statistical handbooks.  I really didn’t think that there was anything controversial about it when I made the first post. After some of the response, it is clear that there is some confusion about acceptance sampling and control chart sampling and on the affect of sample size for control charts.  As sample quantity increases the sensitivity of the chart increases.  I’m sure that Dr. Wheeler would also tell you that using too high a sample quantity will lead to over control of the process.  Higher sample quantities result in tighter control limits and increase the sensitivity of the chart.  I would therefore definitely not advise someone to use as many samples as possible (you stated up to 20), but to determine the least quantity that gives them the sensitivity that they require.
     
     
    Regards,
     
    Rick

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    #80571

    Ron
    Member

    The main function of your control chart is an indicator and predictor. At high volume production you may wish to utilize a EWMA chart and take 5 samples every 10 or 15 minutes until you are comfortable that the process is in statistical control.
    Once you have substantiated that statistical control is in effect you can adjust your sampling scheme. The number of pieces going through the system is not as important as the state of variation within the system.
    Another consideration is the speed at which you can perform the inspection function. If it take you say 2 minutes per samples the pull 5 pieces every 15 minutes.
    The problem with most statistics books is that they are hung up on population sampling. Very few statisticians have real life experience in a manufacturing setting. In Wheelers Understanding Statistical process control which I consider a great topical reference book you won’t find any equations for calculating subgroup sampling. Shewhart coined the term rational subgrouping. Enough said on that.
    The key to obtaining good data is the intent of your use of that data. In your case for a period of two or three days I would continuously sample subgroups based on the time to inspect function. Record all pertinent data  shop temperature machine operator time of day inspector, gage number etc. then create your control chart based on these data. Then spend a lot of time slicingand dicing the data to look for sources of variaiton.
    You did not say how well your process is performing. If it is very good you would want to take more samples per sub-group than if it is very bad. The logic is obvious in that it is ore difficult to detect defects in a good running process. 
    Key me posted on your results.
     

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    #92020

    Lin
    Participant

    Rick,
    Could you

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