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Topic Formula for D2 constant

Formula for D2 constant

Home Forums Old Forums General Formula for D2 constant

This topic contains 64 replies, has 43 voices, and was last updated by  SemiMike 8 years, 3 months ago.

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    I need to be able to calculate the D2 constant for any subgroup size, therefore, I want to know the formula that I would use to do so. (I would need a table larger than any that I have found so far)


    Jeez, can these questions get any easier???  Try this on for size and take it from here:
    Tables for d2 factors run from samples of 2 to over 100 (reference Statistical Quality Control, seventh edition, Grant and Leavenworth, ISBN 0-07-024162-7, Table C, page 717).I do not know the derivation of the d2 factor however it is the expected value of R bar divided by sigma of the universe at different sample sizes. For a discussion of the d2 factor see the text mentioned above or Quality Control and Industrial Statistics By Acheson J. Duncan. My edition is pretty old but I am sure it is discussed in newer revisions of the text.


    Hi C SnodgrassI’ve created an Excel spreadsheet that calculates d2 for n=2 to n=100 which I’ll be happy to send you if you post an e-mail address. There was some correspondence on this topic earlier this year.Best WishesBower Chiel


    Hello Bower. Thanks I hope this is right.  I know I saw this somewhere before but could not find it.
    I did a regression and came up with this formula, where x is the sample size: .3x  -.037x^2 + .0003x^3 – .0000017x^4 + 3.1 log(x)
    It is accurate to within .04 for all values of x, and the worst cases are the small sample sizes.
    I wish I could find that fornmula again.


    This is what I got with an r sq of .999
    d2 = – 0.3721 + 0.9419X – 0.09940x**2 + 0.003983x**3


    Darth, I tried a linear regression, and I had high r^2. But I looked at a scatter plot and saw that is was a log function. I had to play a little with the factors, sort of trade them off to get the answer I had. Take a look at your residuals, you will find taht some of the data points are quite a bit off. Probably the low numbers.  A regression has the best fit in the middle of the range.


    Michael, I used Minitab and fitted a cubic line. Didn’t get as far as a log.


    When Darth is showing up,Stan disappears


    Forum: 1
    Stanly: 0


    HB, how do you import a photo?


    right click/copy on an html version of the pic as you would find in an image search.  Paste in the message field


    You don’t, they just sneak across the border when the sun goes down.


    While admiring the pragmatic aproaches of Michael Mead and Darth might I suggest getting back to the underlying theory? It was published by the English industrial statistician L H C Tippett in the 1920’s, so on hand when Shewhart was developing the control chart. (So why is it often called Hartley’s constant?)To obtain d2 for sample size n you have to integrate the function: -1-(1-F(x))^n-[F(x)]^nfrom minus infinity to plus infinity. F(x) is the distribution function of the standard normal. There is no analytical answer so you have to resort to numerical integration. I’ve created an Excel spreadsheet to do the calculations. It is certainly accurate up to n = 100 and hopefully beyond. Anyone wishing a copy please post an e-mail address or send me an e-mail at WishesBower Chiel


    Thanks Bower,
    I just couldn’t find the formula anywhere. Where would we be without guys like Tippett and Duncan–not the most recognized names, but certainly pillars of today’s quality control.


    Could you please forward?
    Kind Regards,
    Shannon Mc


    Not a problem, my pleasure. I have sent it to the address provided so please be on the lookout for it. You won’t be disappointed.


    Darth,Could you please send me your spreadsheet of d2 values. I have some charts that have large sample sizes and cannot find the d2 values for the calculations….Thanks


    Shaun, my spreadsheet is for illustrative purposes and does not have the ability for handling large samples.  Another poster indicated he had one for large samples.  Possibly he will read this and respond to you.


    Darth, I am also interested in the work you’ve done computing d2 values.  Could you also please send me your spreadsheet of d2 values. ….Thanks


    Can anyone provide an Excel Shhet with D2 valuse in for subgroups of n=2 TO n=100


    hi G!!
    Did you get any valuable information or spreadsheet regarding the d2 value that you can share with me because I have the same dilema?
    my e-mail is 
    thanks in advance


    Hi LeeIf you post an e-mail address I’ll send you a copy of my spreadsheet.Best WishesBower Chiel


    I would be interested in that spreadsheet also.


    Can I also get the spreadsheet? I am desperately looking for formula for d2 as I need to find the relation between Cpk and Ppk (or/and Cp vs. Pp).  As the difference is only standard deviation I assume that d2 factor was calculated based on z-shift assumption of 1.5 – do I get it right?


    Hi InvestorPost an e-mail address and I’ll happily send you, or anyone else who is interested, a copy of my Excel spreadsheet that calculates the value of d2 for any sample size you enter.For random samples of size n from a normal distribution the mean sample range equals the appropriate d2 times the standard deviation, sigma. You therefore get an estimate of standard deviation from a series of sample of size n by dividing the mean range Rbar by d2. It has absolutely nothing whatsoever to do with a 1.5 sigma shift.Best WishesBower Chiel


    Please send d2 formula to my email address:


    Is there any way then to find relation between Cpk and Ppk? I will apparently be able to answer the question as soon as I getthe spreadsheet from you but maybe you Bower Chiel or anybody else can also help me with this by a quick answer?


    Cp and Pp (or Cpk and Ppk) are relates by formula only.
    Recall that Cp = (USL-LSL)/(6*std. dev) and Ppk is given by min(((x-LSL)/(3*Std. dev)), (ULS-x)/(3*Std. Dev)).
    For the Cp and Cpk, the value for Std. Dev is given by MR/d2  This is getting an estimation of the Std. dev using the part to part range.
    The Pp and Ppk uses the overall standard deviation (from excel, this would be like using the =STDEV( ) formula) to estimate the long term standard deviation.
    From my learnings, there is no other relationship, maybe someone else can shine a brighter light on this…?


    This is a message to all the people posting on this message.  If you’re doing Cp / Cpk analysis  to support product launch, product optimization, etc…why would you not go to a statistical package like Minitab or JMP or alike.
    Their formulas / macros to clculate these statistcs have been verifed, and in some cases even validated for accuracy.
    Just a thought…?


    It very simple.  Use six sigma table to calcluate sigma and this make it easy to draw control chart.


    Would like to get a copy of your d2 spreadsheet.thanks in advance for the help.


    Hi GordonPost an e-mail address and I’ll happily send you, or anyone else who is interested, a copy of my Excel spreadsheet that calculates the value of d2 for any sample size you enter.Best WishesBower Chiel


    Thanks in advance for the spreadsheet. Email is


    could you send me a copy of that D2 spreadsheet too?  appraciate it!


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    Its on the way


    Hi Need
    Post an e-mail address and I’ll send you the spreadsheet.
    Bower Chiel


    Hi Bower,
    Can you send me your speadsheet for calculation of d ?
    adress :
    Best regards


    Hi DenisI’ve sent the spreadsheet that computes d2. I’ve also sent a table of values of both d2 and d3 for n =2 up to n = 100 that I’ve created with the help of a mathematician at Edinburgh Napier University. This table also includes corresponding values of d2* for 1 to 20 subgroups. I’ll happily send both to anyone posting an e-mail address.Best WishesBower Chiel


    Hi Bower Chiel,
    Please snd me a copy of the Excel SS.


    Dear Bower Chiel
    I have great interest to know your an Excel spreadsheet that calculates d2 for n=2 to n=100. If you think it is  possible to send me a copy I’ll apreciate for sure.
    Bes Regards,


    Hi OscarPost an e-mail address and I’ll send it.Best WishesBower Chiel


    Thank You Bower for your promptly response. This is my e-address: 


    Please I should like not only to know about the tables on “constants for control charts.
    Dr. Wheeler and Dr. Chamber book, probably brings something about this question.
    But by books on the matter, most of them, refer to Standards like: ASTM, Mil, ISO. But which of these explain well the nature and how to get the constants. But not only on constants, but also on the matter: “Sampling acceptance on quality control”.
    My best regards,
    Mario Rio 


    dear Bowel Chiel,
    Please send me the table also. My e-mail address is
    Thanks and best regards,


    BC:I have done the Monte Carlo calculations for n=1-15 and and my results agree with the ASTM guidelines. Your spreadsheet for N=2 to N=100 AND 1-20 subgroups must be huge.This hobby project has turned out to be quite big, but has really grabbed my attention. I should be done by mid next week.Cheers, Alastair


    BC:I have calculated the table values for normal and non-normal distributions for n=2-50. Could you send your spreadsheet so I could compare the results?6sigmaguru(at)gmail(dot)comCheers, Alastair


    Bower Chiel,
    I’d love to take a look.
    heebeegeebeebb (at) gmail (dot) com


    Hi Bower,I am very interested to know how d2 is calculated for n=2 to n=100.
    I would highly appreciate if you could email me a copy of the Excel file you have.
    My email is ab.ssbb at gmail.comThanks


    Hi Bower,
    Can you send me your speadsheet for calculation of d ?
    Best regards


    Hello Bower Chiel:  I would appreciate a copy of the spreadsheet that computes d2.


    Hi Bower,
    I’d love to see that spreadsheet for d2 calculations too.  Thanks.


    I’d like a copy of the spreadsheet too.  ealanni(at)windstream(dot)net.
    Now a question:  To keep life simple on the production floor, I almost always (40 or 50 instances) have a sample size of 5.  In one or two instances a sample size of 10.   Because of the desire to keep the sample size small enough to keep things simple I do not need D2 values beyond what any readily availble source provides.  So, my question is this: What causes the need for the sample sizes of >100 and abandoning simplicity/speed of analysis on the plant floor?


    Hi Eugenie
    I’ve e-mailed you a copy of the spreadsheet as requested.
    With automatic data capture in some industries it is easy to monitor processes with sample size in excess of 25, the maximum value for which d2 is usually give in text books.  I wrote the spreadsheet in response to a query a while back and not because I had a need for d2 values for sample size in excess of 25. Hopefully others will be able to add further comments.
    Best Wishes
    Bower Chiel


    Hi Bower,
    Could you please send me copy of the d2 spreadsheet?
    Thank you in advance.
    Kind Regards,


    I want to know about hartley’s constant


    Hallo Bower Chiel,
    Thanks please send me the excell sheet for D2. I will also like to know How this Emperical relation deviced for Rbar / d2 = std deviation.
    My email Adress is


    Hi Bower,
    Pls send me the excel sheet. My email id is
    Thanks a lot,
    Mamta Sharma


    Hi Dear Bower Chiel,
    I am interested in D2 values for N2….N100
    My email address is or
    Please mail.


    Formula for D2 Constant
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    Rank - Aluminum

    Hi Bower Chiel,
    I am also interested in d2 values for n=2 to 100. Please send me the excel sheet.
    My email address is


    Hi Bower,
    could you send your D2 spreadheet on email
    Thank you in advance
    Best David


    could you send your D2 spreadheet on email Thank you in advance
    Best JP


    Let’s see, if I want to get a message to Bower I’m going to post a reply to a posting by… Skala, yea, that’ll do it.
    I heard that they can’t send this via e-mail.  Please tie a red bandana around your mailbox and they’ll print out a copy and drop it off.


    Hi Bower,I am interested also with a copy of your excel spreadsheet that calculates the value of d2..My email address: dpr6963@yahoo.comThanksDaniel


    I would be grateful if somebody could send me an excel copy of the d2 constants.  Many thanks


    Amazing interest in this subject. Hope every knows that MUCH of the time, subgroups in modern industry do NOT follow Shewhart’s assumptions of IID, naming INDEPENDENT data. MUCH of the time in MANY industries, the values in the subgroup are CORRELATED and if sample size is 10 or 100, the real EFFECTIVE sample size is usually 1 or 2 or slightly higher if studied carefully over 20 or so batches. I have personally NEVER seen large subgroup sample sizes showing effective sample size even NEAR those numbers.
    r-bar/D2 is dangerous much of the time.

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