Statistical Explanation of Six Sigma

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    Lin Guo

    I have the following calculation based on a normal curve. The central area is the area enclosed by the number of sigma. For example, the central area is about 68% when the area is defined by 1 sigma on both sides of the area. The side areas are the two small areas outside the central areas. The total area under a normal curve is 1. Therefore, the side area is obtained by 1 – central area. The DPMO is the side area divided by 2.


    Central Area

    Side Areas





























    Motorola engineers argue that the center of the distribution is often drifted to either side of the center by 1.5 sigma. Thus, the DPMO for 4.5 sigma instead of 6 sigma should be used. Also, the reason to divide the side areas by 2 to obtain DPMO is that customer requirement is usually one-sided. If the normal curve represents waiting times at a drive-by window of a fast food restaurant, for example, customers care about the fast deliver of the order.  In other words, waiting times located in the upper side area are the defects because they do not meet the customer requirement while those in the lower side area are not a concern.



    You are mistaken.
    This will explain it for you :



    Lin Goa:
    When I grow beans in my garden, some years I have a lot of beans and some years less. I never have found ‘defect’ beans – how do you explain that?
    If my yield is less then my expection – are the missing beans all defects?
    Also, releatively few Motorola engineers bought into Dr. Harry’s vision of process shifts because unlike Harry they actually had process data!!!



    Hi Trev,
    supposing you had a contract to deliver a given amount of beans to a customer each year?



    I’d put something in the contract about ‘seasonable’ yield variation.
    A similar problem arises with demand variation. Taking a previous example about Pizza delivery, when there are too many customers workers can’t just shut the door and let no one else come in until they catch up. Why penalise the workers? The problem is in the system to coin a phrase!
    There are many examples in industry where workers are beat up for not meeting targets when there is no possiblity of corrective action. Sales is just one of them!
    Why measure someone’s performance if there is nothing they can do to change or improve it. 
    The point of my email is low yield is not always the same as a defective performance. Having found this anomoly, quality professionals should address it instead of just regurgitating something a training manager wrote in a business book.



    Hi Trev,
    I think we are looking at this from very different perspectives and we could both be right :D.Concerning the yield: your customer might not agree that you simply download the risk of seasonal variations to him, s/he might even refuse to sign such a contract. (I know I would..).There are other things you could do to reduce your risk – from buying options to analysing your yield and contracting only the amounts that you know based on your data that you can deliver with a 99,99 probability etc. etc.Thinking about your pizza example .. I think the difference between our viewpoints is that I see the defect as “process problem” and you seem to imply that the defect is seen as a fault of the people working in the process. If this is the case I cannot agree with you more – it is such a stupid things to do that I don’t even want to talk about it.On the other hand I can not agree with your conclusion that it is pointless to measure the performance in this case – because we can use the measurement to improve the work process instead of using the data to “beat up” someone.What do you think?



    It is good to be able to debate this important issue with a reasonable man.
    I have no problem with your take on the situation – my problem is the notion of excessive wait time as a defect.
    My counter argument is a low yield does not necessarily imply many defects. I can support my contention with another argument; one which you also alluded to.
    Chemical reactions have a yield that depends on the  reaction. This yield is not limited by defects in the conventional sense.
    By the way, I agree with your Pizza process argument.
    In the case of wait time – why not just use yield? Why focus on defects? The very notion of a defects implies someone didn’t do their job properly when the problem could equally be due to unexpected demand :-)



    I would still stick to the defect term, because from the customer POV long wait times will be defects. And I agree that this means someone did not do their job properly – but I strongly think it it was the designer of the process and not the people actually working in the process.For the sake of an example imagine having to wait at the security check at an airport. The demand can be forecasted with high accuracy – and still sometimes (or at some airports :D) you just see the crowd in the queue growing and growing.
    No one would blame the poor guys doing the checks but I often do have some warm thoughts regarding the people who decide the number of checkpoints etc. I think the long waiting times are a sign of poor organisation and high disregard for the customer – a defect in short :)regards



    I really don’t understand your position. You seem to assume the only way to have a high regard for customers is to call an exception a defect?
    Your point about ‘poor organisation’ goes to the very heart of the matter, which is exatly my point. Low yield implies a common cause of varation, and not a special or localised cause, in contrast to the term defectm which seems to imply a special cause, or localised cause, of variation, presumably based on the principle of ‘command and control.’
    The fact of long wait times at airports appears to be a planning, management, and forecasting, limitation, and also a consequence of political correctness, since most people in my country believe it is wrong to use statistical profiling to screen passengers.



    Good analogy.  Have you read the article I suggested ?
    Harry’s “shifts” were theoretical, based on tolerances in stacks of disks … believe it or not … but after all Harry is just a mad psychologist. The “shift” was orginally Bill Smith’s and based on processes that were out of control !!!



    As a witness, I agree with 98% of the article.


    Dead Horse

    Has anyone counted how often this six sigma shift horse has been beaten to death on this site? It’s dead, leave the caucus alone …


    Dead Horse

    congratulations. u passed the language test :-))))))) … good to see that u at least know the meaning of some basic words. now it’s time to catch up with what’s going on in the real world. … i think the blessing is for people like u. ur obviously totally living in the world of yesteryear …



    Do you know what a “caucus” is, my silly six sigma friend ?
    “Any group or meeting organized to further a special interest or cause.”
    Perhaps this caucus does have an interest in what the term “six sigma” means and just how much utter nonsense it really is !!!
    God bless


    Behind the times

    “The fact of long wait times (great grammar) at airports appears to be a planning, management, and forecasting, limitation, and also a consequence of political correctness, since most people in my country believe it is wrong to use statistical profiling to screen passengers”. what words of wisdom (since most people in my country …) … so many words, so little substance … . please keep lecturing us, we’re in need of such profound observations … not to speak of the theory of low yield and defect … wow, that’s a brainstorm!



    ‘Behind the times?’ What does that mean! What a joke!!!
    I bet you didn’t even know chemical reactions have a yield! Go back to your catapault and disolving asprin you silly, little prat!


    Pete B

    I guess you lost the argument …



    Hi Trev,
    I only think that a defect is a non-conformance to customer requirements. Now, whether this happened just once by accident (what you might call an exception, or a special cause) or it is by design (common cause, low yield etc.) is something that comes afterwards.It seems we have very different backgrounds and that is kindof colouring our arguments :). I do not associate any further meaning to the term “defect” – on the other hand I strongly belive that calling a duck a duck is the right thing to do :)Regards



    It is sad to see that you fail the language test my dear friend.  We cannot all be blessed with intelligence.  Perhaps you should learn some basic words so that you can start to communicate. 
    God bless, my little man.


    Dead Horse

    Trev, regarding language “La-Kot-A” … Kot means “crap” in another language (your people couldn’t have given you a better name to describe the type of theories that you disseminate on this site. be happy to have an iq of 16, a dead pig has one of 25 :-). but you win the official price of being the bxxlshxx artist of the site :-)))). go back and play with your little chemical reactions …


    Dead Horse

    What irony: With all of her wisdom in alchemy, lakota/trev cannot even differentiate between questions of customer specification and process control. Ooops, what an intellectual, cute litte short circuit in the intellectual brilliance of our new Mother Superior. And what contribution to the collective knowledge of this site. I am waiting for more nonsense to come forth from “La-KOT-A” … kot, nomen est omen.



    Hi Sandor,
    Yes, we do seem to have different backgrounds. I’ll make this my last post on this subject and leave the last word with you.
    My local rail company post their performance as follows:
    Last month 95% of trains arrived no more than 9 mins. late, which is just inside customer’s (government) requirements.
    This looks like a yield to me, and it seems easy to understand. Furthermore, it doesn’t imply 5% of trains were defective!
    I won’t even bother to put it into Six Sigma terms – I’ll leave that to you. If you need the number of trains, the number of carriages, the number of seats per coach, the number of signals, the number of guards, etc., let me know.
    Just kidding :-)


    Gaga talk

    Thanks for ending this trail of baby gaga talk about misconceptions of “yields” and “defects”, “pizzas”, “chemical reactions”, “wait times” and “trains” etc. It takes quite a confused mind to call this “an important discussion”.



    Here we have it. The typical response of a Six Sigma consultant. No wonder it’s called SS.



    Loser …



    In your example, the trains weren’t defective.  That wasn’t what was being measured.  The system’s ability to meet the goal of 9 minutes was the measure.  In this manner, you were 5% defective in meeting the customer’s goal.
    I tried to stay away from this thread, but just couldn’t.  Time can be a defect if it is a KPI for a customer and you don’t meet the expectation.  The rest of the argument is irrelevant.


    Tony W

    I also tried to stay away from this argument ..
    What you seem to be saying is a difference of a half a millisecond could separate a non-defect from a defect based on the customer’s KPI.
    I thought Six Sigma subscribed to the Taguchi Loss function?



    “I thought Six Sigma subscribed to the Taguchi Loss function?”
    No.  Six sigma is based on defects – good at one point, bad at the next – a step function.  Taguchi assumes a continuously varying function, which of course is much closer to reality.


    Tony W

    Thanks for the explanation.
    I wondered how two wait times with a similar loss could be considered so differently.
    I assume Taguchi’s loss function also allows us to weigh the loss in terms of importance – unlike Six Sigma which considers all defects as ‘critical’ – even a slight delay.



    I also have been trying to stay away from this . . . but there seems to be some confusion here.  I have been using Six Sigma for almost 10 years and never have considered ‘Six Sigma’ as a step function.  Six Sigma is a process for understanding and improving process variation.   All process results vary — both Taguchi and ‘Six Sigma’ consider all variation to have a related cost — there is no difference here.  Process variation is the ‘voice of the process’.  And all variation has a cause — (further, as Deming stated, sometimes the causes are ‘normal’, and some are considered ‘special’.)
    The customer determines the acceptability of the variation, and puts requirements on what is acceptable or not acceptable. Since the customer ‘has to draw a line in the sand’, there can only be one minimum and/or maximum acceptable limit to the variation — if you want to call this a step-function, then do so — even Taguchi understood that the customer defines the requirement.  And Six Sigma, like Taguchi, and even the customer, understands that those results slightly out of specification (nano-seconds in one case) are not much different than those within the specification — but that is the nature of ‘voice of the customer’.  If the customer has a time expectation, the trains or the beans are not being measured (although the customer could have a quality expectation about the size of the beans or the cleanliness of the trains . . ), but the process’ ability to meet the time expectation. 
    I don’t understand the hang-up about calling a process result not meeting expectation as a ‘defect’.  This has been standard terminology for many, many years prior to Six Sigma.  Anytime a product did not meet an expectation it was considered ‘defective’ — there should be no reason to develop new terminology if we are now applying specifications and requirements to other types of processes (e.g., timeliness of trains).  If you understand statistics and quality, you also understand that there is small/no difference between some result at the extreme, but still in spec, and another similar result just slightly out of spec — but again, there has to be a separation process — unfortunately, while statistics can give us std. dev., ‘Taguchi’s Loss Function’ and 256 shades of gray — customers have to separate things (process results) as ‘good’ or ‘bad’ which is binary. 
    Six Sigma is focused on understanding the cause for the variation, and reducing and controlling it. Although the statistics (and the customer, or Taguchi LF or Deming) may consider those results slightly within spec and those results slightly out of spec having similar attributes, in reality, they may have different causes for variation — this is true whether the process variation is normal or special. I think that having an ability to investigate and understand the various causes (or the same causes) for these results represents the success of Six Sigma — maybe someone thinks they have better method for getting to this level, and I’m not that much of a zealot to try something new, but I haven’t seen it yet.


    GrayR, a refreshingly unideological post, that is supported by well-informed experience with the subject-matter.
    The real issue that needs to be addressed by (hopefully) the next generation of six sigma (or its replacement) is the question of how to define and deal with defects when process input, throughput and process output are not separated, i.e. in services settings. In the transactional world we continue to see the ideological divides between proponents of Shainin, Taguchi, Deming and Harry which are all good and well in tangible product settings. However, this ideological discussion misses out on the insights from marketing oriented efforts (service quality, loyalty, marketing relationship theory etc.) that have also gained interesting insights into how to deal with the measurement and impact of service failures as opposed to product failures. I find it interesting that quality had very dynamic discussions up to the beginning of the 1990s, and has since then ossified into groups of opposing schools that literally have missed out on the fact that the world has moved on. Anyway, I enjoyed your post.



    Looks like you’re a believer of six sigma. I have a question about sigma level, are the sigma levels for counts & variable data the same?



    What is a delay called in Lean? I think it is either called a delay or muda. (Is muda the same as a defect?)
    What is a delay called in Six Sigma? You wrote it is called a defect.
    What is a delay called in Lean Six Sigma?
    Is this site only about Six Sigma? Is there a site for Lean Six Sigma?


    ex SME

    So far you guys have figured:
    1. Mike Harry the mad psychologist came out with the “1.5 shift”
    2. Bill Smith came out with the name “Six Sigma”, which is wrong because it is plus or minus 6 sigmas, so it is actually twelve sigma.
    You guys are missing who came up with the Methodology, and that was:
    3. Mario Perez-Wilson came up with the 5-Stage statistical methodology (MPCpS) that eventually became the Six Sigma DMAIC methodologySo, from these three who is the true father of the six sigma child?
    This is better than finding the father to Ana Nicole Smith’s child! :-)Cast your vote:[1]


    Manjit, You are really dealing with three issues:
    1. Product quality: This is where the “classical” defect definition comes from which are derived from customer driven engineering specifications and a pass/fail of the requirements. The costs associated with these defects are the cost of quality indicatores developed by Juran.
    2. Process throughput: This is where the “classical” waste (muda) definitions come from. In this context, a defect is a waste to the degree it requires rework in the process.
    3. Service Quality: These are “defects” related to the service aspects of the product (you order a book at the book itself is in great shape, but you paid for next day delivery and receive the book in four days). Within the service quality framework of for example Zeithaml, Barry and others, delays would fall under the umbrella of “reliability”. The service quality model assumes five types of service failures: tangibles, reliaibility, emphathy, assurance and individual caring attention. However, in the service quality type of framework, defects are not as statically defined as in the production world, because there are compensating factors, most importantly service recovery. Thus, ironically, in a service type of environment a “defect” when it is dealt with appropriately can actually lead to higher satisfaction, which is the goal of the quality movement no matter if it is service or product related.
    My observation is that in the current quest (craze?) for differentiating between lean and lean six sigma, which is really an American distinction that the Japanese never made (it was always lean and kaizen … I posted that in a previous post), Americans have forgotten one of its own key contributions to the idea of “quality”, i.e. the extension of product related quality, optimal production processes and the service aspect of product production and marketing. In my opinion, it is time to tie the differentiation of defect/non-defect back to its original idea, i.e. improved satisfaction and its link to profitability through variables such as loyalty, retention, improved share of wallet etc.
    In my opinion, six sigma and lean six sigma is currently stagnating because it has become involved in an internal struggle between “heretic” schools of thought (TQM, Six Sigma, Lean, Lean Six Sigma etc.) that has lost sight of the fact that in the end, the initiative needs to be tied to the value creation process of a company. There has been an earth shift over the past ten years in the business world and the business literature that Six Sigma is hopefully going to wake up to soon, before it becomes one of those “initiatives” from yesteryear.


    Ex-SME, your question is a type three type of error: it’s the wrong question. It is totally irrelevant today who “fathered the child”. The key question is “Where is this child going” and “How will it survive and grow to the next level”. Obsession with history, unless used to bring about current change, has always been a sign of decline be it in an empire, nation-state or a once burstling with life type of initiative such as “Six Sigma”. Old people squabble about a father’s child because they want to profit from the inheritance. But inheritances (= low risk, low annuities) are different from entrepreneurial investment (high-risk, high yield) capital venture …



    OK..I think I get yr drift. Theres a Japanse guy sitting at a bench near me, I’ll try to ask him again – but everytime I ask him a question about this he just laughs and shakes his head
    BTW he thinks all Americans are experts. Are you a Pom by any chance?


    The Japanese is probably internally rolling around about all the heavy-duty “Western” rational thinking about something that is quite intuitive and in his world doesn’t need a lot of semantic explanations, explications, implications and non-referential discursive thinking … Not sure what you mean by POM, but I have spent quite some time in the countries that invented the “conditions of the possibility of synthetic judgments a priori” and the “non-referential signifiers”. So from the angles of some of the 191 insignificant non-American countries in this world the American scholastic debates about the differences between lean and lean six sigma sound quite amusing. Especially, when these debates are driven by powerpoint slides that some consultants put together (sometimes over a couple of cold beers late at night:-). Have a great day … and say hi to your Japanese friend!



    If you think this discussion is a waste of time take a look at this link.
    You might want to take care of the ‘defect’ between your ears!



    That article is a piece of idiocy and conjecture.  I can’t believe it was published, little less cited.



    Which  article??


    Rev. Elgier

    On the contrary, I thought it expressed a number of real concerns. Just because you can’t see it doesn’t mean other people are ‘obscured’ by smoke and mirrors.
    Rev. Elgier


    Rev. Elgier

    Presumably the one posted by Len.
    Rev. Elgier



    Rather than making wild unsubstantiated assertions as typifies six sigma, perhaps you would like to describe in detail exactly what in the article is “idiocy and conjecture” ?? 
    The article looks at least 95% accurate from what I’ve read.



    As I expected … rather than provide any intelligent critique of the article, you’ve just crawled off into your hole.

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