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To Z or not to Z?

Six Sigma – iSixSigma Forums Old Forums General To Z or not to Z?

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  • #52862

    MBBinWI
    Participant

    I recently heard some comments deriding the humble Z-score and thought that I’d throw it out to the forum community to see what folks think.
    Now, just so nobody takes things the wrong way, there is good history and deep understanding within many manufacturing organizations (although I have had this opinion challenged over the past few months) of Cp and Cpk.  Much of the organization can easily interpret the difference between a Cpk of .8 and 1.4 and these are clearly understood.  However, outside of the core manufacturing ranks, I’ve often found it difficult to get folks to come to grips with “Cpk is the ratio of the distance between the mean and the closest spec limit and 3 std dev’s,” often hearing comments like “so you’re telling me that 6 sigma is actually Cpk of 2?”
    To newbies (and product engineers who believe that Cp and Cpk are just manufacturing measures) it is so much more clear to speak in terms of a Z-score:  The number of std dev’s that you can fit between the mean and the closest spec limit.  It is a simple fomula (abs value of (spec limit – mean) / std dev), easily graphically described, and makes sense to everyone (so, 4 sigma is Z=4, that makes sense to me).  [try to get someone not familiar with Cpk to understand that it is a ratio of 3 std dev’s – I wish that I had all the time that I have spent in those discussions back – I could take a nice vacation].
    Additionally, by definition, Cp and Cpk only look at one side of the spec limit.   This can oftentimes provide a false level of prediction on the actual capability as lower levels can have significant percent of items outside the limit that is further away.  These are lost with Cpk.
    However, recognizing this situation, somewhere along the line the Z-score was developed with a Z-Bench adaptation which takes into account the total percent outside both spec limits and converts to a single side equivalent capability measure so that a more accurate and “apples to apples” comparison can be made between lower capability and off-center performance situations.
    So, what do you in Forum land think?  Is there utility in the Z-score or should we try to get everyone to learn to deal with Cp and Cpk?

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    #186504

    StuW
    Member

    There are some subtle advantages to keeping the concepts of standardization and capability indices somewhat different.  From a teaching perspective, one can introduce the concept of standardizing a value from a N(mu,sigma^2) distributionto a N(0,1) distribution in order to determine how many standard deviations a value is from the mean, without ever having to raise the concept of a specification limit.  Most basic Stats, or Six Sigma, training classes start with the standardization concept, and then move forward eventually toward capability.   I have found if people understand the concepts of standardization well, it is generally fairly easy to relate that to the capability metrics, and the factor of 3 translation that occurs between a Z-score and a Cpk value.
    I’ll grant there is not much beyond a scaling difference between a Z-score and capability indices, however.   Of course, the same is true between working  with capability indices, or z-scores, and then the scale translation to a Six Sigma score for a process with a built in 1.5-sigma shift.

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    #186506

    Szentannai
    Member

    Hi,
    to me Cp and Cpk have only historical value. The problems the ones you described, together with the fact that one has some pretty serious hidden assumptions by the interpretation of the numbers : for Cp we assume that the process is centered and symmetrical and for Cpk that it is symmetrical. E.g. imagine a process with a skewed distribution – even the Cpk numbers will be unrelated to the DPMO.Working with non-normal distributions in Minitab, it is a simple thing to calculate a z-Bench that has real meaning.
    Moreover, the information you bneed to calculate Cp and Cpk is pretty much the same as the one you would need for Z-bench. So, why waste time and effort to gather the data and then to report a measurement that actually hides information?Regards
    Sandor

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    #186507

    Mikel
    Member

    z-scores are the Harry cult’s version of Shainin speak.Engineers are taught +/- 3 standard deviations as the definition of
    capability. When stats classes start teach z-scores as defined in
    Harry’s nonsense, I’ll accept z-scores.Otherwise it’s needless complication.

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    #186508

    Mikel
    Member

    How does Cp only look at one side of the spec? This is the first I’ve heard of this and I’ve been using Cp for > 25
    years. 25
    years. 25
    years.Explain how you get more information from all of the z nonsense than
    you do from the standard capability output of Minitab.And what about when the target is not the center? Cpm gets you there
    and none of the z stuff does.

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    #186509

    Mikel
    Member

    Just remember that z scores are brought to you by the same person
    that brought you the 1.5 shift and spent several years through several
    changing stories trying to say it is a fact.It takes about a month of real experience to know the 1.5 is not true,
    so why did the guy spend years trying to convince people? And you
    want to buy his z score? Yikes.

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    #186512

    Szentannai
    Member

    Hi Stan,
    imagine a strongly right-skewed distribution – something like maybe processing times. The problems start at the calculation of the mean and of the standard deviation – neither is very intuitive for the skewed distribution – eg. the majority of your measurements will be greater then the mean.Even worse – the Cpk will give you a blissfully reassuring picture – the distance between the LSL and the mean will be nice and acceptable, however the amount of defects on the LSL end will be much larger then what you would expect, based on the (unstated) assumption of a symmetrical distribution.Now, in Minitab I can identify the distribution and then calculate a Z-Bench number which will be the Z – value of a normal distribution that has the identical number of defects as my skewed process. Not very nice I admit, but at least all the defects are accounted for in the measurement – i.e. if my process gets worse (it grows a large, heavy tail on the right side) I shall see a deterioration in the z-score, something I will not see with the Cpk IMHO.Regards
    Sandor

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    #186513

    Mikel
    Member

    If I can identify the distribution, I can also get an accurate Cpk based
    on the distribution. The calculations are exactly the same only now
    we are speaking a strange language that hardly anyone inside the SS
    community speaks and no one outside has a clue about. I don’t know
    how that solves a thing.There is a reason we give pictures with numbers, that’s the real
    solution, not some cult like language created by a buffoon.

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    #186514

    MBBinWI
    Participant

    Stan:  Happy to see that I piqued your response.
    1. Let’s not get wrapped around the math on Cp.  I know you’ll agree that if the distribution is centered between the limits, that Cp and Cpk are the same, and that Cpk only looks at 1 side of the distribution.  So, if A=B, and B=C, then A=C.
    2. I never said that there was more or less info (in fact they are just a mathematical translation of one to the other Z=3Cpk), merely that it’s easier and more straightforward for folks to understand Z-score than Cpk.  Again, try to explain to someone not familiar with Cpk that 1.33 equals 4 sigma.  I’d much rather tell them that they have Z=4, and that is 4 sigma because that makes sense to them.
    3. Your point about the target not being centered shows the failing of Cpk.  Since it only looks at the closest spec limit, when you have lower capabilities, you lose what’s happening on the opposite side of the distribution.  This is where I see Z-Bench as being far superior as it aggregates all probability outside of the limits and computes an equivalent capability as if they were all on one side.  This provides a truer picture across multiple processes – those well centered, those with low relative variation but off-center, and those with higher variation whether centered or not.
    4. I’m not as familiar with Cpm – none of the orgs that I’ve been associated with have used it – but looks kind of like Taguchi loss.  I’m open to learning more.
    BTW, it has been 24 yrs since I graduated with a BSME deg.  I went to what many consider a fairly good engineering school, and I know that capability was not a major emphasis (stats was a generic study not tied to processes and mfg capabilities).  I recently contacted the head of the engineering dpt at my alma mater to see if they were interested in integrating DfSS methods into the curriculum, and the response was that they had Lean/SS as a week long topic area and that that was enough, thank you very much.  Having interviewed recent college grads for jobs over the years, I’ve queried on their curriculum’s integration of capability analysis into the engineering body of knowledge and have continually been dismayed at the lack of such.  So I’m not sure that engineers (maybe IE, but certainly not ME) are being taught +/- 3 sigma either.

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    #186515

    MBBinWI
    Participant

    Sandor:  Cpk is only valid for normally distributed data.

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    #186516

    Mikel
    Member

    We have z short term which is actually 3*Cp. We have Z long term which is actually 3*Ppk. We have z bench which is just calculating what is out on the upper
    and lower sides of a spec and coming back with z value as if all
    were on one side – no thought to what would happen if the
    process were centered (maybe we can add yet another layer of
    confusion by creating a z bench ideal). If we know the data is
    predictable but not normal, we can substitute the appropriate
    distribution (just like with the traditional metrics).We have z upper and z lower.I can do the same thing with Cp, Cpk, Pp, Ppk, and Cpm all of
    which existed when our friend Mikel created this other nonsense.
    And all of them better be accompanied by a picture of the data
    that was used for a sanity check. Minitab also gives estimates of
    defect levels (the intent of z bench), and an output of defect levels
    contained in the data as sanity checks. Anyone using the z
    nonsense in Minitab should turn them off and see the information
    is more complete without them. Assuming we are teaching semi
    intelligent people, they can learn either.I teach people the traditional metrics, but also teach to give a
    picture (histogram) and mean and standard deviation even if there
    is no spec. Most teach oh no I don’t have specs therefore I can’t tell
    you how my output or input is behaving – pretty dumb don’t you
    think?

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    #186517

    Taylor
    Participant

    MBBinWI
    I have read through this and only have a couple comments. One, Z Scores are pointless. Two, Capability Indicies, when done correctly, gives signals which can be used to control the process. It is not just historical data. As an Engineer, I never cared much for Cpk, I wanted to know the Cp, this told me how much variation I had in my process and whether it was stable. Now the shop floor supervisors and operators typically didn’t care about either. All they knew to do was keep it between the lines and above 1.5 Cpk for example. A Z score cant do either for you.

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    #186519

    StuW
    Member

    MBBinWi,
    A couple of points:
    Cpk does include both ends of the distribution, by definition it is min(Cpl,Cpu).  Your complaint is more that it does not capture probabilities from both ends, which is not the purpose of Cpk to begin with.  Cpk is a figure of merit on whether a distribution is capable to the spec limits.   It started with 1.0 as a comparison value, and was then increased to 1.33, 1.5, etc. in order to provide additional protection against long term shifts.
    Cpk formulas have long had proposed modifications for non-Normal distributions.  It was often proposed to use percentiles of the distribution such as (p0.005, p0.995) to replace sigma for cases where the distribution was not Normal.   Also, the example cited for a skewed distribution to the right is not interpreted properly.   In that case, the sigma of the distribution is driven higher by the readings on the upper end.   The net result is the probability cited for the percentage of items outside the lower spec will overstate the actual probability, which is often zero, while it will tend to understate the percentage on the high side, where the skew exists. 

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    #186520

    MBBinWI
    Participant

    StuW:  So, if Cpk is the minimum of the upper or lower, it only is representative of one side.  I stand by my statement.

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    #186521

    Darth
    Participant

    I don’t know why the big deal with z value. It is merely the number of standard deviations a value of interest is away from the mean. It has nothing to do with Harry or the 1.5 shift. It is the basis for the standard normal table.Cp is the potential capability should the process be centered between an upper and lower spec. If you only have a one sided spec, then Cp has no meaning. And what are the odds that a process will ever be centered between two specs!!!!!Cpk is the worst case scenario which is why you pick the closest spec. If the closest spec is far enough away not to worry then who cares about the spec that is farther away?The basic Mini function for capability assumes stability and normality. If the data is not normal then you try to identify the distribution and if appropriate run a non normal capability.Cpm tells you how well you are doing against a target. I may be on target but the distribution may be wider than the specs.All the capability indices are directional hints and should be evaluated carefully.We do a cool demo in the classes I teach. I create a random population of normal data of 10,000 values. Then I have the students randomly select 25 values and do a capability assessment all using the same specs. Because of the randomness of the data, all the students come up with different values for Cpk and in many cases some will be below 1.0 and some above 1.0. That is, we may have judged the process to be capable in some cases and not in others yet the population is exactly the same. Randomness can have an influence on your judgment so use your brains and don’t just rely on Mini’s output.

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    #186523

    StuW
    Member

    Darth:  That is a good exercise because Cp and Cpk are both driven by the sample variability in the estimate of sigma.   My recollection is a spread of +/-0.2 on Cpk for sample sizes of about 30.  This is often overlooked when a Cpk is determined and then immediately converted to a defect level.
    MBBinWi:  I was responding to provide some additional clarity on Cpk, given the array of people who may use this site.  While I’m sure you realize that the Cpk formula does capture both sides of the distribution, the implication of the note contents would lead one to believe that it does not.  A minimum of two values certainly considers both entries, so I also stand by my comments, too.

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    #186526

    Mikel
    Member

    Cool exercise? How about if we just have them generate 25 random
    numbers?The real demonstration is the effect of such a small ample size. And it does have to do with Harry, the z nonsense was created to
    make people think there was something new and better. It is not.

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    #186527

    Szentannai
    Member

    MBBinWI:You are right IMHO. If we take the skewed distribution, things might go terribly wrong on the right side and you might get a significant deterioration of your yield but the Cpk number will stay the same.

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    #186528

    Szentannai
    Member

    Hi Stan,Even if you make corrections for the non-normality, the Cpk number will be a rough measure of the process quality, whereas the Z number is directly linked to the yield. That is something everybody would understand, right? So, how do you translate a Cpk number to something immediately meaningful?Regards
    Sandor

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    #186530

    Szentannai
    Member

    MBBinWI: That was exactly my point.

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    #186531

    Mikel
    Member

    Oh my! The dreaded skewed right distribution!Your z score will stay the same as well.We might actually have to use our brains!

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    #186532

    Mikel
    Member

    What nonsense. Have you ever done this?z scores and Cpk are different by a factor of 3. how can one be better
    than the other?

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    #186533

    Szentannai
    Member

    “z scores and Cpk are different by a factor of 3.”
    Stan, this is plainly wrong, in general. You need a centered symmetrical process for this work, in all other cases it doesn`t.

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    #186534

    Szentannai
    Member

    Ooops,
    if we are talking about a meaurement that delivers irrelevant numbers so that we use our judgement anyway then I guess Cpk is ok.My original point was, why not use a number that actually has some informnation in it, especially that it costs about the same effort … then use our judgement to improve instead of second guessing the measurement?

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    #186538

    Darth
    Participant

    Sorry Stan, the Standard Normal Distribution was developed while Harry was a little tot.  The Z Bench crap may be his but the original Z value converted the normal distribution into something generic.
    The term “standard normal” which denotes the normal distribution with zero mean and unit variance came into general use around 1950s, appearing in the popular textbooks by P.G. Hoel (1947) “Introduction to mathematical statistics” and A.M. Mood (1950) “Introduction to the theory of statistics”.

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    #186540

    Darth
    Participant

    I do the exercise that way for three reasons:1. It lets the students “visualize” their “population” of “product” produced. 2. Each “inspector” selects a random sample from that “population” for their inspection and capability analysis. Like real life there are multiple inspectors taking samples at different times.3. It teaches them the Mini function of sampling from columns.The QA/QC guys freak the most when they see the impact of randomness and variation on the decisions they might make regarding the capability of the process. Population doesn’t change and the process is stable so why should the capability be judged differently based on random samples? Very impactful. And 25 isn’t that small a sample in many applications. So, as you suggest, just imagine if we selected 5-10 items for a sample.

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    #186547

    Mikel
    Member

    Wow.Thanks for the clarification. That’s a point I had not caught the first
    800,000 times I looked at the table. Actually I’ve been confused trying to figure out the relationship
    between k in Juran’s tables and z.Duh. Maybe one of us is not answering the original question. Guess
    who?

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    #186548

    Darth
    Participant

    OK, so that you are now clear as to the answer. The Z value from the original derivation measures the distance, in s.d., from the mean and some value of interest. It has nothing to do with Harry and can be used to calculate some percentage beyond that value of interest. Therefore, if the value of interest is a spec then you can estimate the % beyond that spec, either high or low or both. This is useful.This z score and z bench stuff is the same but gets confusing if you try to distinguish between long term and short term variation and then gets really screwed up when calculating a sigma value which includes the shift. Z is Z. Z by itself is much like Cpk in that it is relative. I like to report out how much is estimated out and how much is estimated in. We already discussed the impact of randomness and variation on the calculations of all these metrics.Bottom line, as I see it, is to do your calculations to get a directional pointer and then use your friggin brain to give some meaning to it. Who cares which one you use, just be consistent.Stan, hope this clarifies it for you.

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    #186549

    Mikel
    Member

    Clear as hell There never was and never will be a need for z bench, z shift,
    or any of the other z tripe. It’s just noise

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    #186550

    Darth
    Participant

    It is clear now that you understand. Hope you and the family had a good Halloween.

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    #186551

    Mikel
    Member

    Now if I just knew the relation of Juran’s k to z. The shift
    appears to be small. Is that right?Cheers. I hope all is well.

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    #186559

    Darth
    Participant

    Stan,
    To tell the truth, I don’t have Juran’s book handy and can’t seem to find any reference to the k you mention for his version of process capability. Can you enlighten?Also, we never got into the discussion of transforming the skewed data in order to do process capability.

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