To Z or not to Z?
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 This topic has 31 replies, 6 voices, and was last updated 12 years, 9 months ago by Ken Feldman.

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October 31, 2009 at 9:59 pm #52862
MBBinWIParticipant@MBBinWI Include @MBBinWI in your post and this person will
be notified via email.I recently heard some comments deriding the humble Zscore 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 Zscore: 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 Zscore was developed with a ZBench 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 offcenter performance situations.
So, what do you in Forum land think? Is there utility in the Zscore or should we try to get everyone to learn to deal with Cp and Cpk?0November 1, 2009 at 2:00 am #186504There 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 Zscore and a Cpk value.
I’ll grant there is not much beyond a scaling difference between a Zscore and capability indices, however. Of course, the same is true between working with capability indices, or zscores, and then the scale translation to a Six Sigma score for a process with a built in 1.5sigma shift.0November 1, 2009 at 11:17 am #186506Hi,
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 nonnormal distributions in Minitab, it is a simple thing to calculate a zBench 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 Zbench. So, why waste time and effort to gather the data and then to report a measurement that actually hides information?Regards
Sandor0November 1, 2009 at 1:25 pm #186507zscores 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 zscores as defined in
Harry’s nonsense, I’ll accept zscores.Otherwise it’s needless complication.0November 1, 2009 at 1:49 pm #186508How 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.0November 1, 2009 at 1:52 pm #186509Just 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.0November 1, 2009 at 2:30 pm #186512Hi Stan,
imagine a strongly rightskewed 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 ZBench 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 zscore, something I will not see with the Cpk IMHO.Regards
Sandor0November 1, 2009 at 3:01 pm #186513If 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.0November 1, 2009 at 3:02 pm #186514
MBBinWIParticipant@MBBinWI Include @MBBinWI in your post and this person will
be notified via email.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 Zscore 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 ZBench 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 offcenter, 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.0November 1, 2009 at 3:06 pm #186515
MBBinWIParticipant@MBBinWI Include @MBBinWI in your post and this person will
be notified via email.Sandor: Cpk is only valid for normally distributed data.
0November 1, 2009 at 3:30 pm #186516We 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?0November 1, 2009 at 5:37 pm #186517
TaylorParticipant@ChadVader Include @ChadVader in your post and this person will
be notified via email.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.0November 1, 2009 at 5:54 pm #186519MBBinWi,
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 nonNormal 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.0November 1, 2009 at 6:05 pm #186520
MBBinWIParticipant@MBBinWI Include @MBBinWI in your post and this person will
be notified via email.StuW: So, if Cpk is the minimum of the upper or lower, it only is representative of one side. I stand by my statement.
0November 1, 2009 at 9:02 pm #186521
Ken FeldmanParticipant@Darth Include @Darth in your post and this person will
be notified via email.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.
0November 2, 2009 at 12:35 am #186523Darth: 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.0November 2, 2009 at 5:32 am #186526Cool 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.0November 2, 2009 at 7:01 am #186527MBBinWI: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.
0November 2, 2009 at 7:06 am #186528Hi Stan,Even if you make corrections for the nonnormality, 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
Sandor0November 2, 2009 at 8:28 am #186530MBBinWI: That was exactly my point.
0November 2, 2009 at 10:26 am #186531Oh my! The dreaded skewed right distribution!Your z score will stay the same as well.We might actually have to use our brains!
0November 2, 2009 at 10:28 am #186532What 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?0November 2, 2009 at 11:23 am #186533“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.0November 2, 2009 at 11:27 am #186534Ooops,
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?0November 2, 2009 at 12:37 pm #186538
Ken FeldmanParticipant@Darth Include @Darth in your post and this person will
be notified via email.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.0November 2, 2009 at 2:10 pm #186540
Ken FeldmanParticipant@Darth Include @Darth in your post and this person will
be notified via email.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 510 items for a sample.
0November 2, 2009 at 7:32 pm #186547Wow.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?0November 2, 2009 at 8:20 pm #186548
Ken FeldmanParticipant@Darth Include @Darth in your post and this person will
be notified via email.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.
0November 2, 2009 at 9:53 pm #186549Clear 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 noise0November 2, 2009 at 10:05 pm #186550
Ken FeldmanParticipant@Darth Include @Darth in your post and this person will
be notified via email.It is clear now that you understand. Hope you and the family had a good Halloween.
0November 2, 2009 at 10:33 pm #186551Now 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.0November 3, 2009 at 1:31 pm #186559
Ken FeldmanParticipant@Darth Include @Darth in your post and this person will
be notified via email.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.0 
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