1.5 Zshift

Fernando the inflation factor of 1.3 between StDev Short term and StDev Long Term corresponds with a 1.5sigma shift. You can verify this is you generate random data from normal distributions that shift around the mean 1.5 sigma. If the shift would be 3 sigma then the inflation factor would become 2.
I agree with the comment that you should only use numbers that are based on real data not some kind of rule of thumb. If you only have short term then don't make any statements about long term data and vice versa.
If you have subgrouped data then having both standard deviations allows you to calculate Zshift by looking at the inflation factor. Using for instance the difference between Zbench long term and Zbench short term does not work (just play around with USL and LSL and see what happens to that difference). The six sigma module in Minitab (an add-on module created for Six Sigma Academy) also calculates a Zshift but it is based on a different reasoning coming from Michael Harry; it is a number that is dependent on the spec limits where the real Zshift is not.
I have through simulation found the following relationships
tau = 0.8070 + 0.3254 z + 0.03125 z**2 - 0.001250 z**3 (where tau is the inflation factor) z = - 2.045 + 2.899 tau - 0.2442 tau**2 + 0.01641 tau**3
These equations are based on the following table that was generated through simulation
z tau
0.5 1.04
1.0 1.16
1.5 1.33
2.0 1.54
2.5 1.76
3.0 2.00
3.5 2.27
4.0 2.53
4.5 2.81
5.0 3.08
5.5 3.36
6.0 3.64
6.5 3.92
7.0 4.20
7.5 4.49
8.0 4.77
8.5 5.05
9.0 5.34
9.5 5.63
10.0 5.91
10.5 6.20
11.0 6.50
11.5 6.77
12.0 7.10

Regards,
Filiep Samyn
Lean Six Sigma Master Black Belt

If you have attribute data then the accepted standard is to assume that your data is long term and to add 1.5 to attain short term sigma. For variable data, I would suggest that if you do not have rational subgroups then you should look at the data you have, if the data has been collected over a short period of time/inference space then the data you have is short term. If you collect data over a longer period of time covering all factors of variation you have long term data (and if you were to look at subgrouping by time then you could get a good estimate of short term
sigma.) I agree with Fernando that the arbitrary 1.5 shift is not ideal, it is my understanding that this shift was obtained from the electronics industry based on an AVERAGE of the shift from a couple of thousand processes (and we know how we feel about the accuracy of using averages!).

When it comes down to it the only reliable way of getting short and long term sigma is by having short term (from subgrouping)and long term data. If this is not possible then I always recommend declaring the sigma value as declared for overall capability from Minitab and also noting whether the data is short or long term. I see no value in adding an arbitrary figure to get an estimate. The method based on a calculation based on the standard deviation may be a slightly better approach than the 1.5 sigma shift, but at the end of the day every process works differently. I have even seen business processes where the short term sigma is greater than the long terms sigma (not something you get very often but it can happen)

Regards,

Liz Ferguson

Business Improvement Manager
6 Sigma Master Black Belt

Thanks for all your replies.

This is the type of answers I expected to receive.
I fully agree about the fact that it would be better not to make any assumption in case we don't have rational subgroups.

Fernando

Hi,

The validity of 1.5 sigma shift factor is something really debatable issue among many academics across the world. Motorola has come up with this information based on its extensive research. The challenging question to ask is " Do all processes shift by 1.5 Sigma"? Is it valid to assume 1.5 sigma shift factor for all processes? We need to see more research in this topic. Moreover, you need to think about the probability distribution of CTQ's as well. We really need to understand the distribution of our processes prior to carrying out rational subgrouping and calculating the Sigma Shift factor. Do all companies (especially service) look into this issue - I doubt it !!! One of the research areas at the Six Sigma Research Centre here at Caledonian Business School is to understand more about the type of distributions and then develop strategies to calculate the Sigma Shift factor.

Regards

Dr Jiju Antony
Director of Six Sigma Research Centre

[quote:8bf434bf25="Fernando"]In an other forum there's a debate about the zshift. I didn't receive any answer to my question, I'll try to post it here.

As you know the six sigma methodology states that when we don't have rational subgrouping we can shift the Z value by 1.5. I have some concerns about this.[/quote:8bf434bf25]

So do I, it's non value added.

[quote:8bf434bf25="Fernando"]As alternative to the Zshift = 1.5, sometimes I’ve heard talking about an inflation rate for the standard deviation. In particular: stdev LT=1.3*stdev ST [/quote:8bf434bf25]

Yes indeed - and sometimes higher; between 1.3 and 1.8 depending upon your knowledge of the state of control of the process; if you know the process has lots of special causes you would use a high factor and vice versa. (This logic could also be applied to the 1.5 shift.) But this then leads to subjectivity - at least with the 1.5 or 1.3 you know where you stand.)

[quote:8bf434bf25="Fernando"]To me it makes more sense than the Zshift, because, given a dataset with rational subgroups, the Zshift can strongly change depending on the spec limits, while the stdev ST and LT are always the same. [/quote:8bf434bf25]

If you have rational subgroups it's a moot point; always calculate the real Z.shift value - never, ever use the 1.5 (or 1.3) when it's unnecessary.
They make a similar amount of 'sense', in that they are both somewhat arbitrary constants which don't represent any real process accurately.

[quote:8bf434bf25="Fernando"]If you inflate the Z LT with a fixed Z shift independently of the value of Z LT itself (1.8, or 2.5, or 4.5 etc) the result could be misleading. [/quote:8bf434bf25]

It is only misleading if too much is read into it. It is a CONVENTION and should be treated as such and understood that real Z.shift could be anything.

The standard deviation inflation factor is susceptible to the same problem as the 1.5 shift; it's an arbitrary constant and should be treated as a convention. The real danger is when some people use Z.shift of 1.5 and some people use the stdev inflation factor. As long as people use the same convention everyone knows where they stand - even if it's on shaky ground!!

[quote:8bf434bf25="Fernando"]If you have some data with rational subgroups you can try to calculate Zst and Zlt. Changing the spec limits you'll see that the Zshift changes[/quote:8bf434bf25]

The same is true for the 1.3 inflation factor method.

The graph below was generated for different specification limits, for mean on target, slightly off target and very off target (diamonds, squares and triangles respectively). See http://www.onesixsigma.com/experience/forum_archive/zshift.php

The short term standard deviation and target values were fixed and the long term standard deviation calculated using the 1.3 inflation factor.

The lines are only valid for this particular data set, of course, but you get the point...it shows that Z.shift is dependent upon the short term process capability (spec limits) and the deviation from target (mean).

The diamonds are for a process which is on target, with various values of Z.st.bench due to different spec limits around the fixed target.

At around 6 Sigma performance the shift is around 1.5 (which is no coincidence, by the way).

For high potential process capability (10 Sigma), the shift is almost 2.5, whilst it is <<1.5 for a process with poor potential capability.

This variation is a result of the properties of the normal distribution; it's non-linearity.

For a process with a mean which is significantly off target (2 standard deviations away in this example), the triangles show how Z.shift varies with Z.st.bench. For a given Z.st.bench, Z.shift , as calculated here, is always higher than for a process which is on target.

So, you see that the inflation factor doesn't really help - in fact it could be harder to explain the value of Z.shift than for the constant 1.5.

[quote:8bf434bf25="Fernando"]So, if we don’t have rational subgroups, don’t you think it would be better to correct the standard deviation with an inflation rate and calculate the corresponding Zshift?[/quote:8bf434bf25]

If there is only short term data available, I would prefer people to quote Z.st.bench ONLY.

If there is only long term data available, I would prefer people to quote Z.lt.bench ONLY. There is really no need to add or subtract constants which have no real practical application, but it has become part of 6 Sigma mythology (I choose the word carefully!)

However, if people must quote both then it doesn't matter which convention is used from a practical point of view, provided that they stick to it & everyone else in the organisation uses the same one.

Hope this helps. These are my initial thoughts, although I'd be glad to develop them further if need be.

Phil

Ps in the above I have defined Z.st as the capability when the process is on target - this is different to the default used by Minitab, which uses the mean, and so does not represent the true 'best' the process could achieve with only common cause variation.

Dear Fernando,
I am a strong advocate of keeping it simple. I heard once that the reason for a 1.5 sigma shift in calculating "Sigma Level" was becasue a Six Sigma process has much more of a ring about it when selling the concept to non-technical management with an interest in profit rather than statistics than a 4.5 Sigma Level.

I think the key is to fix a methodology and metrics that work for an organisation, then get on with process improvement work, reducing variation and putting processes on aim to reduce cycle time and defects to make money, rather than worry about measurement systems.

If the idea you present finds favour with the people who use it, then use it. If it spares time answering awkward questions about the arbitrary nature of the 1.5 Sigma shift, great. The organisation will have to make the same changes to improve whatever scheme is used to measure the processes.

Aaaagghhh not that 1.5 shift thing again. Since people seem to be avoiding your question, I think I'll stick my neck above the parapet and see if I get shot!

Unfortunately however (and in true consultant fashion) I'm afraid I'm going to avoid your question about whether the 1.5 "shift" or "inflation" method is better for estimating Zst (although your arguments in favour of the "inflation" method sound very rational).

This is because I see very little value in estimating Zst using any generic method at all (be it "shift" or "inflation"). Unfortunately the whole 1.5 shift has created untold confusion, and has been used too often in the consulting/training world to baffle, confuse and generally over complicate the whole issue (I'm completely above the parapet now - exciting!).

So, we need to go back to why we are interested in measuring Zshift. We're interested because it gives us an indication of the potential performance of the process.
In other words, a high Z shift indicates that the process has "potential" and might be able to be improved through increased control. A low Z shift indicates that the process is at the limit of its "potential", and may need some substantial improvement in design or technology in order to be improved.

This understanding of Z shift then gives you an understanding of the types of improvement / direction that may be required in your project.

So, if you can't measure Z shift (maybe because you don't have data that represents both the long and short term or perhaps rational subgroups), then I don't think we should estimate it - at all!

Far better to fix the issue by collecting more data so that you can measure the actual Z shift, rather than estimate it through some generic method.

Hope this helps/makes sense etc.

Quentin