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10.6 Off-Centre Distributions

It may feel like a long time ago but I am going to go back to the peanuts where we had a 200 sample distribution and spec limits of 200 to 220g for the fill weight. When we calculated our original Pp  it came out at 1.111. What if a clumsy person accidentally knocked into the controls of the filling machine and it started to put more peanuts into each bag. Let’s say that the mean increased to 215g. If the mean has moved then the whole distribution will also move but in my story the width of the distribution stays the same. Looking at the plot we can see that many of the bags would be out of specification. Let’s work out the process capability for this distribution.

Earlier in this module I said that ‘we are going to limit ourselves to normally distributed data or distributions where the underlying data is normally distributed’. You might have thought that was an odd way to phrase the sentence and that there might be something more to that statement. If you did you would be right. Give yourself a pat on the back !  When conducting a process capability study you can expect to conduct or see the results of a normality test. If the results show that the data is normally distributed you can proceed without any issues. However, if the results show the distribution is not normal then things get a bit sticky.

Take a look at Distribution1 which is made up from 200 data points. We see that the fitted normal curve matches the data pretty well. And when it is tested for normality a P value of 0.68 is returned with the Anderson-Darling test which indicates that the data probably came from a normally distributed population. We would not have a problem using the capability tools for normal data described within this module for this data.

Now take a look at Distribution2 which is also made up from 200 data points. The fitted normal curve does not match the data.  The data returns a P-value of less than 0.005 indicating that the data probably does not come from a normally distributed population.  With a bit more experience or with help from a Black Belt you could try transforming the data or applying a different distribution to the data. If I ever get round to writing it that will be covered in the Black Belt edition of this book. It is covered in my last book, Problem Solving and Data Analysis using Minitab.

Distribution2 can be fitted with a Gamma distribution and the VOP part of the capability calculation can be modified to cope with a Gamma distribution. The point for Green Belts is that the shape of the histogram is clearly not meant to be aligned with a normal distribution.

Finally we get to the sticky bit I mentioned earlier. Distribution3 is made up from 200 data points again. It is fitted with a normal curve which matches the data pretty well. The Anderson-Darling test for normality returns a P-value of 0.02 and as you know  this indicates that the data probably does not come from a normally distributed population.

This bit is sticky as it relies on guess work, however, the risk of making a mistake is reduced by experience and combining our knowledge of SPC. The guess that must be made is whether the underlying distribution is normal and is being affected by special cause variation or whether the underlying distribution is of a different type. If your judgement and SPC indicate the problem is due to special cause variation then you can use the tools described in this chapter in order to calculate capability metrics. But remember, due to the random nature of special cause variation any metrics calculated when special cause variation is present should not be used for predicting future trends. They should only be used to describe the sample data set.

However, if you feel it is another distribution or the data requires transforming please wait for the Black Belt edition or ask a suitably experienced and qualified Black Belt for help.

10.7 Overall and Within Capability

1. Process capability is described as the Voice of the Customer, VOC, divided by the Voice of the Process, VOP.

VOP is taken to be six StDevs. VOC is the tolerance.

2. An overall capability of 2 describes a six sigma process. The specification limits will be six StDevs away from the mean.

3. We can use look up tables to find the Yield% or DPMO for some capability levels. However, we need to be aware of the 1.5 sigma shift that is applied to represent the deterioration of capability over time.

4. Pp is helpful as a learning tool but it is not very good when the process is not centered. We therefore start to use Ppk

which looks at the distance of the mean to the nearest specification limit.

5. Ppk is used to represent overall capability. Cpk represents within capability, the calculation is based on the within subgroup StDev. If special cause variation is present Cpk is aspirational.

6. The methods and examples shown in the Green Belt Edition are for continuous data related to situations where the underlying distribution is normally distributed.

10.8 Normality

10.9 Summary of Process Capability

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As this is the same capability value as before we can conclude that an off-centre mean does not affect this type of process capability calculation. We need a modified calculation that can cope with means that are not centred. The calculation that we need to use is based on the distance from the mean, Xbar, to the nearest specification limit divided by half the process width. It looks like

Notice that the new symbol is Ppk. Our new process capability becomes

If you see both Ppk and Pp  in a capability report and their values are similar you will know that the distribution is at the centre of the spec limits. Otherwise,  Ppk  will always be lower than Pp.

In section 9.4 we said a subgroup was a group of items that are made under similar conditions and that the subgroup was captured in a short space of time. We subgroup in order to reduce the amount of analysis we have to do and it can protect us from non-normal distributions. We also said that our subgroup should only contain short term random variation and not special cause variation. We saw that an IMR chart was used when subgrouping was not possible and  Xbar R & S were used when it was possible. Process capability also uses subgroups when possible but rather than employ different tools it gives you different metrics.

Overall process capability is Ppk , but we are given another measure of capability for just the short term random variation seen within subgroups. This is Within Process Capability or Cpk.

If you remember we said that the Voice of the Process, VOP, was 6σ. The VOP for Cpk is calculated using six times the StDev within the subgroups, 6σwithin . As 6σwithin will be less than 6σOverall this means that Cpk  will be greater than  Ppk  .This is even true when there isn’t any special cause variation due to the way within StDev is calculated.

The key point is that if special cause variation is present Cpk  will be greater than  Ppk  . But the Cpk  value is aspirational as it can only be achieved if you work hard to remove all the special cause variation. Now you know the secret that a good Cpk  value does not always indicate a capable process. Therefore, if you have a supplier telling you that there Cpk  is 2.0 always ask what is the  value of the overall capability as it may be much worse.