Make Valid Control Chart and Subgroup Assumptions

Control Charts Subgrouping By Machine Nozzle
My final example is one that often occurs in manufacturing organizations. Many manufacturing plants contain machines that pack material into bags or boxes. In this case, the machine had three nozzles. The organization was contemplating application of control charts to measure the process performance.

The engineers were contemplating using a subgroup of three to represent the three nozzles of the machine. The data would be collected on an hourly basis for control chart plotting.

What was wrong with the application described above?

The engineers are assuming that there is no special cause acting between the nozzles, and that only common cause variation exists between the three nozzles. This seems to be a dangerous assumption.

In addition, the engineers wanted to take three consecutive samples from each nozzle - for subgroup purposes - to estimate the population standard deviation without special cause variation. They would then calculate the control limits, sample one bag from each nozzle, and use a subgroup of three to plot the control charts. Subgroups made in this fashion meet the assumption of common cause variation within the subgroups. The R bar chart will give information about within nozzle variation and the X bar chat will give information about variation in time. But such an application would be a very conservative control charting approach, and the chance of conducting an alpha risk would be very high.

Proper planning, as well as an understanding of the process, will help with the application of control charts. In this situation it would be a good idea to actually decide on a sampling plan - like collecting five consecutive readings from each nozzle each hour until 25-30 subgroups per nozzle are collected.

Various tools and techniques can be applied to confirm the assumption of common causes between the nozzles. A good Six Sigma black belt would never only rely on the results presented from one method, but would instead apply various methods and confirm the hypothesis based on the analysis.

For example, Figure 1 displays a simple box plot that can be used to verify whether there is any variation between the various machine nozzles.

The above box plot graphically displays whether there is any difference between the various nozzle heads. The graph above indicates that there is no apparent difference between the various nozzles.

This analysis can then be confirmed with a multi-vari chart. A multi-vari chart can yield insights on whether the variation due to nozzles is high or the variation due to time factor is high. Such an analysis can be seen in Figure 2.

The graphical multi-vari chart indicates that there is no cause of concern regarding the variation due to the nozzle heads.

The variation can now be quantified by using a nested analysis of variance (ANOVA) as shown below in Figure 3.

The ANOVA further confirms the fact that of the total variation nozzle head contributes to only to 0.18%. A hypothesis test for equal variances can also be used, the results of which are shown below in Figure 4.

All the various tools shown above will help confirm the hypothesis. Once these assumptions are confirmed, it would be safe to use three as the subgroup size as representatives of each of the nozzles. In case the nozzles are found to behave differently, it would make sense to focus on the particular nozzle behaving differently by creating control charts for that particular nozzle so that more understanding of the process can be gained.

As we've seen from the four examples presented above, it is imperative that Six Sigma black belts understand which statistical tools can help understand process variation, and what underlying assumptions are associated with each tool.

About The Author
Shree Phadnis is a Master Black Belt at KPMG India, and is an ASQ Certified Quality Manager and Engineer. Mr. Phadnis is a member of American Society for Quality.