Interpreting An SPC X-bar And R Chart

In the last few articles we have looked at the basics behind SPC and the completion of an SPC chart. Now we have a completed chart we will look at how to interpret it. Firstly, however, we need to lay a few more foundations.

Types Of Distribution

We will cover the most common types, which are Gaussian, Bimodal and Skewed distributions. We will also consider the other descriptors, being the location (mean, mode and median) as well as spread.

Skewed Distribution - this is where the data is skewed to one side. the mode (highest occurring data point), the mean (or average) and median (middle number) do not coincide.

Bimodal - here we have a distribution that typically contains two of something (two processes, two tools or cavities, two measuring stations). As will be seen later, this can be misleading when trying to interpret the SPC chart. It is important to ensure that data is collected from one source only.

Gaussian - a distribution that is equally dispersed about the mean value. The mean, mode and median values are the same. In SPC we use this form of distribution by taking several samples (the sub-group), as this will tend any distribution towards a Gaussian curve. This enables probability to be assessed. 

The problem with some processes is that they do not behave in a way that presents the data in a normal (Gaussian) distribution. As mentioned in previous articles, we use sub-groups of data, which have the tendency to present the data in a normalised format.

Control Limits

Control limits are calculated based upon what the process is generating. As previously mentioned, control limits should only be calculated when the process is devoid of special causes of variation, and when all the normal causes of variation have been included. This includes but is not limited to:

• Changes of shift
• Normal changes in the environment (internal and external)
• Change of supply of materials (under normal circumstances)
• Autonomous maintenance
• A minimum of twenty data sub-sets

This applies to both control limits for range (R) and for averages (X-bar) of sub-sets.

Interpreting The Sample X-bar And R Chart

As part of the last SPC article you will have given access to a sample chart (if you have not yet downloaded this you can find it here). We will be referring to this chart during the remainder of this article so having it to hand will be useful.

The sample chart has been produced based on data from a machining operation, producing a small shaft.

The 7 Point Rule

Well, not really a rule, more of a recommendation. However, in the article titled 'Probability And SPC', we discussed what we can expect from a process under normal conditions.

The possible outcomes could be any one of the following eight:

• All above the mean
• All below the mean
• All within the +/- 1 σ
• All within the +/- 2 σ
• In one half of the distribution
• All going up
• All going down
• In the outer 1/3 area

The 7 Point Rule suggests that under normal circumstances we would expect that each sub-set of data would not  have more than seven successive points that are in one of the eight possible outcomes.

This is based upon the probability of this occuring of 1 / 128 or 2⁷. 

In other words, each data set has two possible outcomes, either a repeat of one of the above possible outcomes or any other outcome.

For example, if the last data set was in the +/- 1 σ area, you would not expect to see a further six based upon the probablity theory.

The Range Plot

In this example we can see there are no seven points that could be considered to be 'out of control' conditions.

Out of control conditions are those situations that are brought about due to a change. A special cause of variation is present.

The X-bar Plot

Again this is taken from the chart previously provided

At first sight all would appear to be good. However, there are a number of successive points within the +/- 1 σ area, this would need further investigation.

In this example, the control limits have been calculated based upon the first set of twenty sub-sets of data, so 'Whats the problem?' you may ask.

We have seen a lot of instances where the individual data values have not been plotted as part of the SPC chart.

Not plotting this information, as in this case, can lead to misinterpretation of the X-bar plot.

Here we can see that there are two distinct distributions of the individual data points, in other words a bimodal distribution. As previously stated, this cannot be used in an X-bar and R chart, as it will be masked by the process of calculating the mean values of each sub-set of data.

The investigation would need to centre on what has caused this bimodal distribution (special cause)

This example clearly shows the need for plotting of the individuals

Summary

Reviewing the example X-bar and R chart shows how powerful this tool can be for reviewing the control of processes.

We have used an example of a manufacturing machining process but SPC is just as applicable to many other processes, not just manufacturing. In fact, any process from which data can be gathered can be controlled using SPC.