Xbar R Chart
Xbar R Chart versus Xbar S Chart
I recently got a question from a reader that wanted to know when to use an Xbar R chart versus Xbar S chart. In this post, I will answer that question.
Well, the decision to use either of these Statistical Process Control Charts comes down to how the average range and the average standard deviation estimate the population.
Since we use the average range and the average standard deviation to compute the control limits for the Xbar Chart, then having a standard deviation that estimates the population best is critical.
Let’s do a simulation…
So, I decided to conduct a simulation. I generated 50 subgroups with 15 samples in each subgroup using a Mean of 50 and standard deviation of 5. In total, there were 750 values. I show a partial table in Figure 1.
Simulated Values Using Mean = 50 and Standard Deviation = 5
Figure 1: Simulation of 50 subgroup with up to 15 values per Subgroup
Let’s Estimate the Standard Deviation based on the Range
This simulation examined 50 subgroups were each subgroup had n = 5 to 15 values. For each subgroup, of n=5 to 15 values, I computed the Range and then calculated the Average Range across all 50 subgroups. The Average Range was then divided by the appropriate d2 constant for each subgroup made up of n = 5 to n = 15 values. I show this shown in table 1.
Table of Standard Deviation Estimates Based on the Range
Table 1: Standard Deviation Estimated From Average Range for n = 5 to 15
Let’s Estimate the Average Standard Deviation
For the next part of this simulation I computed the standard deviation for each subgroup, of n=5 to n=15 values, and then calculated the average standard deviation across all 50 subgroups. Now I had the average standard deviation for n = 5 through n = 15 values per subgroup.
Table of Average Standard Deviation Estimates
Table 2: Standard Deviation Estimates for n = 5 to 15 values per Subgroup
Let’s Plot the Results
I then plotted the data in tables 1 and 2 in the following graph shown in Figure 2.
Plot of Standard Deviations Estimate for n=5 to n=15 Subgroups
Figure 2: Plot of Standard Deviation Estimates Based on the Range (BLUE) and Average Standard Deviation (RED) for n=5 to n=15 values per subgroup
Now we can Interpret the Results
In Figure 2, I show the population standard deviation as a horizontal line through a standard deviation at 5. Notice that the standard deviation, based on the Average Range, for n = 5 to 11 estimates the population standard deviation well. But, for the same range of subgroups (n = 5 to 11), the average standard deviation under-estimates the population standard deviation.
Based on this observation, we’re inclined to believe that the Average Range estimates the population standard deviation well between n = 5 through n = 11 values per subgroup.
But what happens when you have n = 12 or more values in each subgroup? We can see that the standard deviation based on the Average Range over estimates the population standard deviation. In this case, the control limits for the Xbar Chart would be wider.
But notice that the average standard deviation for n = 12 or more values per subgroup estimates the population well. In the case, the control limits for the Xbar Chart would be just right.
Based on this simulation, we would suggest that we use the Xbar R Chart for n = 11 or less values per subgroup. Or, if we had n = 12 or more values per subgroup we would suggest the use the Xbar S Chart.
The Xbar R chart and Xbar S chart are awesome tools. For additional information on these Statistical Process Control Charts (Xbar R Chart versus Xbar S Chart) check out this resource.
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It is interesting to note that back in the day, before the proliferation of computers…or even scientific calculators, the range was an easy calculation to perform by hand. This facilitated the creation of hand-plotted SPC charts by the actual operators running the process, which in turn imparted a greater sense of ownership of that process by the operator…elevating the intrinsic motivation for the operator to track down the source of any detected special cause events. Smaller subgroups (i.e. < 12) would naturally be favored by the hand calculation, which is a limitation for the hand calculation approach. However, when data is gathered remotely and control charts are displayed on a network, process managers must be diligent to ensure the process operators remain engaged in problem-solving efforts.
So true!
How will you estimate and create control chart for the part having three outer diameters
Xbar Rchart or Sbar will be suitable.
Hello Naseem.
Please confirm that I understand your question.
You have 1 part, where there are 3 features and each feature has a different diameter?
If this interpretation is correct please reply.
Yes , your interpretation is correct.
Hello Nasseem.
In response to your question.
How would I setup a control chart to monitor three different diameters on the same part?
This is a really interesting question. but to answer this question, I will do so in the context of some assumptions.
Let’s suppose the part is made using a single form tool. As such, there may be a relationship between different diameters and how they change. If this is correct, and a relationship can be shown to exist, and the relationship is strong, then monitoring one diameter and using this as a predictor of the other diameters may be possible.
Assuming this is the case, sampling one diameter and then predicting the others using a reduced sampling program for these diameters may be one way to approach your problem.
Please note, there are several assumptions in what I am suggesting. For example, a single form tool that forms different diameters may have different linear contact profiles with the part. This potentially means that a higher degree of wear may exist with that diameter (always need to understand the system). So, if the wear is different between the diameters, then I would concentrate on that diameter that tends to wear out faster then the others.
Secondly, if diameter wear is a feature you wish to monitor, then you might want to use a linear regression technique that establishes control limits about the rate of change in that diameter that changes faster then the other. This then becomes the early warning feature that signals you that something is wandering out of control. Furthermore it provides an potential means of predicting how long it takes the diameter to wear prior to going out of specification.
Best regards,
Andrew