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.