## 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

**and the**

*average range***estimate the population.**

*average standard deviation*Since we use the ** average range** and the

**to compute the**

*average standard deviation***for the**

*control limits***, then having a**

*Xbar Chart***that estimates the population best is critical.**

*standard deviation*### 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

**was then divided by the appropriate**

*Average Range***for each subgroup made up of n = 5 to n = 15 values. I show this shown in table 1.**

*d2 constant***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

**across all 50 subgroups. Now I had the**

*average standard deviation***for n = 5 through n = 15 values per subgroup.**

*average standard deviation***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

**at 5. Notice that the**

*standard deviation***, based on the**

*standard deviation***, for n = 5 to 11 estimates the population**

*Average Range***well. But, for the same range of subgroups (n = 5 to 11), the**

*standard deviation***under-estimates the population standard deviation.**

*average standard deviation*Based on this observation, we’re inclined to believe that the ** Average Range** estimates the population

**well between n = 5 through n = 11 values per subgroup.**

*standard deviation*But what happens when you have n = 12 or more values in each subgroup? We can see that the ** standard deviation** based on the

**over estimates the population**

*Average Range***. In this case, the**

*standard deviation***for the**

*control limits***would be wider.**

*Xbar Chart*But notice that the ** average standard deviation** for n = 12 or more values per subgroup estimates the population well. In the case, the

**for the**

*control limits***would be just right.**

*Xbar Chart*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.**

Naseem Akhtar says

September 15, 2017 at 9:31 amHow will you estimate and create control chart for the part having three outer diameters

Xbar Rchart or Sbar will be suitable.

Andrew Milivojevich says

September 22, 2017 at 4:38 pmHello 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.

Naseem Akhtar says

October 10, 2017 at 7:29 amYes , your interpretation is correct.