## 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.**

George S. baggs says

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.

Andrew Milivojevich says

So true!

Naseem Akhtar says

How will you estimate and create control chart for the part having three outer diameters

Xbar Rchart or Sbar will be suitable.

Andrew Milivojevich says

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.

Naseem Akhtar says

Yes , your interpretation is correct.

Andrew Milivojevich says

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