Xbar-R Chart

An Xbar-R Chart is a control chart used in quality control to watch how a process behaves over time.

The Xbar-R Control Chart consists of two parts:

Xbar Chart: Plots the mean of small subgroups of measurements. Shows if the process average is staying steady or drifting up/down.

R Chart: Plots the range (difference between the highest and lowest) within each subgroup. Shows if the process consistency (spread or variability) is stable.

With the help of a Xbar-R Control Chart you can identify trends, shifts, or any unusual patterns that might indicate a problem with the process. Therefore, Xbar-R Charts give important information about the stability of the respective process.

When to Use Xbar-R Charts?

Depending on how your data are structured, you'll select the appropriate control chart—for example, an Xbar-R chart is ideal when you're working with small subgroups of two to ten observations and your measurements are continuous and approximately normally distributed.

Example Xbar-R Chart

Imagine we work in quality management at a fulfillment center, where products are stored, packed, and shipped. To ensure efficient order processing, we monitor the time from order receipt to shipment. Our goal is to track the average processing time and ensure it remains within acceptable limits.

Of course, we need data to monitor the process. To obtain data, we take a random sample of 5 orders per day. So at the first day we measure the processing time of five orders.

For example, the processing time for the first order was 12 minutes, the processing time for the second order was 14 minutes and so on and so forth.

Similarly, we measure the processing time on the second day on the third day and so on. Let's say we measure the times on a total of 25 days.

Create an Xbar Chart

To Create an Xbar Chart we first calculate the mean values of the 5 orders from all 25 days.

Now we can create the xbar chart. To do this we plot the 25 days on the x-axis and the mean values we just calculated on the y-axis.

Now we are almost finished, we only need to calculate the three lines. The center line is simply the mean value of all values. So we just calculate the Mean Value of all points. The red lines are the upper and lower control limits.

Upper and Lower Control Limits

The Upper Control Limit (UCL) and Lower Control Limit (LCL) define the boundaries of expected process variation, typically set at ±3 standard deviations (sigma) from the mean.

• UCL = Mean + 3σ
• LCL = Mean − 3σ

Here, σ (sigma) refers to the standard deviation. But how do we calculate 3σ? This is where it gets interesting: instead of using 3σ directly, we use A₂ times R̄.

So, what are A₂ and R̄, and why do we use them? Let's break it down.

You might think let's just use the “normal” formula for standard deviation. But that approach can be pretty unreliable when you're dealing with small subgroup sizes.

For example, in our case, each subgroup only has 5 samples, each time we have 5 orders, and when we calculate the standard deviation for each subgroup, it tends to vary significantly.

R: Range

That's where the range comes into play. The range is just the difference between the highest and lowest value in a subgroup.

And, when we have a small sample size the range is less sensitive to fluctuations in each subgroup.

So, lets calculate the range for each sample. For example on day 1, the lowest value is 12 and the highest is 15, so we get a range of 3.

When we have the range for each sample, we can calculate the mean of this Ranges R̄ (Rbar).

So we have the overall mean which is 13.38 and we have the mean of the ranges R bar which is 3.56. But what about A₂?

We simply look up the value of A₂ in a table. Here, n is the subgroup size. So for a sample size of 5, we get an A2 value of 0.577.

And with that we can calculate the control limits. We get an upper control limit of 15.44 and a lower control limit of 11.33.

But where do the values in the table come from? Researchers found that for any subgroup size n in normally distributed data, there's a consistent relationship between the average range and the true standard deviation. This relationship is represented by the constant A₂.

So Multiplying the average range by A₂ gives us a reliable estimate of 3 Sigma. And for small subgroups, this method works better than simply computing the standard deviation directly.

Create a Xbar-R Chart

Now we have the so-called Xbar chart, in most cases the xbar chart is extended by the R chart. As we know, R stands for range.

To create the R chart, we simply need the Ranges we have just calculated.

Ok, but how do we calcaulte the Upper control limit and the lower Control limit in this case?

You can calculate the control limits using this formula. Here, R bar is again the average range, and D₃ and D₄ are constants that depend on your subgroup size.

Create an x bar R Control chart online

If you want to create an x bar R Control chart online with numiqo, simply copy your data into the table and click on statistical process control. Now you only need to select the variables and you will get an xbar-R chart.