#### Control Chart Constants, where did the A2 and E2 constants come from?

In statistical process control (SPC) charting, we use the A2 and E2 constants to calculate control limits for an Average (X-bar chart) and Individuals charts. But where do the A2 and E2 constants come from? Let’s look at the following example, for an X-bar chart, that will explain how we derive the A2 constant.

#### The Generalize Control Limit Equation for Variable Charting

The expression used to compute the control limits for an X-bar chart is:

In this expression parameters μ, σ, and n represent the mean, standard deviation, and sample size. The expression, σ/√n is also called the standard error of the mean.

#### Control Chart Constants – X-bar Chart

When using an X-bar Chart we collect several consecutive samples of size, n, to form a homogeneous subgroup and compute a subgroup average. Once we have a enough subgroups, say 30 or more, we can estimate the population average. To do so, we compute the average of the subgroup averages. We call this estimate of the mean X-double bar. The modified expression appears below.

Next we need to estimate the standard error of the mean. Recall in a earlier post** (Estimating Gage Repeatability Using Range Statistics)**, I showed you how to estimate the standard deviation using the average range from the following expression.

#### Estimating the Standard Deviation Using the Average Range

To estimate the standard deviation we compute the range for each subgroup. Recall the range is the difference between the smallest from the largest value. Once we have the range for each subgroup we then calculate the average range and divide by the d2 constant. To learn more about the d2 constant read the following post **(Range Statistics and the d2 Constant)**.

#### Control Limit Equation for X-bar Chart

We can now substitute equation (3) into equation (2) to get equation (4) as shown below.

We now have the final equation to compute the control limits for the X-bar Chart based on the average range (R-bar). Take special notice of the expression 3/d_{2}√n. * This is the A2 constant*.

The A2 constant is a function of the sample size n. Once we know the sample size, n, we can find the value for d2 and compute the value for A2.

#### Control Chart Constants for A2 at n=5, n=7

Let’s assume that we want to build control limits using a sample size of n=5. In this case the d2 constant is d2=2.326. Substituting these values into equation (5) we have:

Let’s assume that we want to build control limits using a sample size of n=7. In this case the d2 constant is 2.704. Substituting these values into equation (5) we have:

#### Control Chart Constants Depend on d2

The X-Bar chart and Individuals chart both use A2 and E2 constants to compute their upper and lower control limits. In both cases we need the d2 constant. Without it we cannot estimate the control limits using equation (4). To learn more about the d2 constant and how you can derive the d2 constant read the following post **Range Statistics and d2 Constant – How to Calculate Standard Deviation**.

#### Control Chart Constants for A2 at n=2 thru n=7

In Table 1, shown are the d2 and A2 constants for various samples sizes, n=2 through n=7. We can use these d2 and A2 values to calculate the control limits for the X-Bar Chart.

#### Control Chart Constants – Individuals Chart

Let’s apply this new-found knowledge to derive the E2 constants used to compute the control limits for an Individuals Chart. When using an Individuals Chart the subgroup sample size is n=1. In this case, we can change equation (4) and use the following expression shown in equation (6).

Since n=1, notice that the sample size, n, does not appear in equation (6). Take special notice of the expression 3/d_{2}. ** This is the E2 constant**.

#### Control Chart Constants – E2

Because d2 is a function of the Average Moving Range (MR-Bar), we often compute MR-Bar based on a Moving Range of MR=2. For example, the first moving range (MR_{1}) is the absolute value of the difference between the 1^{st} and 2^{nd} values. Likewise, the second moving range (MR_{2}) is the absolute value of the difference between the 2nd and 3rd values and so on.

We can also compute MR-Bar based on a Moving Range of MR=3. In this case, the first moving range (MR_{1}) is the absolute value of the difference between the 1^{st} and 3^{rd} values. Likewise, the second moving range (MR_{2}) is the absolute value of the difference between the 2^{nd} and 4^{th} values and so on.

#### Control Chart Constants – Individuals Chart

Let’s assume that we want to build control limits using a Moving Range=2. In this case the d2 constant is d2=1.1.128. Notice this is the same d2 constant used for a subgroup size of n=2. Substituting this value into equation (7) we have:

Let’s assume that we want to build control limits using a Moving Range span of 3 values. In this case, we use the d2 constant for a sample size of n=3 which is 1.693. Notice this d2 value is the same used for a subgroup size of n=3 for an Xbar chart. Substituting this value into equation (7) we have:

#### Control Chart Constants for E2 at MR=2 thru MR=5

In Table 2, shown are the d2 and E2 constants for various Moving Ranges, n=2 through n=7. We can use these d2 and E2 values to calculate the control limits for the Individuals Chart.

This post on Control Chart Constants is a subset of the broader topic of Statistical Process Control Charting. To learn more about Control Charts, please refer to the following link: **What are Control Chart**?

#### Control Chart Constants Explained!

So if you ever wondered where the A2 and E2 constants came from – now you know! I trust you enjoyed this post on Control Chart Constants.

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Now, I’d like to hear from you. If you enjoyed this article or have other comments please let me know. I enjoy hearing from my readers!

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Ger de Waard says

Dear Andrew, I came recently to your article, but I have a question. How can i generate in Excel a Relative Efficiency of the Range to estimate the variance, s2 tabel.. You have presented it till n=6, but Minitab advices and uses 2 <n< 9 hence 2-8 for Xbar-R. I would like to make a table that shows when it would really be advisable to either do a Xbar-R or go to Xbar-S chart, but al teat till n=8. Much appreciated. If not possible to you have a more comprehensive table that you could share with me?

Andrew Milivojevich says

Hello Ger de Waad.

I addressed your question via a simulation in the following post.

https://andrewmilivojevich.com/xbar-r-chart-versus-xbar-s-chart/

I trust you find it helpful.

Best regards,

Andrew

gebreyohannes says

Hello Andrew Milivojevich

I have a question about the Control Chart Constants. When is A2 and A3 used?

When would we use A2 as opposed to A3? The same for D3, D4?

With best regards?

Andrew Milivojevich says

The A2 constant is used when computing the control limits for the Xbar or Individuals Chart when the data in a subgroup is based on the Range or Moving range. However, A3 is used when calculating the control limits for the Xbar chart when the data in a subgroup is used to compute the standard deviation.

With respect to D3 and D4. D3 is used to compute the lower control limit for the ranges. However, the LCL = 0 when the subgroup size of n=6 or smaller. Alternatively, D4 is used to compute the upper control limit for the ranges.

Suyash Bansal says

Thanks! It helped 🙂

Steve says

Why do they use A2 and E2 for the constants? That is, why not A1 and E1 or A3 and E3? Who came up with using A2 and E2 and why did they decide to use those for the constants? Thanks.

Andrew Milivojevich says

Hello Steve.

I’m not sure why they call it A2 and E2.

But, the A2 and E2 constants depend on the d2 constant.

As such, I believe the “2” in A2 and E2 is borrowed from the d2 constant.

Best regards,

Andrew Milivojevich

David Gallois says

Thank you for your time and response. This solves the issue in my mind completely.

Thank you again.

Dave G.

Andrew Milivojevich says

Hello David.

I am glad I could help.

If you have any other questions or would like me to write about a topic that is of interest to you then please let me know.

In the meantime, I hope you enjoy the website!

Best regards,

Andrew

Dave Gallois says

Dear Andrew,

Thank you for explaining that E2 = 3/d2. However we are struggling to find a table of constants that will give us either the E2 or d2 constants for a subgroup size of (1). We are using a XMR control chart for samples that are homogeneous per batch.

Andrew Milivojevich says

Hello Dave.

Please refer to Table 2, in the post, it shows the constants for E2 and d2 for a moving range span of 2, 3, 4, and 5 values.The E2 values, in that table, only apply for an XMR.

Sometimes there is some confusion when we talk about XMR charts, since 1 value is collected at each sampling period n=1. However, to examine the variation, in the data series, we compute the moving range. As such I like to refer to this as the MOVING RANGE SPAN. When the moving range span is n=2, we are looking at a moving range of 2 adjacent values. When the moving range span = 3 we are looking at the moving range between the 1st and 3rd sample. Even though we collect 1 value at a time we are using “subgroups” when we compute the moving range.

I trust this explanation gets to the heart of your question.

Best Regards,

Andrew

Martina says

Dear Andrew,

thanks for sharing the text.

I am interested also in derivation of other constants for calculating limits in control charts (specially A3, D4, D5, B4, B6 constants). Could you please recommend me a literature that deals with the problem?

Andrew Milivojevich says

Hello Martina.

I am currently writing a post on how to derive the D3 and D4 constants used to compute control limits for the range chart. Once I post this article I will look to prepare another post that discusses the other constants you requested. I apologize that I cannot promise an exact date I will post such an article. Please stayed tuned!

Best Regards,

Andrew

Premlata says

Thank you so much sir for explaining in easy way.

Ali says

Hi Andrew, Happy New Year – let 2016 be full of exciting getaways and unforgettable experiences!

Andrew Milivojevich says

Hello Ali,

Thank you! And Happy New Year to you as well! I trust 2016 will be a great year for all!

Vinay Prakash says

Thank you very much for sharing and explaining in easy language.