### D2 values for the distribution of the average range appear in the following table.

The columns and rows represent the subgroup size (n) and number of subgroups (k). For a given subgroup size, say n=2, notice that the value of d2 changes as the number of subgroups, k, increases. As an example, notice that d2=1.150 for n=2 and k=15. When k is infinite, the value of d2=1.128. Notice the difference in the value of d2 is small.

When we use control charts, such as the average and range chart, we often wait to collect k=20 to 30 subgroups. The reason we do so is simple. The value of d2 when k=30 is close to the value of d2 derived from a continuous distribution of subgroups having the same subgroup size. This is why the table of d2 values is published up to k =15.

Control charts use range statistics and d2 values to estimate the standard deviation to compute control limits. The average and range chart is a perfect example. Such a chart is often used to track the behavior of a product feature during production.

### Sample Parts made Under Like Conditions

Often production personnel sample n=5 consecutive parts from a process and plot the average and range. Sampling in this manner often assures we have parts made under like conditions. Once we have many subgroups we can compute control limits about these averages. When the subgroup average or range falls outside a control limit we should question if the conditions changed. So the calculation of control limits based on a subgroup of parts made under like conditions is an important concept.

### How many subgroups do we need to compute control limits?

To compute the control limits, we need to collect a critical number of subgroups. Often we use 30 or more subgroups. Once we compute the range for each subgroup we then calculate the average range. To compute the average range we add the ranges and divide by the number of range values. We use the following expression to compute the average range.

Once we know the average range we need to look up the correct d2 constant. As an example, suppose we collected k=30 subgroups where each subgroup contains n=5 parts. Also assume that the average range is 10.82. As such, the appropriate d2 value for n=5 is 2.326. Using the expression below we can estimate the standard deviation for a subgroup of parts. This standard deviation is a often referred to as measure of within subgroup variation.

So there you have it. This is how we estimate the within subgroup standard deviation for a collection of parts made under like conditions. In another post, I’ll discuss how we can compute the control limits for the average and range charts using within subgroup variation. If you want to learn more about range statistics then click on the following links.

**Range Statistics and d2 Constant | How to Calculate Gage Repeatability Using the Average Range**

If you liked this post or have any questions then please let me know by adding your comments below. I look forward to hearing from you.

Anju says

Okay Andrew.

Thank you a lot for the clear explanation. It was really halpful for me to understand the concept in a better way.

Also, looking forward for your posts on the same.

Thanks once again.

Anju

Anju says

Okay Andrew.

Thank you so much for your clear explaination. It was really helpful for me to understand the concept a little more.

Also, looking forward for your posts on the same.

Thanks,

Anju

Anju says

Thank you so much Andrew.

But my actual requirement is to get d2 values for k=1 and n>20…….

Currently, a case arised where, I am in need of d2 value for k=1 and n=40.

So I wanted to know, if their is any formula that can get us through values of d2 for k=1 and n> 20 …ie., n=21, n=22, … till n=100.

Once again, Thanks in advance.

Andrew Milivojevich says

Hello Anju. I am not aware of an empirical expression that can answer your question. In the absence thereof, I would suggest that you try to derive the values via simulation.

Best Regards,

Andrew

Anju says

Hello Andrew,

Thank you so much. Let me try that then.

Best Regards,

Anju

Andrew Milivojevich says

Hello Anju.

I found a reference and it might have an expression that estimates the d2 value. I will get back to you shortly.

Andrew

Anju says

Hello Andrew,

Wow! That’s great. Thank you. Looking forward for your reply.

Anju

Andrew Milivojevich says

Hello Anju.

I was only able to derive a regression model that estimates d2 to two decimal places. However, this regression model provides a d2 estimate based on an infinite number of k subgroups. In your case, you need corrected d2 values based on k=1 subgroup. Such d2 values are slightly larger. So I did find an expression that estimates these corrected d2 values. These corrected d2 values depend on the degrees of freedom for k=1 subgroup and n samples within that subgroup. So, I need to do some more digging to see how these degrees of freedon are estimated for the range of n you requested. If I find something I will post it here. Sorry I could not have better news for you.

Best Regards,

Andrew Milivojevich

Anju says

Hello,

Thanks a lot for the page.

For k=1 subgroup, n>20 subgroup size, can we get the d2 value? Please help.

We just have the d2 value for n>20 only for infinite subgroups.

Thanks in advance.

Andrew Milivojevich says

Hello Anju.

For k=1 and n=20 i believe the the d2 value is 3.80537.

I hope this helps.

Andrew

Christopher says

Hello Andrew,

Many thanks for this quick reply. I have found the table AIAG’s 4th Edition MSA Manual as you said, exactly what I was looking for.

Christopher

Andrew Milivojevich says

Hello Christopher.

Glad I could help!

Andrew

Christopher says

Dear Mr Milivojevich,

First of all thank you for this post. It helped me better understand the way to use de d2 index.

However, I have a case were I have 10 subgroups (k) with a size of 20 for each one of them (n). How shall I proceed to determine the d2 index?

Thank you in advance for your return.

Yours sincerely,

Christopher Thorne

Andrew Milivojevich says

Hello Christopher.

For k = 10 subgroups and n = 20 samples per subgroup, the d2 = 3.74205. If you have an infinite number of k subgroups then d2 = 3.735.

You can get these values from AIAG’s 4th Edition MSA Manual, page 203.

Andrew