**How to Calculate Gage Repeatability Using the Average Range.**

I’m often asked to help Quality Engineers better understand how to perform Gage R&R calculations. So I decided to write several posts about the topic. Over the next few weeks I plan to roll out a number of posts I trust will enlighten everyone. In this post we’ll discuss how to calculate Gage Repeatability using the average range.

The best place to start is with a couple of definitions. When someone refers to a Gage R&R (Gage RR) they’re referring to Repeatability and Reproducibility.

### **The definition for Repeatability is:**

**Repeatability is the variation in values obtained with one measuring instrument when used several times by an operator while measuring the identical feature on the same part.**

** The definition for Reproducibility is:**

**Reproducibility is the variation in the average of values made by different operators using the same measuring instrument when measuring the identical feature on the same parts.**

The Automotive industry Action Group (AIAG) publishes a **Measurement System Analysis (MSA) manual**. Many consider this manual an authoritative source in conducting a Measurement Systems analysis. In conducting a Gage RR there are two methods the MSA manual refers to; they are the Average and Range and ANOVA methods. We will be discussing the Average and Range method. It is the method most often used, but inferior to the ANOVA method. The ANOVA method accounts for the interaction between the operator and the parts measured. The Average and Range method does not take this interaction in consideration.

### **Step 1: How to Calculate Gage Repeatability – Collect the Data**

In this post, lets consider the following data set. Here we show ten parts selected at random from a process. The parts represent the typical variation exhibited by the process. The Quality Engineer conducting the study presents the sample to two lab operators. Each operator measures a specific feature of a part that’s presented randomly and records a value. Once the operator completes their measurements they repeat the process again. The data in the table represent two values for the same part feature measured twice by the same operator using the same gage.

Let’s examine the data in the table above. Here we have two values for each part. Each number, for a part, represents a value for the same feature, from the same gage measured by the same person. This is the definition of repeatability and we say that we have r=2 trials for each part where “r” represents the number of values.

Now that we have our data how do we compute repeatability? One way to do so is to compute the standard deviation for each of the 10 parts in this study. Then, pool all 10 values and compute a single standard deviation. But such a calculation is somewhat cumbersome unless you use a computer program. Fortunately we can use simpler methods to perform these calculations. Using the Range method we can estimate the standard deviation.

### **Step 2: How to Calculate Gage Repeatability – Calculate the Range**

The Range method is easy; just subtract the smallest value from the largest value for each part. This difference represents the range. I show the range for all 10 parts in the table below.

Now that we have the Range for each part we can now compute the Average Range. To do so we add all the Range values and divided by the number of parts. In this case, the average range equals:

Before we compute the standard deviation based on the average range we need to stop and think for a moment. We need to make sure that all the range values used to compute the average range are homogeneous. This means that the range values must be free of special causes. We typically check this by examining the range values using a Range Chart. Such a chart displays the range values, average range and upper control limit for the ranges. Any range value that exceeds the upper control limit for ranges would be unusual. This is a signal that one of the two values for a part may be in error. The person conducting the study should identify the part in question and have it re-measured. If the range value falls below the upper control for ranges then we use the new values. The process that examines the suitability of the ranges is critical. If not performed we risk having an average range that’s inflated. This will increase gage repeatability and we may falsely reject the use of the gage.

### **Step 3: How to Calculate Gage Repeatability – Check for Unusual Range Values**

To compute the Upper Control limit for the ranges we use the following expression:

**UCL _{ranges} = D_{4}R_{bar} (R_{bar} = average range)**

The D4 constant is a function of the number of trials, r. In this case, we had two values for each part so r=2. I show the values for D4 for r=2 and r=3 in the table below.

In our case, we use D4 = 3.268. Since the average range (Rbar) is 4.2 we can compute the Upper Control Limit (UCL) for the ranges. In this case, the UCL = 3.268(4.2) = 13.73. Since none of the range values exceed the UCL we can use the average range to estimate the standard deviation. (In another post we’ll also discuss how we can use the range values to determine if we have a sufficient number of distinct data categories).

To compute the standard deviation based on the average range we use the following expression.

In this expression, we divide the average range (Rbar) by d2. Here d2 is a function of the number of values (r) per part. In this example, we have r=2 values per parts. From the table below the value for d2 when r=2 is 1.128. For those of you that have an interest in knowing where d2 comes from and how it’s derived then refer to the following **post**.

### **Step 4: How to Calculate Gage Repeatability – Estimate the Standard Deviation**

So, now we can estimate the standard deviation due to gage repeatability. The answer is:

So now you know how to compute gage repeatability for a single operator. In the next post we’ll look at how to pool the average range across other operators and estimate the standard deviation due to gage repeatability.

I welcome your questions and your feedback. If you enjoyed this post then please let me know by leaving your comments below. I value your feedback – they help me help you.

Luca says

Hi,

I was wondering what the intuitive meaning is of D4? I know the value depends on the number of trials and decreases if the number of trials increases, but how can I intuitively understand what this value is?

Kinds regards,

Luca

Andrew Milivojevich says

Hi Luca.

The D4 constant is a value when multiplied by Rbar computes the UCL for the Ranges. It is a simpler way to compute 3 sigma limits.

I am publishing a FREE ebook soon that will show this in more detail. In the mean time you can read the post: Xbar and Range Charts. I go into the derivation of the D4 constant in that blog post.

Best Regards,

Andrew

Skarlett says

Hi Andrew.

In the this MSA model: y = x + e (where x is the true value of the measurement and e is the measurement error), we estimate the variance of the total observed measurement as: sigma^2_total = sigma^2_process + sigma^2_gage.

An we calculate sigma_gage= Rbar/d2.

However, in Control Charts, we estimated sigma_total = Rbar/d2.

Why is the overall variability of the measurement different in the two models?

Thank you!!

PS: This is base on page 380, Montgomery : Introduction to Statistical Quality Control (7th edition)

Andrew Milivojevich says

Hello Skarlett.

The estimate of Repeatability for the Gage and the X-bar/Range Control Chart both estimate the standard deviation based on subgroups (R-bar/d2).

When the X-bar/Range chart is in a state of statistical control we estimate the standard deviation using R-bar / d2. In this case, if you compute the total standard deviation by taking the square root of the sum of squared deviations divided by the degrees of freedom you will have the same answer. This implies that the variation between the subgroups cannot be distinguished from the variation within the subgroups. In this case, we have a system of common causes. Therefore R-bar/d2 is a measure that estimates a system of common causes.

When the variation between the subgroup averages increases to a point that it exceeds the variation contained within the subgroups, then we have a system of special causes. Now, the total variation is the sum of both special and common causes. In this case. the total variation increases and the R-bar/d2 estimate no longer estimates the total.

I hope this helps.

Best Regards,

Andrew

Tom Booth says

Hi Andrew, AIAG computes repeatability by multiplying the average range by K1 (4.56 for 2 trials and 3.05 for 3 trials), but you divide by d2, which is 1.128 for two trials or 1.693 for three trials. What is the difference? Thanks, Tom

Andrew Milivojevich says

Hello Tom! Thanks for the question.

The first and second editions of the AIAG manual used K1 = 4.56 or 3.05 form=2 or 3 replicate observations. When computing reproducibility K2 = 3.65 or 2.70 for 2 or 3 operators respectively. By the third edition K1 = 0.8862 and 0.5908 for m=2 or 3 replicate values and K2 = 0.7071 and 0.5231 for 2 or 3 operators.

In the second edition, the general expression to compute the K values is: K* = 5.15 / d2*. The value 5.15 corresponds to 99% of the area under a normal distribution curve. As such, when estimating the standard deviation for repeatability (EV), reproducibility (AV), and part variation (PV) we would multiple these standard deviations by 5.15 to estimate the spread (ie. 5.15 times the standard deviation of each estimate – EV, AV and PV) corresponding to 99% of the area beneath a normal distribution curve.

By the third edition, the 5.15 value was no longer needed because it was a common value that does not alter our result when we compute the percentage of the total standard deviation (the 5.15 value appears in the numerator and denominator and thus cancels out). Therefore by removing it, the general form for the K expression becomes: k* = 1/ d2*. In the third edition the d2* values are found in Appendix C. For K1, the d2* values, in Appendix C, depends on the number of range values. So if we have 10 parts and 3 operators we have a total of 30 subgroups. If each subgroups has m=2 values then we can enter Appendix C and look up the value for d2. In this case, m=2 (two measurements per part) and g = 30 (number of range values = parts time number of operators). In this case, the value of d2 = 1.12838 and if we take the inverse of this value we have 1/1.12838 = 0.8862. If we have m=3 values per subgroup and g=30 groups, the value of d2 in Appendix C is 1.69257 and the inverse of this value yields K1 = 0.59089. So that takes care of the K1 values. How about K2?

When we compute the range across operators we only form a single subgroup that has only 2 or 3 value depending on the number of operators. When we use Appendix C we have m=2 or 3 and g = 1 because we only have 1 subgroup of ranges value for the operators. This results in d2 values equal to 1.41421 and 1.91155. Taking the inverse of these values we have K2 = 0.7071 and 0.5231.

Finally, how about K3? In the AIAG manual displays the K3 values 2 through 10 parts. Since the range is computed across all parts we only form 1 subgroup so g=1 in Appendix C. As such, only the number of values in that subgroup change from m=2 through 10. In this case, the d2 values are: 1.41421, 1.91155, 2.23887, 2.48124, 2.67253,2.82981, 2.96288, 3.07794, 3.17905. Taking the inverse of these values result in K3 values of: 0.7071, 0.5231, 0.4467, 0.4030, 0.3742, 0.3534, 0.3375, 0.3249, 0.3146.

Thanks for the question! I appreciate it.

Best Regards,

Andrew

Emine says

Hello Andrew!

I was curious about the answer to this question. It’s nice. Thanks for your reply.

Dokuz Eylül University

Emine

chris says

Hi Andrew, I would like to conduct a gage type 1 study on a process where we measure the bulk density of soil. Our normal process is to hold a standard cup under a conveyor to collect soil as it falls to a WIP bin, then level the cup off and measure the bulk density. (note: Since its a natural product the bulk density can vary minute to minute)

I have designed the gage type 1 repeatability study by collecting a qty of soil into a bucket, to use as my trial sample (this was to ensures the bulk density within this sample remains consistent throughout the trials)

An operator was then tasked to measure the bulk density 25 times as per SOP using the trial sample.

My question is I don’t have a range for each sample measurement, only the range for the entire 25 samples. Hence how can i calculate if the measurement is repeatable. Example bulk density trial measurements attached

694

679

700

670

680

687

671

687

699

690

670

673

672

691

691

675

690

697

694

682

679

690

695

678

695

Thanks

Andrew Milivojevich says

Hello Chris.

The process of measurement, you described, is consistent with the definition of Repeatability. In your case, 1 sample, having the same feature, measured by the same person, using the same operating procedure.

Having 25 measurements, you would use the sample standard deviation, instead of the Range, to estimate Repeatability. Assuming I entered the values you provided correctly, the sample standard deviation = 9.78. Using a value of 5.15 to describe the spread in observations we have 50.4 (5.15 x 9.78).

We know need to ask if this sample standard deviation and corresponding spread in observations is suitable. Please let me know if these values, that describe the Repeatability of your measurement process, are to large. If so, I would be happy to describe a procedure that would improve the Repeatability of this measurement process.

Best Regards,

Andrew

chris says

Hi Andrew, thanks for the response. I arrived at a process spread of 59 (3 sigma either side of the mean, +/- 9.78 x 3), where the mean is 685. I am not sure how (5.15 x 9.78 calculation works, can you explain?)

I would like to reduce the current spread significantly, before I undertake a reproducibility study, which I suspect would generate a larger spread, so any suggestions would be most welcomed.

Cheers

Chris

Andrew Milivojevich says

Hello Chris.

The process spread you computed is fine.

The value, 5.15, is an value used in a prior edition of AIAG’s Measurement Systems Analysis Manual. It corresponds for 99% or the area under a Normal Curve. Using 6, as you did, corresponds to 99.73%.

The link below is a blog post that discusses how to use the distribution of averages to reduce Repeatability. It is a technique that is used in the Chemical Processing industry.

https://andrewmilivojevich.com/reducing-gage-rr/

If you have questions, after reading this post, please let me know in the comments sections of that post.

Best Regards,

Andrew

Johari says

Good day

s=R-bar / d2 @ 4.2/1.128 = 3.72

This answer (3.72) is in Percentage? if not in percentage ,how to know our final answer is pretty consistent . ie

0<2 is pretty consistent

2<3 is so so

3<5 can't be use.

May i know,acceptance level for Repeatability as per this equation .

Andrew Milivojevich says

Hello Johari,

The units of “s” are the same as the units for “R-bar” (since d2 is unit-less). For example, if the response of interest is Force, then “s” may be expressed in Newtons (s=3.72 N).

To know if s=3.72 N is acceptable depends on the context of how you apply this result. For example, Gage Repeatability is often expressed as a percentage of the “process spread” or “tolerance range”. For example, assume the standard deviation of a process is s=5 N. The process spread then equals: 6 x s = 6 x 5 = 30 N. The Gage Repeatability as a percentage of the process spread is: 3.72 N / 30 N = 0.124 or 12.4%. In this case, the AIAG MSA manual would consider this acceptable if measuring a non-critical part feature since it is less than 30%. When the customer part feature is deemed critical than 10% or less is acceptable. In that case, the Gage Repeatability Percentage at 12.4% would need to be improved.

Now, assume the “tolerance range” for a part feature of interest is +/- 20 N. The tolerance spread is 40 N. In the this case, the Percent Gage Repeatability is 3.72 N / 40 N = 0.093 or 9.3%. As such, the Gage Repeatability Percentage would be deemed acceptable according to the AIAG MSA manual.

Going forward, knowing how to interpret Gage Repeatability depends on how you want to apply it. If the need is to improve a “process”, you might consider activities to reduce Gage Repeatability. On the other hand, if your customer whats to know if you can meet their “tolerance range” you may need to report that number.

Please note: AIAG = Automotive Industry Action Group and MSA = Measurement Systems Analysis Manual (the latest is the 4th edition).

Thanks for the question, and I hope I have addressed it sufficiently.

Best Regards,

Andrew Milivojevich