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:
UCLranges = D4Rbar (Rbar = 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.
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