Gage Repeatability and Reproducibility (R&R) (2024)

Gage Repeatability and Reproducibility are often referred to as Gage R&R. It’s a method to assess the repeatability and reproducibility of a measurement system. In other words, Gage R&R studies are carried out to discover how much of the process variation is due to the measurement system.

What is a Measurement System Analysis?

Measurement System Analysis (MSA) is a tool for analyzing the variation present in each inspection, measurement, and test equipment type. It is the system used to assess the quality of the measurement system.

What is a Gage and Gage Repeatability and Reproducibility?

A gage, in this context, is a tool for measurement. A gage could be simple, like calipers and rulers. Or it could be a complex piece of machinery. It could even be a piece of software.

Gage R&R focuses on two key aspects of measurement:

• Repeatability: Repeatability is the variation between successive measurements of the same part or trait by the same person using the same gage. In other words, how much variation do we see in measurements taken by the same person, on the same part, using the same tool?
• Reproducibility: Reproducibility is the difference in the average of the measurements made by different people using the same instrument when measuring the identical characteristics on the same part. In other words, how much variation do we see in measurements taken bydifferentpeople on the same part using the same tool?

Looking at these two metrics helps us to understand variation in our measurements. When we understand it, we can combat it.

Why is Gage Repeatability and Reproducibility Important?

Gage Repeatability and Reproducibility measure the amount of variability in measurements caused by the measurement system itself. Then, it compares this variability with the total to determine the actual variability of the measurement system. Gage R&R is very important when new workers are assigned; new tools are used, or any significant process changes.

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For example, imagine a situation where our performance metrics show a serious problem in our manufacturing process. We spend a lot of time and money trying to fix it and improve the performance of a process. But we’d have noticed serious measurement variations if we’d spent some time looking at gage repeatability and reproducibility instead. The problem wasn’t in the process at all; it was in the measurements. Checking this first would have saved time, money, and stress.

Types of Gage Repeatability and Reproducibility Study

Based on the available data and data type, there are basically three types of Gage R&R available:

Crossed Gage R&R

Select crossed-gage R&R when each operator measures each part, and it must have a balanced design with random factors. It is used for non-destructive testing.

Nested Gage R&R

Select nested gage R&R when only one operator measures each part. It is used for destructive testing. Since it is not crossed with other factors, it is called nested gage R&R. It must have a balanced design with random factors.

Expanded Gage R&R

Select expanded gage R&R when we need to include more factors (maximum of eight) than operator and part. Typically crossed and nested deal with only two factors (operator and part). Design can be balanced or unbalanced.

Methods to Perform Gage Repeatability and Reproducibility Study

There are basically three methods that exist to perform Gage R&R:

• Range method
• Average and range method
• Analysis of variance method

Range Method: The range method will provide a quick approximation of measurement variability but does not compute the measurement system repeatability and reproducibility separately.

Average and Range method: The Average and Range method quantifies the measurement system’s variability and provides repeatability, reproducibility, and part variation. Only crossed Gage R&R can be performed with the Average and Range method.

Analysis of Variance method: It is the most widely used and accurate method for measurement system repeatability and reproducibility. It also quantifies the variability of the interaction between the operator and the parts. Gage R&R (crossed, nested, and expanded) can be done with the ANOVA method.

Gage Repeatability and Reproducibility Using the Average and Range Method

The Average and Range method determines the total measurement system variability, which can be separated into components like repeatability, reproducibility, and part variation. Furthermore, this method requires multiple parts, operators, and trials. The Average and Range method is easy to compute; however, the ANOVA method is more accurate than the Average and Range method.

Example of Gage R&R using Average and Range method in a Six Sigma project

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Step 1:Calibratethe gage(s) being used.

Step 2:Record a lot of different measurements taken by various people on the same units using the same gage. For each measurement, ensure that the unit, the person, and the gage are all recorded.

Step 3:Interpret your results to find sources of variation.

When you’ve finished your measurement recording, your results should look similar to this:

Step 4:Find the range and mean for each combination

For Operator A and Part x , the Range = Max – Min = 0.33 – 0.29 = 0.04

Operator A and Part x, Mean = (0.29 + 0.31 + 0.33 + 0.32) /4 = 0.3125

Similarly, compute the range and mean for each combination.

Step5: Calculate the mean of means and range of each operator

For operator A: the mean of range = (0.04 + 0.04 + 0.04) /3 = 0.04

Similarly, the mean of mean of operator A = (0.3125 + 0.2875 + 0.2875) /3 = 0.2958

Similarly, compute the mean of range and mean of each operator

Now find the total mean range and difference in means (Xdiff)

Total mean range = (0.04 + 0.02 + 0.01) /3 = 0.0233

Range of means = 0.2991 – 0.2958 = 0.0033

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Step 6: Find the Repeatability- Equipment variation (EV)

EV = R̅*k₁

• Where R̅ = Total mean range = 0.0233
• k₁ =1/d2

To find the k₁, we need the d2 value from the table

The d2 value can be found in the table based on the subgroup size and the number of combinations of parts and operators (g).

• Subgroup size = number of trials = 4
• Number of combinations of parts and operators (g) = 3 parts and 3 Operators = 3*3=9

Then, we have to see the d2 value from Tables 4 and 9

refer d2 table

• d2 =2.080
• k₁ =1/d2 = 1 /2.080 = 0.480
• EV= R̅ *k₁= 0.0233 *0.480 = 0.0112

Step7: Find the Reproducibility– Appraiser Variation (AV)

• x̅_diff is the range of means = 0.0033
• Where n = number of parts = 3
• r = number of trials = 4

To find the k₂, we need the d2 value from the table

The d2 value can be found in the table based on the number of parts and the number of combinations of parts and operators (g)

• Number of parts =3
• Number of combinations of parts and operators (g) = 3 parts and 3 Operators = 3*3=9

• d2 =1.718
• k₂=1/d2 = 1 /1.718 = 0.5820

If the number is negative, set AV=0

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Step8: Calculate Gage R&R

Then, interpret the results. According to the Automotive Industry Action Group (AIAG), below are the guidelines for measurement system assessment using %GRR.

Since the Total Gage R&R is 1.12%, it is in the green zone. So, it is considered to be an acceptable measurement system based on application and cost factors.

How to Measure Gage Repeatability and Reproducibility Using ANOVA Method

• Measuring the Gage R&R with a minimum of 10 parts is recommended.
• Select two technicians to measure the parts.
• Have each technician measure each part 2 or 3 times.
• It is recommended that each technician take three measurements per part.
• Collect the measurements of parts in random order and measure the overall average of all measurements (x̿).

• t= number of technicians
• r= number of trials or replications
• p= number of parts

Step 1: Calculate the technician sum of squares

This provides the sum of squares by determining the squared deviations between the technician average and the overall average.

Step 2: Compute the parts sum of squares

It provides the sum of squares by determining the squared deviations between the part’s average and the overall average.

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Step 3: Calculate the total sum of squares

SS_Total = SS_Technician + SS_Part + SS_Tech*Part + SS_Equipment

It is the squared deviation of each individual result from the overall average.

Step 4: Compute Equipment within the sum of squares

It uses the deviation of all trials for a given part and given technician from the average for that part and technician.

Step 5: Find the interaction sum of squares

SS_Total = SS_Technician + SS_Parts + SS_{Technician*Part} + SS_Equipment

SS_{Technician*Part} = SS_Total – (SS_Technician + SS_Part+ SS_Equipment)

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Step 6: Create an ANOVA table

Step 7: Evaluate repeatability, technician, part, and interaction variance

• σ² _{Repeatability} = MS_Equipment
• σ² _{TechnicianxPart} = (MS_{TechnicianxPart} – σ² _{Repeatability})/ number of trials
• And, σ² _Part = (MS_Part – MS_{TechnicianxPart})/ (number of trials * number of technicians)
• σ² _Technician = (MS_Technician – MS_{TechnicianxPart})/ (number of trials * number of parts)

If any of the values are negative, then consider it as zero.

Step 8: Compute Gage R&R and interpret the results

• Gage R&R = σ² _{Repeatability} + σ² _Technician
• Equipment Variation (Reliability) = σ² _{Repeatability}
• Technician Variation (Reproducibility) = σ² _Technician + σ² _{TechnicianxPart}
• Part to Part = σ² _Part
• Total Variation = σ² _{Repeatability} + σ² _Part+ σ² _Technician+ σ² _{TechnicianxPart}

Calculate the % Contribution Variance and interpret the results. Below are the criteria for acceptance of Gage R&R.

Then, find the standard deviation and % study variance. Interpret the results. According to the Automotive Industry Action Group (AIAG), below are the guidelines for the measurement system assessment using % GRR.

What is the Number of Distinct Categories (NDC)

The number of distinct categories is a metric. In gage R&R, the goal is to identify the measurement system’s ability to detect a difference in the measured characteristic. It represents the number of non-overlapping confidence intervals that span the range of product variation.

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The formula for the number of distinct categories

Number of distinct categories = (Standard deviations for parts / standard deviation for gage) * √2

So, the number of categories depends on the ratio of the variability in the measuring parts and the variability in the measurement system.

Guidelines for the number of distinct categories

According to the Automotive Industry Action Group (AIAG), the number of distinct categories should be greater than 5 for an adequate measuring system.

• >=5: Adequate measuring system
• =2: Data can be divided into two: say Low and High
• =3: Data can be divided into three: say Low, Medium, and High
• <2: Measurement system of no value for controlling the system

Example of Gage Repeatability and Reproducibility (R&R) using ANOVA method in a Six Sigma project

Example: A testing engineer selected ten parts representing the expected range of process variation. Three technicians measured the ten parts three times a part in random. Assess the measurement system Gage R&R.

• t = number of technicians = 3
• r = number of trials or replications = 3
• p = number of parts = 10

Average of all the measurements = (2.78+1.87+1.87+2.36+2.36+2.21+……….+2.44+1.8+1.72+4.12+3.25+3.69)/90= 3.066

Step 1: Calculate the technician sum of squares

Calculate the average measurement for each technician

For technician A: Average value of 3 trials for ten parts = ((2.78+1.87+1.87+2.36+2.36+2.21+……….+4.1+3.88+3.56)/30 = 3.05

Calculate the squared deviation of each technician: squared deviations between the technician average and the overall average

For technician A: (3.05-3.066)² =0.0003

Similarly, calculate for technicians B and C

Add all squared deviation for technicians = 0.0003+0.0022+0.0008=0.0033

For the ten parts and three trials, the sum of deviations = 3*10*0.0033 =0.0999

Step 2: Calculate the parts sum of squares

Calculate the average of measurement for each part for all the trials for part 1: (2.78+1.87+1.87+2.56+2.22+2.14+2.56+2.22+2.15)/9=2.263

Calculate the squared deviation of each part: squared deviations of each part and the overall average.

For part 1: (2.263-3.066)² =0.644

Similarly, calculate values for 10 parts

Add all squared deviations for 10 parts = 0.644+0.577+0.537+1.542+0.357+0.833+0.677+5.914+1.165+0.703=12.9477

For 3 trials and 3 technicians, the sum of the deviations = 3*3*12.9477 =116.5294

Step 3: Calculate the total sum of squares

Calculate the squared deviation for each individual result from the overall average.

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For the first part of trial 1: (2.78-3.066)² = 0.082

Similarly, conduct for all the 90 trials and sum the squared deviation of 3 trials = 0.082+1.430+1.430+……..+1.111+0.034+0.389 =120.682

Step 4: Compute Equipment within the sum of squares

It uses the deviation of all trials for a given part and given technician from the average for that part and technician.

Take the average of the first part of the first technician’s three trials = (2.78+1.87+1.87)/3 =2.173

Calculate the Squared Deviation Trial 1 for first part = (2.78-2.173)² = 0.368

Similarly, calculate the squared deviation for all trials for each part.

Sum of 90 values = 0.368+0.092+0.092+…………………+0.188+0.191+0.000=3.606

Step 5: Find the interaction sum of squares

SS_{Technician*Part} = SS_Total – (SS_Technician + SS_Part+ SS_Equipment)

SS_{Technician*Part} = 120.682 – (0.0999 + 116.5294+ 3.606) =0.447

Step 6: Create an ANOVA table

Since the p-value for interaction is more than 0.05, we need to consider the repeatability without interaction values.

Step 7: Evaluate repeatability, technician, part, and interaction variance

• σ² _{Repeatability} = MS_Equipment =0.05196
• Then, σ² _{TechnicianxPart} = (MS_{TechnicianxPart} – σ² _{Repeatability})/ number of trials = (0.02482-0.05196)/3 = -0.00904 =0
• σ² _Part = (MS_Part – MS_{TechnicianxPart})/ (number of trials*number of technician) = (12.9477-0.05196)/(3*3) = 1.4328
• σ² _Technician = (MS_Technician – MS_{TechnicianxPart})/ (number of trials*number of part) =(0.04996-0.05196)/(10*3) = -0.00006667 =0

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Step 8: Compute Gage R&R and interpret the results

• Gage R&R = σ² _{Repeatability} + σ² _Technician =0.05196+0 = 0.05196
• Equipment Variation (Reliability) = σ² _{Repeatability} =0.05196
• Technician Variation (Reproducibility) = σ² _Technician + σ² _{TechnicianxPart} = 0+0 =0
• Part to Part = σ² _Part=1.4328
• Total Variation = σ² _{Repeatability} + σ² _Part+ σ² _Technician+ σ² _{TechnicianxPart} = 0.05196+1.4328+0+0 = 1.4848

From the above values, compute the % contribution variance.

Since the Total Gage R&R is 3.5%, it is in the yellow zone. So it may be acceptable depending on the application and cost factors, but the team needs to improve it further.

Find the standard deviation and % study variance.

According to the Automotive Industry Action Group (AIAG) measurement system assessment using %GRR.

Since the Total Gage R&R is 18.71%, it is in the yellow zone. So it may be acceptable depending on the application and cost factors, but the team needs to improve it further.

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Calculate the number of distinct categories (NDC)

According to the Automotive Industry Action Group (AIAG), the number of distinct categories should be greater than 5 for an adequate measuring system.

Number of distinct categories =(Standard deviations for parts / standard deviation for gage) * √2 = (1.9702 /0.2279) * √2 = 7

Gage Repeatability and Reproducibility ANOVA Excel Template

Gage Repeatability and Reproducibility using Minitab

Step 1: Copy the data in the Minitab sheet

Step2: Select Stat –> Quality Tools –> Gage Study –> Gage R&R (Crossed)

Select Part numbers, Operator, and Measurement data, and select the ANOVA method.

Under options: Enter process tolerance, the default alpha to remove interaction term would be 0.25. click on ok.

Step 3: Interpretation of results (session window)

• The p-value for the operator (0.163) is greater than 0.05.
• The p-value for Parts * Operator (0.98) is greater than 0.05. Hence, Minitab ignores the values and considers the values in the without interaction table.
• The part-to-part variation is 96.5x, which is much greater than the total Gage R&R (3.5%). This tells us that there is a lot of variation between the parts.
• Total Gage R&R is 3.5%. So, it may be acceptable depending on the application and cost factors, but there is a scope for improvement.
• Similarly, in the % study variance, the total Gage R&R is 18.71%. According to the AIAG, it may be acceptable depending on the application and cost factors, but the team needs to improve it further.
• The number of distinct categories is 7, which is greater than the acceptable number of 5.

Step 4: Interpretation of results (graphs window)

• First Graph- Components of Variation: It clearly shows too much variation is between the part to part, but not due to Gage R&R.
• Second Graph- Measurement values by parts: Clearly indicate the variation between the parts.
• Third Graph- R chart by the operator: All the values are within control limits.
• Fourth Graph – Measurement value by the operator: The difference between operators is small.
• Fifth Graph – X bar chart by the operator: Most of the points are outside of control limits. Hence, it indicates that variation is basically due to the parts.
• Sixth Graph – Operator * Parts Interaction: There is not much difference between the operators, and also there is no interaction between the parts and the operator.

Videos of Gage Repeatability and Reproducibility Charts

Gage R&R Definitions

Precision

• Getting consistent results repeatedly.
• The repeatability of the gage.

Accuracy

• Unbiased true values are obtained.
• Must be assured before an R&R can be performed.
• This is why wecalibrate.

Sensitivity

• Ability to detect differences in measurement.

Reproducibility

• We compare the results of different operators at different times.
• We examine variation between the averages of each operator.

Repeatability

• We look at the variation between individual operators.
• We look at the variation within their readings.

Traceability

• The accuracy of a measuring instrument mapped to US national standards.

Attribute Gage Repeatability and Reproducibility

Attribute data is a form of discrete data. Counts rather than measurements denote it. Such as Yes or no, Pass or fail and GO or no GO. It is a complex measurement system because human judgment is involved in most cases. Attribute gage R&R helps to perform such analysis. Ideally, the target for gage R&R would be 100%; however, getting 100% is not always possible. Hence, anything above 90% percent is acceptable.

Attribute Gage R&R Example in a Six Sigma project

• To measure the Gage R&R, it is recommended to measure a minimum of 20 to 30 parts
• Select 2-3 technicians to measure the parts
• Have each technician measure each part 2 or 3 times

Step 1: Take master appraiser readings

Step 2: Select three Operators and have them categorize each transaction (2 trials) without knowing what the master readings are.

You will use these trials as a sample.

Step 3: Repeatability of the operator- count the number of times the operator readings agree (between two trials). Divide the total agreed number by the total transactions to obtain the percentage of agreement.

• In Excel, we can use the “IF” formula to check that both the trial data sets are agreed upon (C5 and D5 cells). If both are agreed, use 1; otherwise, 0. For Example: =IF(C5=D5,1,0)
• Operator Jack: Out of 30 Transactions, 29 values are agreed upon between trial 1 and trial 2. So, repeatability of Jack =29/30 = 96.7%

Step 4: Compute each operator vs. Master readings

• Now compare Jack’s two trials values with Master transactions
• In Excel, we can use the “IF” formula to check both the trial data sets agree with the Master reading (C5, D5, and B5 cells). If both are agreed, use 1; otherwise, 0. For Example: =IF(AND(C5=B5,D5=B5,1,0).
• Operator Jack: Out of 30 Transactions, 27 values are agreed upon between trial 1, trial 2, and the master readings. So, the % agreement of Jack’s data set with the master transactions =27/30 = 90.0%.

Similarly, compute for the other the operator’s repeatability and agreement with master readings.

Step 5: Compute reproducibility between operators

• In Excel, we can use the “IF” formula to check reproducibility between operators (C5, D5, G5, H5, K5, and L5 cells). If all are agreed, use 1; otherwise, 0. For Example: =IF(AND(C5=D5,C5=G5,C5=H5,C5=K5,C5=L5,1,0)
• Reproducibility: Out of 30 Transactions, 23 values are agreed upon between 3 operators. So, the % Reproducibility =23/30 = 76.7%.

Step 6: Overall Effectiveness (All operators vs. Standard) – Compute the percentage of the time that all the operator’s transactions are agreed upon among each other and with the master transaction.

• In Excel, we can use the “IF” formula to check overall effectiveness (C5, D5, G5, H5, K5, L5, and B5 cells). If all are agreed, use 1; otherwise, 0. For Example: =IF(AND(B5=C5,B5=D5,B5=G5,B5=H5,B5=K5,B5=L5,1,0)
• Overall Effectiveness: Out of 30 Transactions, 23 values are agreed upon between 3 operators. So, the % Overall Effectiveness =23/30 = 76.7%

Step7: Conclusions

As per AIAG acceptance criteria of attribute data MSA:

MSA will be failed as the overall efficiency is only 76.7%. MSA Should be greater than 90%. We need to take the appropriate actions for improvement.

Attribute Gage R&R using Minitab

Step 1: Copy the data in the Minitab sheet

Step2: Select Stat –> Quality Tools –>Attribute Agreement Analysis

Select the Multiple Columns option and include Operator trials 1 and 2 data, add the number of appraisers as 3, and the number of trials as 2. Add the master values under “Known standard/attribute.”

Step 3: Interpretation of results (session window)

Additional Attribute Gage R&R Articles

• https://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/making-sense-attribute-gage-rr-calculations/
• https://www.spcforexcel.com/knowledge/measurement-systems-analysis/attribute-gage-rr-comparing-appraisers
• https://asq.org/quality-resources/articles/attribute-gage-rr?id=c9deedbb83da4d248e4f06108e44c006

What You Need to Know for Your Six Sigma Exam

Green Belts

The IASSC Green Belt BOK lists understanding Gage R&R under their Measure Phase.

The ASQ Green Belt BOK describes the following requirements for Gage R&R under Measurement system analysis.

Calculate, analyze, and interpret measurement system capability using repeatability and reproducibility (GR&R), measurement correlation, bias, linearity, percent agreement, and precision/tolerance (P/T). (Evaluate)

Black Belts

The IASSC Black Belt BOK lists understanding Gage R&R under their Measure Phase.

The ASQ Black Belt BOK describes the following requirements for Gage R&R under Measurement system analysis.

Use various analytical methods (e.g., repeatability and reproducibility (R&R), correlation, bias, linearity, precision to tolerance, percent agreement, etc.) to analyze and interpret measurement system capability for variables and attributes measurement systems. (Evaluate)

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