# Gage R&R Methods Comparison

**Introduction**

For Gage R&R, there are generally two types of calculation method, and this post will explain both X-Bar Method and Analysis of Variances (ANOVA) Method with detailed equations/examples.

For gage R&R, the following four critical indicators are listed below:

- Number of Samples (n)
- Number of Appraisers (a)
- Number of Trials per Sample (k)
- Specification (Median, USL and LSL)

In general, the variances within Gage R&R are listed below:

- Equipment/Instrument Variances (EV)
- Appraiser/Operator Variance (AV)
- Measured Part Variances (PV)

**XBar-R Method**

The XBar-R method example consists of 12 samples (n = 12), 4 appraisers (a = 4) and 4 trials per samples (k =4).

For X Bar methods, there are coefficients needed to be determined. Please refer to the following d2 constant based on the example parameter.

And based on the table and calculation equation with the above description, K1 will be roughly 0.486 while K2 is 0.447 and K3 is 0.299 respectively.

After the coefficient calculation, the data can be collected to analyze respective operator’s measurement (with repeated measurement for all tested samples)

Once data are collected, then the calculation for X-Bar R method can be initiated by the following parameters and indicators.

And by analyzing the GR&R and %PV, the results can be described by the criteria listed below

**ANOVA Method**

The ANOVA method example consists of 10 samples (n = 10), 4 appraisers (a = 4) and 3 trials per samples (k = 3).

In gage R&R, interactions between cross factors such as repeated sample’s measurement by individual appraiser will occur (as in MSA Type II study). Therefor ANOVA is required to analyze any potential interaction and verify the significance between the interaction of appraiser and samples.

Before calculating ANOVA method for gage R&R, please refer to the following road map for the different parameters ANOVA will calculate.

After getting the coefficients, the following are the data input for each appraiser along with their trial values.

Throughout the calculation, the only difficulties to calculate would be the reliability and total variance. The toggle section will explain the details of each sum square which involves with summation notation. While the p-value indicates the significant level for the factor to impact interaction level inside the measurement system.

For the Sum of Square definitions, please refer to the following explanation:

Square the differences between appraiser’s average and grand overall average. Then add it up before multiplying with number of trials and number of samples.

Square the differences between sample’s average and grand overall average. Then add it up before multiplying with number of trials and number of appraisers.

Square the differences between each measurement point (in this case there are 192 data points (4 x 4 x 12) and grand overall average. Then add it up together for the total of sum square.

Square the differences between each part’s trial value and their respective part’s overall trial average by the particular operator. Then add it up together for the total of sum square in equipment.

After calculation with the sum of square, mean square and probability values from the given data sets. Then ANOVA method can be calculated based on the following parameters.

And by analyzing the GR&R and variacne contribution% via ANOVA method, the results can be described by the criteria listed below