# Methods and formulas for gage R&R table in Crossed Gage R&R Study

Select the method or formula of your choice.

## VarComp for ANOVA method

Variance component contributed by each source.

When the Operator*Part interaction is included in the ANOVA model, the variance components are calculated as follows:

Repeatability

Repeatability = MSRepeatability

Operator

TermDescription
anumber of parts
nnumber of replicates
Operator*Part

TermDescription
nnumber of replicates
Part-to-Part

TermDescription
bnumber of operators
nnumber of replicates
Reproducibility
Reproducibility = VarCompOperator +VarCompOperator*Part
Total Gage R&R
Total Gage R&R = VarCompRepeatability + VarCompReproducibility
Total Variation
Total Variation = VarCompTotal Gage R&R + VarCompPart-to-Part

When Operator*Part interaction is not included in the ANOVA model, the variance components are calculated as follows:

Repeatability

Repeatability = MSRepeatability

Operator

TermDescription
anumber of parts
nnumber of replicates
Part-to-Part

TermDescription
bnumber of operators
nnumber of replicates
Reproducibility
Reproducibility = VarCompOperator
Total Gage R&R
Total Gage R&R = VarCompRepeatability + VarCompReproducibility
Total Variation
Total Variation = VarCompTotal Gage R&R + VarCompPart-to-Part

If a variance component is negative, then Gage R&R will report it as zero.

## Variance for Xbar and R method

Variance contributed by each source.

Repeatability
Equipment variation, the standard deviation is calculated as:

TermDescription
anumber of parts
knumber of operators
Rijrange of measurements by operator j for part i
d2d2 = d2* from Appendix C 1. When looking for this value, use g = (# of parts) * (# of operators) and m = (# of replicates).
Reproducibility

Operator variation, the standard deviation is calculated as:

TermDescription
the average measurement for operator i
anumber of parts
rnumber of trials
d2d2 = d2* from Appendix C1. When looking for this value, use g = 1 and m = (# of operators).
Part-to-Part

The standard deviation is calculated as:

TermDescription
Rprange of part average values
d2d2 = d2* from Appendix C1. When looking for this value, use g = 1 and m = (# of parts).
Total Gage R&R

The standard deviation is calculated as:

Total Variation

The standard deviation is calculated as:

## %Contribution

%Contribution is the percentage of overall variation from each variance component. It is calculated as the variance component for each source divided by the total variation, then multiplied by 100 to express as a percentage. %Contribution is calculated by dividing each variance component by the total variation and multiplying by 100. The percentages in this column add to 100.

## StdDev

StdDev (SD) is the standard deviation for each source of variation. The standard deviation is equal to the square root of the variance component for that source.

When you specify to use the historical standard deviation to estimate the process variation, Minitab does the following:
• If then the total standard deviation is σ and .
• Otherwise Minitab uses the data to estimate the total standard deviation and .
TermDescription
historical standard deviation
total gage standard deviation calculated from the data
part-to-part standard deviation

## Study Var

The study variation is calculated as the standard deviation for each source of variation multiplied by 6 or the multiplier that you specify in Study variation.

Usually, process variation is defined as 6s, where s is the standard deviation as an estimate of the population standard deviation (denoted by σ or sigma). When data are normally distributed, approximately 99.73% of the data fall within 6 standard deviations of the mean. To define a different percentage of data, use another multiplier of standard deviation. For example, if you want to know where 99% of the data fall, you would use a multiplier of 5.15, instead of the default multiplier of 6.

## %Study Var and CI

The %study variation is calculated as the study variation for each source of variation, divided by the total variation and multiplied by 100.

%Study Var is the square root of the calculated variance component (VarComp) for that source. Thus, the %Contribution of VarComp values sum to 100, but the %Study Var values do not.

### Confidence interval

If L and U are the lower and upper bounds of a variance component, then the confidence interval for the corresponding percent of study variation is:

## %Tolerance and CI

%Tolerance is the percent of tolerance for each component.

If the tolerance (Upper spec – Lower spec) is given, the %tolerance is calculated by dividing the Study Var for each component by the specified tolerance.

If only one specification limit is given, the percent tolerance is half the Study Var for each component divided by the one-sided tolerance. The one-sided tolerance is the absolute value of the given specification limit subtracted from the average of all the measurements.

Minitab displays this value only when you enter a process tolerance (Upper spec – Lower spec) or a specification limit in the Options subdialog box.

### Confidence interval

If L and U are the lower and upper bounds of a variance component, then the confidence interval for the corresponding percent tolerance is:

TermDescription
kk is the study constant and the default value is 6

## %Process and CI

%Process is the percent of process variation, calculated by the ratio of estimated standard deviation to historical standard deviation.

Minitab displays this value only when you enter a historical standard deviation in the Options subdialog box.

### Confidence interval

If L and U are the lower and upper bounds of a variance component, then the confidence interval for the corresponding process variation is:

TermDescription
kk is the study constant and the default value is 6
1 Automotive Industry Action Group (AIAG) (2010). Measurement Systems Analysis Reference Manual, 4th edition. Chrysler, Ford, General Motors Supplier Quality Requirements Task Force
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