Methods and formulas for variance components in confidence intervals in Crossed Gage R&R Study

Select the method or formula of your choice.

Common notation and rules for Gage R&R confidence intervals

For all variance components, lower and upper bounds for variance components must not be negative values. If the bounds calculated using the formulas are negative, then they are set to zero.

For all ratios between 0 and 1, lower and upper bounds should also be between 0 and 1. If the bounds are outside the range, they are set to 0 or 1 accordingly.

Notation

TermDescription
the α *100 percentile of the chi-square distribution with nq degrees of freedom
Fα(nq, nγ)the α *100 percentile of the F distribution with nq and nγ degrees of freedom
Jthe number of operators
Ithe number of parts
Kthe number of replicates

For degrees of freedom:

Parts: n1=I–1

Operators: n2=J–1

Part Operator Interaction: n3=(I–1)(J–1)

Replicates: n4=IJ(K–1)

MSPart = S12

MSOperator = S22

MSPart*Operator = S32

MSReplicates = S42

Repeatability variance component confidence interval

Minitab calculates the lower and upper bounds for an exact (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Formulas

With operator
Without operator
Without interaction term

Notation

TermDescription
the α *100 percentile of the chi-square distribution with nq degrees of freedom
Jthe number of operators
Ithe number of parts
Kthe number of replicates

Reproducibility (or Operator) variance component confidence interval

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Formulas

With interaction term
Without interaction term

Notation

TermDescription
the α *100 percentile of the chi-square distribution with nq degrees of freedom
Jthe number of operators
Ithe number of parts
Kthe number of replicates

Operator variance component confidence interval

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1–α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Formulas

Notation

TermDescription
the α *100 percentile of the chi-square distribution with nq degrees of freedom
Jthe number of operators
Ithe number of parts
Kthe number of replicates

Interaction variance component confidence interval

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Formulas

Notation

TermDescription
the α *100 percentile of the chi-square distribution with nq degrees of freedom
Jthe number of operators
Ithe number of parts
Kthe number of replicates

Total gage variance component confidence interval

With operator and interaction term

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Without operator term

The lower and upper bounds for an exact (1 – α) *100% confidence interval are:

Without interaction term

Minitab uses the modified large-sample (MLS) method, the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

Notation

TermDescription
the α *100 percentile of the chi-square distribution with nq degrees of freedom
Jthe number of operators
Ithe number of parts
Kthe number of replicates

Part-to-part variance component confidence interval

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

With operator

Without operator

Without interaction

Notation

TermDescription
the α *100 percentile of the chi-square distribution with nq degrees of freedom
Jthe number of operators
Ithe number of parts
Kthe number of replicates

Total variance component confidence interval

Minitab uses the modified large-sample (MLS) method to calculate the lower and upper bounds for an approximate (1 – α) *100% confidence interval. To calculate the one-sided confidence bounds, replace α/2 with α in H and G.

With operator

Without operator

Without interaction term

Notation

TermDescription
the α *100 percentile of the chi-square distribution with nq degrees of freedom
Jthe number of operators
Ithe number of parts
Kthe number of replicates
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