Methods and formulas for Gage Bias

Select the method or formula of your choice.

Bias

Bias is the difference between the part's reference value and the operator's measurements of the part.

Formula

Average bias for each part:

Notation

TermDescription
zi,j jth measurement of the ith part
refireference value of the ith part
minumber of replicates of the ith part

%Bias

%Bias is bias expressed as a percentage of the overall process variation.

Formulas

%Bias = 100 * (|Average Bias| / Process Variation)

Notation

TermDescription
zi,j jth measurement of the ith part
refireference value of the ith part
minumber of replicates of the ith part

p-value for Gage Linearity and Bias Study

Use the p-values to test whether bias = 0 at each reference value, and whether the average bias =0.

The p-value is defined as the area under the sampling distribution to the right of the + |test statistic| and the area under the sampling distribution to the left of the - |test statistic|. The p-value in the output is obtained from using the t-distribution with γ df and the t-statistic.

Minitab provides specific t-statistic calculations for the sample range method and for the sample standard deviation method.

Sample range method

Minitab uses either the sample range (default) or sample standard deviation to estimate repeatability standard deviation. Repeatability standard deviation is used to calculate the t-value, which leads to the calculation of the p-value to test bias = 0 for all reference values and for each reference.

Formulas

For the sample range method, when each distinct reference value corresponds to a unique part, the repeatability standard deviation:

When more than one part has the same reference value, the repeatability standard deviation:

The t-statistic for testing bias is:

The degrees of freedom (γ) is obtained from a table in the AIAG manual1. Minitab uses the t distribution with γ df and the t-value to calculate the p-value.

Notation

TermDescription
Xithe bias of the ith measurement for a part
d2a value from a table1, with sample size = n
average range

Sample standard deviation method

Minitab uses either the sample range (default) or sample standard deviation to estimate repeatability standard deviation. Repeatability standard deviation is used to calculate the t-value, which leads to the calculation of the p-value to test bias = 0.

Formulas

For the sample standard deviation method, when one reference value corresponds to a single part, the repeatability standard deviation:

The t-statistic for testing bias is:

The degrees of freedom are n - 1. The p-value in the output is obtained from the t-distribution using the t-value and the degrees of freedom.

When more than one part has the same reference value, the repeatability standard deviation is the pooled sample standard deviation s across the parts with the same reference value:

The t-statistic for testing bias is:

The degrees of freedom are (n1- 1) + ... + (ng - 1). The p-value in the output is obtained from the t-distribution using the t-value and the degrees of freedom.

Notation

TermDescription
xith measurement of the part
average measurement for the part
nsample size
S1the sample standard deviation for part 1 with n1 measurements
Sgthe sample standard deviation for part g with ng measurements
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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