# 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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