Select the method or formula of your choice.

Bias is calculated as the difference between the known standard value of a reference part and the observed average measurement.

When the actual measurement is unknown, Attribute Gage Study (Analytic Method) estimates the bias as follows:

- Point estimate of bias with a lower tolerance limit
- bias = lower limit + intercept / slope
- Point estimate of bias with an upper tolerance limit
- bias = upper limit + intercept / slope

The intercept and slope for both formulas are from the fitted line on the probability plot.

Minitab regresses the z–score Φ^{-1}(Prob (Acceptance)) on reference values X_{T} to calculate the intercept and slope.

Pre-adjusted repeatability is the repeatability that is calculated before adjusting for over-estimation.

Minitab estimates the pre-adjusted repeatability by:

Term | Description |
---|---|

X_{T} | represents the estimated reference values at the 0.995 and 0.005 probabilities of acceptance, which are calculated from the fitted line on the probability plot. |

Repeatability is the amount of variation in the measurement system that is from the gage. An attribute gage study regresses the probabilities of acceptance on the reference values to obtain repeatability.

Pre-adjusted repeatability is the repeatability that is calculated before adjusting for over-estimation. Minitab divides estimates of repeatability by the adjustment factor 1.08 to calculate the adjusted repeatability.

Minitab estimates the repeatability by:

Term | Description |
---|---|

X_{T} | represents the estimated reference values at the 0.995 and 0.005 probabilities of acceptance, which are calculated from the fitted line on the probability plot. |

The denominator, 1.08, is the adjustment factor given by the Automotive Industry Action Group^{1} Minitab uses the adjusted repeatability value to test bias = 0.

To test bias = 0 using the regression method, Minitab uses the following formula:

The denominator is the formula for repeatability.

Term | Description |
---|---|

X_{T} | represents the estimated reference values at the 0.995 and 0.005 probabilities of acceptance, which are calculated from the fitted line on the probability plot. |

The values 31.3 and 1.08 are simulation results specific to the case when you have 20 trials for all parts and you have all of the following:

- 6 parts that have acceptances greater than 0 and less than 20
- 1 part has 0 acceptances
- 1 part has 20 acceptances

To test bias = 0 using the regression method, Minitab uses the following formula:

Term | Description |
---|---|

a | the intercept of the probability plot's fitted line |

b | the slope of the probability plot's fitted line |

LL | lower tolerance limit |

s | the error standard deviation calculated using the fitted line |

K | the number of parts |

x_{i} | the reference value of each part |

the mean of the reference values |

The degrees of freedom is used to calculate the p-value.

DF = N – 1.

Term | Description |
---|---|

N | number of trials |

The degrees of freedom is used to calculate the p-value.

DF = N – 2.

Term | Description |
---|---|

N | number of points used to obtain the fitted line |

P-values are used in hypothesis tests to help you decide whether to reject or fail to reject a null hypothesis.

To determine whether bias in the measurement system is statistically significant, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that bias exists when there is no significant bias.

The fitted line is a regression line that examines the relationship between the probability of acceptances and the reference values of the measured parts.

The general form of a fitted line is: Y = b_{0} + b_{1} X

Minitab regresses the z–score Φ^{-1}(Prob (Acceptance)) on reference values X_{T} to get the intercept and slope.

Term | Description |
---|---|

b_{0} | the intercept—the constant that determines the vertical placement of the regression line |

b_{1} | the slope of the regression line |

X | the predictor value |

R-sq for Fitted Line is the coefficient of determination, which is used to check whether the fitted line models the data well. The R-sq (R^{2}) value for the fitted regression line indicates the percentage of the variation in the probability of acceptance responses that is explained by the regression model.

R^{2} = 1 - (SS error / SS total)

Term | Description |
---|---|

SS error | sum of square for the error |

SS total | total sum of squares |