Benchmark Z statistics for between/within capability are calculated by finding the Z value using the standard normal (0,1) distribution for the corresponding statistics.

where:

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

LSL | Lower specification limit |

USL | Upper specification limit |

Φ (X) | Cumulative distribution function (CDF) of a standard normal distribution |

Φ^{-1} (X) | Inverse CDF of a standard normal distribution |

Between/within standard deviation |

The calculations for the confidence interval for Z.Bench depend on which specification limit the process has.

Minitab solves the following equation to find *p*_{1}:

where

Minitab solves the following equation to find *p*_{2}:

where

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

LSL | the lower specification limit |

USL | the upper specification limit |

α | the alpha for the confidence level |

the Cumulative Distribution Function (CDF) from a standard normal distribution | |

the inverse CDF from a standard normal distribution | |

the (1 - α/2)^{th} percentile of the standard normal distribution | |

N | the total number of measurements |

υ | the degrees of freedom for s |

the process mean (estimated from the sample date or a historical value) | |

s | the sample standard deviation between/within subgroups |

a random variable that is distributed as a non-central t distribution with degrees of freedom and non-centrality parameter δ |

where

To calculate , substitute the sample estimates for the parameters in the formula for :

where

To calculate a one-sided upper confidence bound, change to in the definition of *U*.

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

the estimated tail probabilities outside of the specificataion limits | |

the (1 - α / 2)^{th} percentile of the standard normal distribution | |

α | the alpha for the confidence level |

the process mean (estimated from the sample date or a historical value) | |

s | the sample standard deviation between/within subgroups |

υ | the degrees of freedom for s |

the Cumulative Distribution Function (CDF) from a standard normal distribution | |

the Probability Density Function (PDF) from a standard normal distribution | |

USL | the upper specification limit |

LSL | the lower specification limit |

the inverse CDF from a standard normal distribution |