The (1 -α) 100% confidence interval for Cp is calculated as follows:

where ν is approximated by

and

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

χ^{2}_{α}_{,}_{ν} | The α percentile of the chi-square distribution with ν degrees of freedom |

α | Alpha for the confidence level |

ν | Degrees of freedom |

Y_{ij} | The jth observation in the ith subgroup |

The ith subgroup average | |

The average across all subgroups | |

I | The number of subgroups |

n_{i} | The ith subgroup size |

The calculations for the confidence interval for Z.Bench depend on the known values of the specification limits.

- When both the lower and upper specifications limits are known, Minitab calculates only the lower bound of Z.Bench.
(1 -α) 100% lower bound = Φ

^{-1}(1 - P_{U})where:

###### Note

The lower bound is displayed only when 1-α is greater than or equal to 0.80.

- When only the lower specification limit is known, Minitab calculates both the lower and the upper bounds for Z.Bench.
(1 -α) 100% lower bound = Φ

^{-1}(1 - P_{U})(1 -α) 100% upper bound = Φ

^{-1}(1 - P_{L})where:

- When only the upper specification limit is known, Minitab calculates both the lower and the upper bounds for Z.Bench.
(1 -α) 100% lower bound = Φ

^{-1}(1 - P_{U})(1 -α) 100% upper bound = Φ

^{-1}(1 - P_{L})where:

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

LSL | Lower specification limit |

USL | Upper specification limit |

α | Alpha for the confidence level |

Φ (X) | cdf of a standard normal distribution |

N | Total number of observations |

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

ν | Degrees of freedom based on the method used to estimate σ^{2}_{B/W}. See the formula for v used to calculate the Cp confidence interval for between/within capability analysis. |

γ_{N, 1 }-_{}_{α} | Gamma value based on the alpha level and number of observations (for more information see the Gamma table section) |

x̅ | Average of the observations |

Between/within standard deviation |

The (1 -α) 100% confidence interval for Cpk is calculated as follows:

If the sigma tolerance value, say k, in the input is not the default 6, then 9 in the formula should be replaced by (k/2)**2.

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

N | The total number of observations |

α | Alpha for the confidence level |

v | The degrees of freedom. For information on the calculation of ν , see the topic on the Cp confidence interval for between/within capability analysis. |

Z_{1}-_{α}_{/2} | The 1-α/2 percentile from the standard normal distribution |

The (1 -α) 100% confidence interval for Pp is calculated as follows:

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

χ^{2}_{α}_{,}_{ν} | The α percentile of the chi-square distribution with ν degrees of freedom |

α | Alpha for the confidence level |

ν | The degrees of freedom (Σn_{i}– 1) |

n_{i} |
The number of observations in the ith subgroup |

The (1 -α) 100% confidence interval for Ppk is calculated as follows:

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

N | The total number of observations |

α | Alpha for the confidence level |

v | The degrees of freedom (Σn_{i} – 1 or N – 1) |

n_{i} | The number of observations in the i^{th} subgroup |

Toler | Multiplier of the sigma tolerance (Minitab uses 6 as the default value) |

Z_{1}_{–}_{α}_{/2} | The 1 – (α/2) percentile from the standard normal distribution |

The (1 – α) 100% lower confidence bound for Cpm is calculated as follows:

Minitab displays only the lower bound for Cpm.

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

ν | Degrees of freedom, defined as N ((1 + a^{2}) ^{2} / (1 + 2a^{2})) |

a | (Mean – Target)/ |

α | Alpha for the confidence level |

N | Total number of observations |