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

where ν is calculated based on the method used to estimate σ^{2}_{within}:

- Pooled standard deviation: ν = Σ (n
_{i}- 1) - Average moving range and Median moving range: ν ≈ k – Rspan + 1
- Square root of MSSD: ν = k - 1
- Rbar: ν = 0.9 k (n - 1)
- Sbar: ν = f
_{n}k (n - 1), where f_{n}is the adjustment factor that varies with n, as shown in the following table.

n | f_{n} |
---|---|

2 | 0.88 |

3 | 0.92 |

4 | 0.94 |

5 | 0.95 |

6, 7 | 0.96 |

8, 9 | 0.97 |

10-17 | 0.98 |

18-64 | 0.99 |

65- | 1.00 |

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 |

σ^{2}_{within} | Within-subgroup variance |

n_{i} | The i^{th} subgroup size |

k | Number of samples |

Rspan | Length of the moving range |

n | Average sample size (Σ n_{i} / k). |

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 |

ν | Degrees of freedom based on the method used to estimate σ^{2}_{within} (for information on the calculation of ν, see the topic on Cp confidence interval bounds) |

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

x̅ | Process mean (estimated from the sample data or a historical value) |

Within-subgroup standard deviation |

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

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

N | The total number of observations |

α | Alpha for the confidence level |

v | The degrees of freedom based on the method used to estimate σ^{2}_{within} (for information on the calculation of v, see the section on Cp confidence interval bounds) |

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% 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 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 |

ν | The degrees of freedom (N – 1) |

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

x̅ | Process mean (estimated from the sample data or a historical value) |

Overall standard deviation |

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 |