Select the method or formula of your choice.

The calculation of sample size, n, and critical distance, k, depends on the number of specification limits given and whether standard deviation is known.

The sample size is given by:

The critical distance is given by:

where:

The sample size is given by:

The critical distance is given by:

The standard deviation is estimated from the sample data:

First Minitab calculates z:

Then Minitab finds p* from the standard normal distribution as the upper tail area corresponding to z. This is the minimum probability of defective outside one of the specification limits.

The method Minitab uses for the calculation of sample size and critical distance depends on this value of p*.

Let p_{1} = AQL, p_{2} = RQL

- If 2p* ≤ (p
_{1}/ 2), then the two specifications are relatively far apart and calculations follow the single limit plans. - If p
_{1}/ 2 < 2p* ≤ p_{1}, then the two specifications are not relatively far apart, but are still not so close that the minimum probability of defective can be found for certain mean values. Minitab performs an iteration to find sample size and critical distance.

Let

μ = μ_{0}+ m * h, where h = σ/100

Let m = 1, 2, ...300. For each μ calculate:

where Φ is the cumulative distribution function of the standard normal distribution. If Prob (X<L) + Prob (X>U) is extremely close to p_{1}, then Minitab uses the larger value between Prob (X<L) and Prob (X>U) to find sample size and accept number.

Suppose Prob (X<L) is the larger value, let p_{L} = Prob (X<L).

The sample size is given by:

The critical distance is given by:

where:

Z_{pL} = the (1 – p_{L}) * 100 percentile of the standard normal distribution

- If p
_{1 }< 2p* < p_{2}, then the specifications of the plans must be reconsidered because the minimum probability defective determined by the two specification limits and the standard deviations is larger than the acceptable quality level p_{1}. Consider a plan with a slightly larger probability of defective than p_{1}. - If 2p* ≥ p
_{2}, then the lot should be rejected; the minimum probability of defective determined by the two specification limits and the standard deviation is larger than the rejectable quality level. You can reject the lot without testing any products.

First Minitab lets the critical distance be the value as given in the case of two separate single-limit plans:

Then Minitab finds the upper tail area from the standard normal distribution, p*, corresponding to the k as the percentile; and the percentile Zp** from the standard normal distribution corresponding to upper tail area of p* / 2.

The maximum standard deviation (MSD) is given by:

The estimated standard deviation is given by:

Minitab tests whether the estimated standard deviation, s, is less than or equal to the MSD.

If the estimated standard deviation, s, is less than or equal to the MSD, then:

The sample size is given by:

The critical distance is given by:

If the estimated standard deviation, s, is not less than or equal to the MSD, then the standard deviation is too large to be consistent with acceptance criteria and you must reject the lot.

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

Z_{1} | the (1 – p_{1}) * 100 percentile of the standard normal distribution |

p_{1} | Acceptable Quality Level (AQL) |

Z_{2} | the (1 – p_{2}) * 100 percentile of the standard normal distribution |

p_{2} | Rejectable Quality Level (RQL) |

Z_{α} | the (1 – α) * 100 percentile of the standard normal distribution |

Z_{β} | the (1 – β ) * 100 percentile of the standard normal distribution |

X_{i} | the i^{th} measurement |

the mean of actual measurements | |

L | lower specification limit |

U | upper specification limit |

σ | known standard deviation |

Let p be the probability of defective which is the x value of a point on an OC curve.

- Single lower specification limit and known standard deviation
- Prob (X < L) = p.
- Single upper specification limit and known standard deviation
- Prob (X > L) = p.

First Minitab calculates z

Then finds p* from the standard normal distribution as the upper tail area corresponding to z. This is the minimum probability of defective outside one of the specification limits.

The method Minitab uses for the probability of acceptance depends on this value of p*.

Let p_{1} = AQL, p_{2} = RQL

- If 2p* ≤ (p
_{1}/ 2), then the two specifications are relatively far apart and calculations for sample size and critical distance follow the single limit plans. - If p
_{1}/ 2 < 2p* ≤ p_{1}, then the two specifications are not relatively far apart, but are still not so close that the minimum probability of defective can be found for certain mean values.

For any given p, Minitab finds the mean, μ, of the measurements using a grid search algorithm. Then,

When you have both upper and lower specification limits, but do not know the standard deviation, Minitab uses the OC curve for the single-limit plan to approximate the double specification limits case. The OC curve derived for a single-limit plan with specified p_{1}, p_{2}, α, and β is the lower limit of the band of OC curves for a two-sided specification plan with the same p_{1}, p_{2}, α, and β and for most practical cases can be taken as the OC curve for the two-sided plan. See Duncan^{1}.

- Duncan (1986). Quality Control and Industrial Statistics, 5
^{th}edition.

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

n | sample size |

k | critical distance |

σ | known standard deviation |

Z_{p} | the (1 - p)^{th} percentile from the standard normal distribution |

Φ | the cumulative distribution function of the standard normal distribution |

T |
is non-central t distributed with degrees of freedom = n – 1, and the non central parameter, |

L | lower specification limit |

U | upper specification limit |

The probability of rejecting (P_{r}) describes the chance of rejecting a particular lot based on a specific sampling plan and incoming proportion defective. It is simply 1 minus the probability of acceptance.

P_{r} = 1 – P_{a}

where:

P_{a} = probability of acceptance

Minitab calculates an accept or reject decision based on your measurements from sampled items and the criteria (sample size and critical distance) of your variables acceptance sampling plan.

First Minitab calculates the mean and standard deviation from your data (if you have not specified a historical standard deviation):

- Mean
- Standard deviation
- Acceptance criteria

Minitab uses σ instead of s in the Z calculations when you provide a historical standard deviation.

- If both specifications are given, the Z-values for each specification are calculated. Accept the lot if Z.LSL ≥ k and Z.USL ≥ k; otherwise reject the lot.
- If only one specification is given, the corresponding Z-value is calculated. If lower specification only, accept the lot if Z.LSL ≥ k; otherwise reject the lot. If upper specification only, accept the lot if Z.USL ≥ k; otherwise reject the lot.

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

X_{i} | measurement data |

mean | |

s | estimated standard deviation |

σ | known standard deviation |

n | sample size |

k | critical distance |

L | lower specification limit |

U | upper specification limit |

The average outgoing quality represents the quality level of the product after inspection. The average outgoing quality varies as the incoming fraction defective varies.

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

P_{a} | probability of acceptance |

p | incoming fraction defective |

N | lot size |

n | sample size |

The average total inspection represents the average number of units that will be inspected for a particular incoming quality level and probability of acceptance.

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

P_{a} | probability of acceptance |

N | lot size |

n | sample size |

The acceptance region is calculated only when both specifications are given and the standard deviation is unknown.

Minitab finds the coordinates of the acceptance region:

- Find the upper tail area from the standard normal p* corresponding to the critical distance as the percentile.
- For any two proportions, p
_{01}and p_{02}, satisfying p_{02}+ p_{01}= p*:

p_{01} = (p* / 100) * h

p_{02} = (p* / 100) * (100 - h)

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

L | lower specification limit |

U | upper specification limit |

Zp_{01} | the (1 - p_{01})* 100 percentile from the standard normal distribution |

Zp_{02} | the (1 - p_{02})* 100 percentile from the standard normal distribution |

p_{01} | (p* / 100) * h |

p_{02} | (p* / 100) * (100 – h) |