Between/within capability analysis is based on the following four standard deviations:

σ_{within} is an estimate of the variation within subgroups (for example, one shift, one operator, or one material batch). Minitab estimates σ_{within} using one of the following methods:

- Pooled standard deviation:
where:

###### Note

If you change the default method and choose not to use the unbiasing constant, σ

_{within}is estimated by S_{p}.Term Description d Degrees of freedom for S _{p}= Σ (n_{i}- 1)X _{ij}j ^{th}observation in the i^{th}subgroupX̅ _{i}Mean of the i ^{th}subgroupn _{i}Number of observations in the i ^{th}subgroupC _{4}(d+1)Unbiasing constant Γ(.) Gamma function - Average of subgroup ranges (Rbar):
where:

If n are all the same:

Term Description r _{i}Range of the i ^{th}subgroupd _{2}(n_{i})An unbiasing constant read from a table (for more information, see the section Unbiasing constants d2(), d3(), and d4() d _{3}(n_{i})An unbiasing constant read from a table (for more information, see the section Unbiasing constants d2(), d3(), and d4() n _{i}Number of observations in the i ^{th}subgroup - Average of subgroup standard deviations (Sbar):
where:

###### Note

If you change the default setting and do not use the unbiasing constant, σ

_{within}is estimated by Σ S_{i}/ number of subgroups.Term Description C _{4}(n_{i})Unbiasing constant (as defined for pooled standard deviation) S _{i}Standard deviation of subgroup i n _{i}Number of observations in the i ^{th}subgroup

σ_{Between} is an estimate of the variation between subgroups (for example, subgroups collected at set intervals, batches, or by different operators).

- Average of moving range:
where:

Term Description R _{i}The ith moving range w The number of observations used in the moving range. The default is w = 2. d _{2}(w)An unbiasing constant read from a table (for more information, see the section Unbiasing constants d2(), d3(), and d4() - Median of moving range:
where:

Term Description MR _{i}The ith moving range Median of the MR _{i}w The number of observations used in the moving range. The default is w = 2. d _{4}(w) - Square root of mean squared successive differences (MSSD):
###### Note

If you change the default setting and do not use the unbiasing constant, σ

_{within}is estimated byTerm Description d _{i}Differences of successive group means C _{4}(n_{i})Unbiasing constant (as defined for the pooled standard deviation) C _{4}'(n_{i})Unbiasing constant ≈ c _{4}(n_{i}). For more information, see the section Unbiasing constant c4'().N Total number of observations n _{i}Number of observations in the i ^{th}subgroup

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

σ^{2}_{Between} | Variance between subgroups |

σ^{2}_{within} | Variance within subgroups |

where:

By default, Minitab does not use the unbiasing constant when estimating σ_{overall}. σ_{overall} is estimated by S. If you want to estimate overall standard deviation using the unbiasing constant, you can change this option on the Estimate subdialog box when you perform the capability analysis. If you always want Minitab to use the unbiasing constant by default, choose and select the appropriate options.

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

X_{ij} | The j^{th} observation in the i^{th} subgroup |

X̅ | Process mean |

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

C_{4} (N) | Unbiasing constant (as defined for the pooled standard deviation) |

N (or Σ n_{i}) | Total number of observations |

The Box-Cox transformation estimates a lambda value, as shown in the following table, which minimizes the standard deviation of a standardized transformed variable. The resulting transformation is Y^{λ} when λ ҂ 0 and ln Y when λ = 0.

The Box-Cox method searches through many types of transformations. The following table shows some common transformations where Y' is the transform of the data Y.

Lambda (λ) value | Transformation |
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