Sigma (*σ*) is the standard deviation of the process. If you enter an historical value for *σ*, then Minitab uses the historical value. Otherwise, Minitab uses one of the following methods to estimate *σ* from the data.

Minitab uses the range of each subgroup, , to calculate , which is an unbiased estimator of *σ*:

where

When the subgroup size is constant, the formula simplifies to the following:

where (Rbar) is the mean of the subgroup ranges, calculated as follows:

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

r_{i} | range for subgroup i |

m | number of subgroups |

d_{2}(·) | value of unbiasing constant d_{2} that corresponds to the value specified in parentheses. |

n_{i} | number of observations in subgroup i |

d_{3}(·) | value of unbiasing constant d_{3} that corresponds to the value specified in parentheses. |

The pooled standard deviation (*S _{p}*) is given by the following formula:

When the subgroup size is constant, *S _{p}* can also be calculated as follows:

By default, Minitab applies the unbiasing constant, c_{4}(), when you use the pooled standard deviation to estimate *σ*:

When the subgroup size is constant, the unbiased *S*_{p} can also be calculated as follows:

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

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

mean of subgroup i | |

n_{i} | number of observations in subgroup i |

μ_{v} | mean of the subgroup variances |

c_{4}(·) | value of the unbiasing constant c_{4} that corresponds to the value that is specified in parentheses. |

d | degrees of freedom for S, given by the following formula:
_{p} |