The process standard deviation is also called sigma, or *σ*. If you enter an historical value for sigma, then Minitab uses the historical value. Otherwise, Minitab uses one of the following methods to estimate sigma 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. |

If you do not use an unbiasing constant, then the Sbar is the mean of the subgroup standard deviations:

If you use the unbiasing constant, c_{4}(*n*_{i}), then Sbar is calculated as follows:

When the subgroup size is constant, Sbar is:

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

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

S_{i} | standard deviation of subgroup i |

m | number of subgroups |

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} |

d_{2}(*N*) is the expected value of the range of *N* observations from a normal population with standard deviation = 1. Thus, if *r* is the range of a sample of *N* observations from a normal distribution with standard deviation = *σ*, then E(*r*) = d_{2}(*N*)*σ*.

d_{3}(*N*) is the standard deviation of the range of *N* observations from a normal population with *σ* = 1. Thus, if *r* is the range of a sample of *N* observations from a normal distribution with standard deviation = *σ*, then stdev(*r*) = d_{3}(*N*)*σ*.

Use the following table to find an unbiasing constant for a given value, *N*. (To determine the value of *N*, consult the formula for the statistic of interest.)

For values of *N* from 51 to 100, use the following approximation for d_{2}(*N*):

For values of *N* from 26 to 100, use the following approximations for d_{3}(*N*) and d_{4}(*N*):

For more information on these constants, see the following:

- D. J. Wheeler and D. S. Chambers. (1992).
*Understanding Statistical Process Control*, Second Edition, SPC Press, Inc. - H. Leon Harter (1960). "Tables of Range and Studentized Range".
*The Annals of Mathematical Statistics*, Vol. 31, No. 4, Institute of Mathematical Statistics, 1122−1147.

N |
d_{2}(N) |
d_{3}(N) |
d_{4}(N) |
---|---|---|---|

2 |
1.128 | 0.8525 | 0.954 |

3 |
1.693 | 0.8884 | 1.588 |

4 |
2.059 | 0.8798 | 1.978 |

5 |
2.326 | 0.8641 | 2.257 |

6 |
2.534 | 0.8480 | 2.472 |

7 |
2.704 | 0.8332 | 2.645 |

8 |
2.847 | 0.8198 | 2.791 |

9 |
2.970 | 0.8078 | 2.915 |

10 |
3.078 | 0.7971 | 3.024 |

11 |
3.173 | 0.7873 | 3.121 |

12 |
3.258 | 0.7785 | 3.207 |

13 |
3.336 | 0.7704 | 3.285 |

14 |
3.407 | 0.7630 | 3.356 |

15 |
3.472 | 0.7562 | 3.422 |

16 |
3.532 | 0.7499 | 3.482 |

17 |
3.588 | 0.7441 | 3.538 |

18 |
3.640 | 0.7386 | 3.591 |

19 |
3.689 | 0.7335 | 3.640 |

20 |
3.735 | 0.7287 | 3.686 |

21 |
3.778 | 0.7242 | 3.730 |

22 |
3.819 | 0.7199 | 3.771 |

23 |
3.858 | 0.7159 | 3.811 |

24 |
3.895 | 0.7121 | 3.847 |

25 |
3.931 | 0.7084 | 3.883 |

N |
d_{2}(N) |
---|---|

26 |
3.964 |

27 |
3.997 |

28 |
4.027 |

29 |
4.057 |

30 |
4.086 |

31 |
4.113 |

32 |
4.139 |

33 |
4.165 |

34 |
4.189 |

35 |
4.213 |

36 |
4.236 |

37 |
4.259 |

38 |
4.280 |

39 |
4.301 |

40 |
4.322 |

41 |
4.341 |

42 |
4.361 |

43 |
4.379 |

44 |
4.398 |

45 |
4.415 |

46 |
4.433 |

47 |
4.450 |

48 |
4.466 |

49 |
4.482 |

50 |
4.498 |

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

Γ() | gamma function |