Select the method or formula of your choice.

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} |

Each plotted point, , represents the mean of the observations for subgroup, .

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

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

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

The center line represents the process mean (). If you do not specify a historical value, then Minitab uses the average from your data, , which is calculated as follows:

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

sum of all individual observations | |

total number of observations |

The value of the lower control limit for each subgroup, *i*, is calculated as follows:

The value of the upper control limit for each subgroup, *i*, is calculated as follows:

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

μ | process mean |

k | parameter for Test 1 (The default is 3.) |

σ | process standard deviation |

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

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 |