Formulas and Tables

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X-bar and sigma chart formulas

X-bar control limits are based on either range or sigma, depending on which chart it is paired with. When the X-bar chart is paired with a sigma chart, the most common (and recommended) method of computing control limits based on 3 standard deviations is:

X-bar

x-bar-formula

n is the number of observations

x-double-bar-formula

k is the number of subgroups

f-x-bar-sigma-ucl

Upper control limit

f-x-bar-sigma-lcl

Lower control limit

Sigma

s-bar-formula

k is the number of subgroups

range-bar-formula

k is the number of subgroups

f-sigma-ucl

Upper control limit

f-sigma-lcl

Lower control limit

Subgroup Size
A3
B3
B4
2
2.659
—–
3.267
3
1.954
—–
2.568
4
1.628
—–
2.266
5
1.427
—–
2.089
6
1.287
0.030
1.970
7
1.182
0.118
1.882
8
1.099
0.185
1.815
9
1.032
0.239
1.761
10
0.975
0.284
1.716
11
0.927
0.321
1.679
12
0.886
0.354
1.646
13
0.850
0.382
1.618
14
0.817
0.406
1.594
15
0.789
0.428
1.572
16
0.763
0.448
1.552
17
0.739
0.466
1.534
18
0.718
0.482
1.518
19
0.698
0.497
1.503
20
0.680
0.510
1.490
21
0.663
0.523
1.477
22
0.647
0.534
1.466
23
0.638
0.545
1.455
24
0.619
0.555
1.445
25
0.606
0.565
1.435