Formulas and Tables

X-bar and range 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 range 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

Upper control limit

Lower control limit

Range

range-bar-formula

k is the number of subgroups

Upper control limit

Lower control limit

Tabular values for X-bar and range charts
Subgroup Size
A2
d2
D3
D4
2
1.880
1.128
—–
3.268
3
1.023
1.693
—–
2.574
4
0.729
2.059
—–
2.282
5
0.577
2.326
—–
2.114
6
0.483
2.534
—–
2.004
7
0.419
2.704
0.076
1.924
8
0.373
2.847
0.136
1.864
9
0.337
2.970
0.184
1.816
10
0.308
3.078
0.223
1.777
11
0.285
3.173
0.256
1.744
12
0.266
3.258
0.283
1.717
13
0.249
3.336
0.307
1.693
14
0.235
3.407
0.328
1.672
15
0.223
3.472
0.347
1.653
16
0.212
3.532
0.363
1.637
17
0.203
3.588
0.378
1.622
18
0.194
3.640
0.391
1.608
19
0.187
3.689
0.403
1.597
20
0.180
3.735
0.415
1.585
21
0.173
3.778
0.425
1.575
22
0.167
3.819
0.434
1.566
23
0.162
3.858
0.443
1.557
24
0.157
3.895
0.451
1.548
25
0.153
3.931
0.459
1.541