Formulas and Tables

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Kurtosis formula

Formulas and Tables

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Skewness formula

Formulas and Tables

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Coefficient of variance formula

Where

sigma-individuals

Formulas and Tables

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Sigma

sigma-individuals

Sigma of the individuals (actual sigma)

sigma-estimated

Estimated sigma

Subgroup Size
d2
Subgroup Size
d2
2
1.128
14
3.407
3
1.693
15
3.472
4
2.059
16
3.532
5
2.326
17
3.588
6
2.534
18
3.640
7
2.704
19
3.689
8
2.847
20
3.735
9
2.970
21
3.778
10
3.078
22
3.819
11
3.173
23
3.858
12
3.258
24
3.895
13
3.336
25
3.931

Formulas and Tables

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Median 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:

Median chart formulas

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 median and range charts
Subgroup Size
A2
D3
D4
2
1.880
—–
3.268
3
1.023
—–
2.574
4
0.729
—–
2.282
5
0.577
—–
2.114
6
0.483
—–
2.004
7
0.419
0.076
1.924
8
0.373
0.136
1.864
9
0.337
0.184
1.816
10
0.308
0.223
1.777
11
0.285
0.256
1.276
12
0.266
0.283
1.717
13
0.249
0.307
1.693
14
0.235
0.328
1.672
15
0.223
0.347
1.653
16
0.212
0.363
1.637
17
0.203
0.378
1.622
18
0.194
0.391
1.608
19
0.187
0.403
1.597
20
0.180
0.415
1.585
21
0.173
0.425
1.276
22
0.167
0.434
1.566
23
0.162
0.443
1.557
24
0.157
0.451
1.548
25
0.153
0.459
1.541

Formulas and Tables

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Individuals and moving range chart formulas

The most common (and recommended) method of computing control limits for an individuals chart based on 3 standard deviations is:

Individuals (X)

Upper control limit

Lower control limit

Moving Range

Upper control limit

Lower control limit

Tabular values for X and range charts
Subgroup
Size
E2
D4
1
2.660
3.268
2
2.660
3.268
3
1.772
2.574
4
1.457
2.282
5
1.290
2.114
6
1.184
2.004
7
1.109
1.924
8
1.054
1.864
9
1.010
1.816
10
0.975
1.777

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

Formulas and Tables

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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

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Formulas and Tables

Variables control charts

Descriptive statistics

Tables

Attributes control charts

Rare event charts

Capability indices