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Actual percent out-of-spec formulas

Actual % out-of-spec = % above upper spec + % below lower spec

To find actual % above spec:

To find actual % below spec:

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Theoretical percent out-of-spec formulas

To find theoretical % above spec:

Locate the value for Zupper in a standard normal table.

To find theoretical % below spec:

Locate the value for Zlower in a standard normal table.

Given:

And assuming:

In theory:

If the Z value from the standard normal table is multiplied by 100, then this process is expected to produce X% of its output outside specification.

Show me:
The standard normal table
The formulas for Target Z

Related topic:
The formula for actual percent out of spec

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Chi-square formula and degrees of freedom table

Alpha value = 1%
Degrees of freedom
Value
Degrees of freedom
Value
1
3.84
1
6.63
2
5.99
2
9.21
3
7.82
3
11.3
4
9.49
4
13.3
5
11.10
5
15.1
6
12.60
6
16.8
7
14.10
7
18.5
8
15.50
8
20.1
9
16.90
9
23.2
10
18.30
10
24.7
11
19.70
11
26.2
12
21.00
12
27.7
13
22.40
13
29.1
14
23.70
14
30.6
15
25.00
15
30.6
16
26.30
16
32.0
17
27.60
17
33.4
18
28.90
18
34.8
19
30.10
19
36.2
20
31.40
20
37.6
21
32.70
21
38.9
22
33.90
22
40.3
23
35.20
23
41.6
24
36.40
24
43.0
25
37.70
25
44.3
26
38.90
26
45.6
27
40.10
27
47.0
28
41.30
28
48.3
29
42.60
29
49.6
30
43.80
30
50.9

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Kurtosis formula

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Skewness formula

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Coefficient of variance formula

Where

sigma-individuals

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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

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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

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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