Formulas and Tables
SPC DEMO
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Standard normal table
| z | x.x0 | x.x1 | x.x2 | x.x3 | x.x4 | x.x5 | x.x6 | x.x7 | x.x8 | x.x9 |
| 4.0 | .00003 | |||||||||
| 3.9 | .00005 | .00005 | .00004 | .00004 | .00004 | .00004 | .00004 | .00004 | .00003 | .00003 |
| 3.8 | .00007 | .00007 | .00007 | .00006 | .00006 | .00006 | .00006 | .00005 | .00005 | .00005 |
| 3.7 | .00011 | .00010 | .00010 | .00010 | .00009 | .00009 | .00008 | .00008 | .00008 | .00008 |
| 3.6 | .00016 | .00015 | .00015 | .00014 | .00014 | .00013 | .00013 | .00012 | .00012 | .00011 |
| 3.5 | .00023 | .00022 | .00022 | .00021 | .00020 | .00019 | .00019 | .00018 | .00017 | .00017 |
| 3.4 | .00034 | .00032 | .00031 | .00030 | .00029 | .00028 | .00027 | .00026 | .00025 | .00024 |
| 3.3 | .00048 | .00047 | .00045 | .00043 | .00042 | .00040 | .00039 | .00038 | .00036 | .00035 |
| 3.2 | .00069 | .00066 | .00064 | .00062 | .00060 | .00058 | .00056 | .00054 | .00052 | .00050 |
| 3.1 | .00097 | .00094 | .00090 | .00087 | .00084 | .00082 | .00079 | .00076 | .00074 | .00071 |
| 3.0 | .00135 | .00131 | .00126 | .00122 | .00118 | .00114 | .00111 | .00107 | .00104 | .00100 |
| 2.9 | .0019 | .0018 | .0018 | .0017 | .0016 | .0016 | .0015 | .0015 | .0014 | .0014 |
| 2.8 | .0026 | .0025 | .0024 | .0023 | .0023 | .0022 | .0021 | .0021 | .0020 | .0019 |
| 2.7 | .0035 | .0034 | .0033 | .0032 | .0031 | .0030 | .0029 | .0028 | .0027 | .0026 |
| 2.6 | .0047 | .0045 | .0044 | .0043 | .0041 | .0040 | .0039 | .0038 | .0037 | .0036 |
| 2.5 | .0062 | .0060 | .0059 | .0057 | .0055 | .0054 | .0052 | .0051 | .0049 | .0048 |
| 2.4 | .0082 | .0080 | .0078 | .0075 | .0073 | .0071 | .0069 | .0068 | .0066 | .0064 |
| 2.3 | .0107 | .0104 | .0102 | .0099 | .0096 | .0094 | .0091 | .0089 | .0087 | .0084 |
| 2.2 | .0139 | .0136 | .0132 | .0129 | .0125 | .0122 | .0119 | .0116 | .0113 | .0110 |
| 2.1 | .0179 | .0174 | .0170 | .0166 | .0162 | .0158 | .0154 | .0150 | .0146 | .0143 |
| 2.0 | .0228 | .0222 | .0217 | .0212 | .0207 | .0202 | .0197 | .0192 | .0188 | .0183 |
| z | x.x0 | x.x1 | x.x2 | x.x3 | x.x4 | x.x5 | x.x6 | x.x7 | x.x8 | x.x9 |
| 1.9 | .0287 | .0281 | .0274 | .0268 | .0262 | .0256 | .0250 | .0244 | .0239 | .0233 |
| 1.8 | .0359 | .0351 | .0344 | .0336 | .0329 | .0322 | .0314 | .0307 | .0301 | .0294 |
| 1.7 | .0446 | .0436 | .0427 | .0418 | .0409 | .0401 | .0392 | .0384 | .0375 | .0367 |
| 1.6 | .0548 | .0537 | .0526 | .0516 | .0505 | .0495 | .0485 | .0475 | .0465 | .0455 |
| 1.5 | .0668 | .0655 | .0643 | .0630 | .0618 | .0606 | .0594 | .0582 | .0571 | .0559 |
| 1.4 | .0808 | .0793 | .0778 | .0764 | .0749 | .0735 | .0721 | .0708 | .0694 | .0681 |
| 1.3 | .0968 | .0951 | .0934 | .0918 | .0901 | .0885 | .0869 | .0853 | .0838 | .0823 |
| 1.2 | .1151 | .1131 | .1112 | .1093 | .1075 | .1056 | .1038 | .1020 | .1003 | .0985 |
| 1.1 | .1357 | .1335 | .1314 | .1292 | .1271 | .1251 | .1230 | .1210 | .1190 | .1170 |
| 1.0 | .1587 | .1562 | .1539 | .1515 | .1492 | .1469 | .1446 | .1423 | .1401 | .1379 |
| 0.9 | .1841 | .1814 | .1788 | .1762 | .1736 | .1711 | .1685 | .1660 | .1635 | .1611 |
| 0.8 | .2119 | .2090 | .2061 | .2033 | .2005 | .1977 | .1949 | .1922 | .1894 | .1867 |
| 0.7 | .2420 | .2389 | .2358 | .2327 | .2297 | .2266 | .2236 | .2206 | .2177 | .2148 |
| 0.6 | .2743 | .2709 | .2676 | .2643 | .2611 | .2578 | .2546 | .2514 | .2483 | .2451 |
| 0.5 | .3085 | .3050 | .3015 | .2981 | .2946 | .2912 | .2877 | .2843 | .2810 | .2776 |
| 0.4 | .3446 | .3409 | .3372 | .3336 | .3300 | .3264 | .3228 | .3192 | .3156 | .3121 |
| 0.3 | .3821 | .3783 | .3745 | .3707 | .3669 | .3632 | .3594 | .3557 | .3520 | .3483 |
| 0.2 | .4207 | .4168 | .4129 | .4090 | .4052 | .4013 | .3974 | .3936 | .3897 | .3859 |
| 0.1 | .4602 | .4562 | .4522 | .4483 | .4443 | .4404 | .4364 | .4325 | .4286 | .4247 |
| 0.0 | .5000 | .4960 | .4920 | .4880 | .4840 | .4801 | .4761 | .4721 | .4681 | .4641 |
Formulas and Tables
SPC DEMO
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Target Z, Zupper, and Zlower formulas
Target Z
Zupper
Zlower
Where:
= the mean of the X-bars
T = specification target
Sigma is an estimated sigma or sigma of the individual values, depending on how the histogram
is configured or whether you are calculating Cpk (estimated) or Ppk (actual).
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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:
Related topic:
The formula for theoretical percent out of 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:
- An X-bar value
- A sigma value and
- A set of specifications
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
Formulas and Tables
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Chi-square formula and degrees of freedom table
|
Alpha value = 5% |
Alpha value = 1%
|
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|
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
|
|
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
Formulas and Tables
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Sigma
Sigma of the individuals (actual sigma)
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
|