Formulas and Tables

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p-chart formulas

The p formula (for the proportion of nonconforming units from subgroups that can vary in size):

f-p-bar

To calculate control limits for the p-chart:

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np-chart formulas

The np-chart formula (for number of nonconforming units from subgroups with a constant size):

Formulas and Tables

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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

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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:

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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

Formulas and Tables

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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

Formulas and Tables

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Skewness formula