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• Roger Rodriguez

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Nonconforming Fraction as a Function of Quality Index Q and Sample Size (standard deviation method) This spreadsheet reproduces Table B-5 of ANSI/ASQ Z1.9 and MIL-STD 414 Custom calculation Enter Q and n to get p

Q 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92

Sample size 3 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Q n p

1.7 5 0.66%

4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

5 3.799 3.606 3.417 3.231 3.048 2.869 2.693 2.521 2.352 2.187 2.026 1.869 1.716 1.567 1.423 1.283 1.148 1.018 0.893 0.773 0.659 0.552 0.451 0.356 0.270 0.192 0.123 0.065 0.021 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

7 5.282 5.126 4.972 4.820 4.671 4.524 4.379 4.237 4.098 3.961 3.827 3.695 3.565 3.438 3.314 3.192 3.072 2.955 2.840 2.728 2.618 2.511 2.406 2.304 2.204 2.106 2.011 1.918 1.828 1.740 1.655 1.572 1.491 1.413 1.337 1.263 1.192 1.123 1.057 0.992 0.930 0.870 0.813

10 5.875 5.731 5.590 5.450 5.313 5.179 5.046 4.915 4.787 4.661 4.537 4.415 4.295 4.178 4.062 3.948 3.837 3.728 3.620 3.515 3.412 3.311 3.211 3.114 3.019 2.925 2.834 2.744 2.657 2.571 2.487 2.405 2.324 2.246 2.169 2.094 2.021 1.950 1.880 1.812 1.746 1.681 1.618

15 6.202 6.065 5.930 5.797 5.666 5.538 5.411 5.286 5.164 5.043 4.924 4.808 4.693 4.580 4.469 4.361 4.254 4.148 4.045 3.944 3.844 3.746 3.650 3.556 3.463 3.373 3.284 3.196 3.111 3.027 2.945 2.864 2.785 2.708 2.632 2.558 2.485 2.414 2.344 2.276 2.209 2.144 2.080

20 6.339 6.204 6.072 5.942 5.813 5.687 5.563 5.440 5.320 5.201 5.085 4.970 4.858 4.747 4.638 4.531 4.425 4.322 4.220 4.120 4.022 3.926 3.831 3.738 3.647 3.557 3.469 3.383 3.298 3.215 3.133 3.054 2.975 2.898 2.823 2.749 2.677 2.606 2.537 2.468 2.402 2.337 2.273

1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.758 0.705 0.654 0.605 0.558 0.514 0.472 0.431 0.393 0.357 0.323 0.291 0.260 0.232 0.206 0.181 0.158 0.137 0.118 0.101 0.085 0.070 0.057 0.046 0.036 0.028 0.021 0.015 0.010 0.006 0.003 0.002 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.557 1.497 1.439 1.382 1.327 1.274 1.222 1.171 1.122 1.074 1.028 0.983 0.940 0.898 0.857 0.817 0.779 0.742 0.706 0.672 0.639 0.606 0.576 0.546 0.517 0.489 0.463 0.437 0.413 0.389 0.366 0.345 0.324 0.304 0.285 0.267 0.250 0.233 0.218 0.203 0.189

2.018 1.957 1.898 1.839 1.782 1.727 1.673 1.620 1.568 1.518 1.469 1.421 1.374 1.328 1.284 1.240 1.198 1.157 1.117 1.078 1.040 1.003 0.968 0.933 0.899 0.866 0.834 0.803 0.772 0.743 0.715 0.687 0.660 0.634 0.609 0.585 0.561 0.538 0.516 0.495 0.474

2.210 2.149 2.089 2.031 1.973 1.917 1.863 1.809 1.757 1.706 1.656 1.607 1.559 1.512 1.467 1.423 1.379 1.337 1.296 1.255 1.216 1.178 1.140 1.104 1.069 1.034 1.000 0.968 0.936 0.905 0.874 0.845 0.816 0.789 0.762 0.735 0.710 0.685 0.661 0.637 0.614

For sample size n and quality index Q, g(Q, n) 

Q n 1  and then the nonconforming fraction is 2(n - 1) 2

p  1  I  g (Q, n), n  w hereI ( g , n) 

1 n n  Β  1,  1 2 2 



g

0

t

n 2 2

Γ(q)Γ(s) (beta function) Γ(q  s) This can be simplified for use on a spreadsheet, using the Β(q,s) 

( 1  t)

n  2

2(n - 1)

2

p  1  I  g (Q, n), n  w hereI ( g , n) 

1 n n  Β  1,  1 2 2 



g

0

t

n 2 2

( 1  t)

n  2

Γ(q)Γ(s) (beta function) Γ(q  s) This can be simplified for use on a spreadsheet, using the Β(q,s) 

familiar F distribution (a transformation of the beta distribution) 1 F(Q,n)  1 and degrees of freedom  n - 2 g(Q,n) Then the nonconforming fraction is the cumulative F distribution for F w ith n - 2, n - 2 degrees of freedom. EXAMPLE : Q  1.59, n  5 as previously calculated. g

1.59 5 1   0.9444 2(5 - 1) 2

I(0.9444,5)  0.9781 1 - 0.9781  0.02

From the previous page, g(1.59,5)  0.9444 1 1 Then F  1   1  0.05885 g(Q,n) 0.9444 The nonconforming fraction is the cumulative F distribution for F  0.05885 and degrees of freedom n 1  n 2  n  2 w hichis 3 in this case.

25 6.414 6.281 6.150 6.021 5.894 5.769 5.646 5.525 5.406 5.289 5.173 5.060 4.948 4.838 4.730 4.624 4.519 4.417 4.316 4.217 4.120 4.024 3.930 3.838 3.747 3.658 3.571 3.485 3.401 3.318 3.237 3.157 3.079 3.003 2.928 2.854 2.782 2.711 2.642 2.574 2.507 2.442 2.378

30 6.463 6.330 6.200 6.072 5.946 5.822 5.699 5.579 5.460 5.344 5.229 5.116 5.005 4.896 4.789 4.683 4.579 4.477 4.377 4.278 4.181 4.086 3.992 3.901 3.810 3.722 3.635 3.549 3.465 3.383 3.302 3.223 3.145 3.069 2.994 2.920 2.848 2.778 2.709 2.641 2.574 2.509 2.445

35 6.496 6.364 6.235 6.107 5.981 5.858 5.736 5.616 5.498 5.382 5.268 5.155 5.045 4.936 4.829 4.724 4.620 4.519 4.419 4.320 4.224 4.129 4.036 3.944 3.854 3.766 3.679 3.594 3.510 3.428 3.347 3.268 3.190 3.114 3.039 2.966 2.894 2.824 2.755 2.687 2.620 2.555 2.491

50 6.554 6.423 6.295 6.168 6.043 5.920 5.800 5.681 5.563 5.448 5.335 5.223 5.113 5.005 4.899 4.794 4.692 4.590 4.491 4.393 4.297 4.203 4.110 4.019 3.930 3.842 3.755 3.671 3.587 3.505 3.425 3.346 3.269 3.193 3.118 3.045 2.974 2.903 2.834 2.766 2.700 2.635 2.571

75 6.597 6.468 6.340 6.214 6.090 5.967 5.847 5.729 5.612 5.498 5.385 5.274 5.164 5.057 4.951 4.847 4.745 4.644 4.545 4.448 4.352 4.258 4.166 4.075 3.986 3.898 3.812 3.728 3.645 3.563 3.483 3.404 3.327 3.252 3.177 3.104 3.033 2.962 2.893 2.826 2.759 2.694 2.630

100 6.619 6.489 6.362 6.236 6.112 5.990 5.870 5.752 5.636 5.522 5.409 5.298 5.189 5.082 4.976 4.873 4.771 4.670 4.572 4.474 4.379 4.285 4.193 4.103 4.013 3.926 3.840 3.756 3.673 3.591 3.511 3.433 3.356 3.280 3.206 3.133 3.061 2.991 2.922 2.854 2.788 2.723 2.659

150 6.640 6.510 6.383 6.258 6.134 6.013 5.893 5.775 5.660 5.545 5.433 5.323 5.214 5.107 5.001 4.898 4.796 4.696 4.597 4.501 4.405 4.312 4.220 4.129 4.040 3.953 3.867 3.783 3.700 3.619 3.539 3.461 3.384 3.308 3.234 3.161 3.089 3.019 2.950 2.883 2.816 2.751 2.688

2.316 2.255 2.195 2.136 2.078 2.022 1.967 1.913 1.861 1.809 1.759 1.710 1.661 1.614 1.568 1.524 1.480 1.437 1.395 1.354 1.314 1.275 1.237 1.200 1.164 1.129 1.094 1.061 1.028 0.996 0.965 0.935 0.905 0.876 0.848 0.821 0.794 0.769 0.743 0.719 0.695

2.383 2.321 2.261 2.203 2.145 2.089 2.033 1.979 1.927 1.875 1.824 1.775 1.727 1.679 1.633 1.588 1.544 1.500 1.458 1.417 1.377 1.337 1.299 1.261 1.225 1.189 1.154 1.120 1.087 1.054 1.023 0.992 0.962 0.933 0.904 0.876 0.849 0.823 0.797 0.772 0.748

g fraction is

 1 



g

0

t

n 2 2

ing the

( 1  t)

n 2 2

dt

2.429 2.367 2.307 2.249 2.191 2.134 2.079 2.025 1.972 1.920 1.870 1.820 1.772 1.724 1.678 1.632 1.588 1.544 1.502 1.460 1.420 1.380 1.342 1.304 1.267 1.231 1.196 1.161 1.128 1.095 1.063 1.032 1.002 0.972 0.943 0.915 0.887 0.861 0.834 0.809 0.784

2.508 2.447 2.387 2.328 2.270 2.214 2.158 2.104 2.051 1.999 1.948 1.898 1.850 1.802 1.755 1.709 1.665 1.621 1.578 1.536 1.495 1.455 1.416 1.378 1.340 1.304 1.268 1.233 1.199 1.166 1.134 1.102 1.071 1.041 1.011 0.982 0.954 0.927 0.900 0.874 0.848

2.568 2.506 2.446 2.387 2.330 2.273 2.218 2.163 2.110 2.058 2.007 1.957 1.908 1.860 1.813 1.767 1.722 1.678 1.635 1.593 1.551 1.511 1.472 1.433 1.395 1.359 1.323 1.287 1.253 1.219 1.186 1.154 1.123 1.092 1.062 1.033 1.004 0.977 0.949 0.923 0.897

2.597 2.535 2.475 2.416 2.358 2.302 2.246 2.192 2.139 2.086 2.035 1.985 1.936 1.888 1.841 1.795 1.750 1.706 1.663 1.620 1.579 1.538 1.499 1.460 1.422 1.385 1.349 1.314 1.279 1.245 1.212 1.180 1.148 1.117 1.087 1.058 1.029 1.001 0.974 0.947 0.921

2.625 2.564 2.504 2.445 2.387 2.330 2.275 2.220 2.167 2.114 2.063 2.013 1.964 1.916 1.869 1.823 1.777 1.733 1.690 1.647 1.606 1.565 1.526 1.487 1.449 1.412 1.375 1.340 1.305 1.271 1.238 1.205 1.173 1.142 1.112 1.082 1.053 1.025 0.998 0.971 0.944

 1 



g

0

t

n 2 2

( 1  t)

n 2 2

dt

ing the

ta distribution)

2

eF om.

ted.

1 - 0.9781  0.0219

444

885

ulative F s of freedom

In Microsoft Excel, FDIST(F,n1,n2) returns the upper tail for F with n1,n2 degrees In this case, 1-FDIST(0.05885,3,3) equals 0.0219 as before.

200 6.650 6.521 6.394 6.269 6.145 6.024 5.905 5.787 5.671 5.557 5.445 5.335 5.226 5.119 5.014 4.910 4.809 4.709 4.610 4.513 4.418 4.325 4.233 4.143 4.054 3.966 3.881 3.797 3.714 3.633 3.553 3.474 3.397 3.322 3.248 3.175 3.103 3.033 2.964 2.897 2.830 2.765 2.702

2.639 2.578 2.518 2.459 2.401 2.344 2.288 2.234 2.181 2.128 2.077 2.027 1.978 1.929 1.882 1.836 1.791 1.747 1.703 1.661 1.619 1.579 1.539 1.500 1.462 1.425 1.388 1.352 1.318 1.284 1.250 1.218 1.186 1.155 1.124 1.095 1.065 1.037 1.009 0.982 0.956

pper tail for F with n1,n2 degrees of freedom. In this application, this is the conforming fraction so the nonconformin 9 as before.

ing fraction so the nonconforming fraction is 1-FDIST(F,n-2,n-2) for any sample size.

Nonconforming Fraction as a Function of Quality Index Q and Sample Size (standard deviation method) This spreadsheet reproduces Table B-5 of ANSI/ASQ Z1.9 and MIL-STD 414 Custom calculation Enter Q and n to get p

Q n p

#NUM!

Tabla B5 MIL STD 414 Tabla de Porcentaje de defectos por lote (estimado) utilizando el método de la desviación standard Q 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40

Tamaño de Muestra (n) 25 30

3

4

5

7

10

15

20

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

3.799 3.606 3.417 3.231 3.048 2.869 2.693 2.521 2.352 2.187 2.026 1.869 1.716 1.567 1.423 1.283 1.148 1.018 0.893 0.773 0.659 0.552 0.451 0.356 0.270 0.192 0.123 0.065 0.021 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

5.282 5.126 4.972 4.820 4.671 4.524 4.379 4.237 4.098 3.961 3.827 3.695 3.565 3.438 3.314 3.192 3.072 2.955 2.840 2.728 2.618 2.511 2.406 2.304 2.204 2.106 2.011 1.918 1.828 1.740 1.655 1.572 1.491 1.413 1.337 1.263 1.192 1.123 1.057 0.992 0.930 0.870 0.813 0.758 0.705 0.654 0.605 0.558 0.514 0.472 0.431 0.393 0.357 0.323 0.291 0.260 0.232 0.206 0.181 0.158 0.137 0.118 0.101 0.085 0.070 0.057 0.046 0.036 0.028 0.021 0.015 0.010 0.006 0.003 0.002 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

5.875 5.731 5.590 5.450 5.313 5.179 5.046 4.915 4.787 4.661 4.537 4.415 4.295 4.178 4.062 3.948 3.837 3.728 3.620 3.515 3.412 3.311 3.211 3.114 3.019 2.925 2.834 2.744 2.657 2.571 2.487 2.405 2.324 2.246 2.169 2.094 2.021 1.950 1.880 1.812 1.746 1.681 1.618 1.557 1.497 1.439 1.382 1.327 1.274 1.222 1.171 1.122 1.074 1.028 0.983 0.940 0.898 0.857 0.817 0.779 0.742 0.706 0.672 0.639 0.606 0.576 0.546 0.517 0.489 0.463 0.437 0.413 0.389 0.366 0.345 0.324 0.304 0.285 0.267 0.250 0.233 0.218 0.203 0.189 0.175 0.163 0.151 0.139 0.128 0.118 0.109

6.202 6.065 5.930 5.797 5.666 5.538 5.411 5.286 5.164 5.043 4.924 4.808 4.693 4.580 4.469 4.361 4.254 4.148 4.045 3.944 3.844 3.746 3.650 3.556 3.463 3.373 3.284 3.196 3.111 3.027 2.945 2.864 2.785 2.708 2.632 2.558 2.485 2.414 2.344 2.276 2.209 2.144 2.080 2.018 1.957 1.898 1.839 1.782 1.727 1.673 1.620 1.568 1.518 1.469 1.421 1.374 1.328 1.284 1.240 1.198 1.157 1.117 1.078 1.040 1.003 0.968 0.933 0.899 0.866 0.834 0.803 0.772 0.743 0.715 0.687 0.660 0.634 0.609 0.585 0.561 0.538 0.516 0.495 0.474 0.454 0.435 0.416 0.398 0.381 0.364 0.348

6.339 6.204 6.072 5.942 5.813 5.687 5.563 5.440 5.320 5.201 5.085 4.970 4.858 4.747 4.638 4.531 4.425 4.322 4.220 4.120 4.022 3.926 3.831 3.738 3.647 3.557 3.469 3.383 3.298 3.215 3.133 3.054 2.975 2.898 2.823 2.749 2.677 2.606 2.537 2.468 2.402 2.337 2.273 2.210 2.149 2.089 2.031 1.973 1.917 1.863 1.809 1.757 1.706 1.656 1.607 1.559 1.512 1.467 1.423 1.379 1.337 1.296 1.255 1.216 1.178 1.140 1.104 1.069 1.034 1.000 0.968 0.936 0.905 0.874 0.845 0.816 0.789 0.762 0.735 0.710 0.685 0.661 0.637 0.614 0.592 0.571 0.550 0.530 0.510 0.491 0.473

6.414 6.281 6.150 6.021 5.894 5.769 5.646 5.525 5.406 5.289 5.173 5.060 4.948 4.838 4.730 4.624 4.519 4.417 4.316 4.217 4.120 4.024 3.930 3.838 3.747 3.658 3.571 3.485 3.401 3.318 3.237 3.157 3.079 3.003 2.928 2.854 2.782 2.711 2.642 2.574 2.507 2.442 2.378 2.316 2.255 2.195 2.136 2.078 2.022 1.967 1.913 1.861 1.809 1.759 1.710 1.661 1.614 1.568 1.524 1.480 1.437 1.395 1.354 1.314 1.275 1.237 1.200 1.164 1.129 1.094 1.061 1.028 0.996 0.965 0.935 0.905 0.876 0.848 0.821 0.794 0.769 0.743 0.719 0.695 0.672 0.650 0.628 0.606 0.586 0.566 0.546

6.463 6.330 6.200 6.072 5.946 5.822 5.699 5.579 5.460 5.344 5.229 5.116 5.005 4.896 4.789 4.683 4.579 4.477 4.377 4.278 4.181 4.086 3.992 3.901 3.810 3.722 3.635 3.549 3.465 3.383 3.302 3.223 3.145 3.069 2.994 2.920 2.848 2.778 2.709 2.641 2.574 2.509 2.445 2.383 2.321 2.261 2.203 2.145 2.089 2.033 1.979 1.927 1.875 1.824 1.775 1.727 1.679 1.633 1.588 1.544 1.500 1.458 1.417 1.377 1.337 1.299 1.261 1.225 1.189 1.154 1.120 1.087 1.054 1.023 0.992 0.962 0.933 0.904 0.876 0.849 0.823 0.797 0.772 0.748 0.724 0.701 0.678 0.656 0.635 0.614 0.594

35

50

75

100

150

200

6.496 6.364 6.235 6.107 5.981 5.858 5.736 5.616 5.498 5.382 5.268 5.155 5.045 4.936 4.829 4.724 4.620 4.519 4.419 4.320 4.224 4.129 4.036 3.944 3.854 3.766 3.679 3.594 3.510 3.428 3.347 3.268 3.190 3.114 3.039 2.966 2.894 2.824 2.755 2.687 2.620 2.555 2.491 2.429 2.367 2.307 2.249 2.191 2.134 2.079 2.025 1.972 1.920 1.870 1.820 1.772 1.724 1.678 1.632 1.588 1.544 1.502 1.460 1.420 1.380 1.342 1.304 1.267 1.231 1.196 1.161 1.128 1.095 1.063 1.032 1.002 0.972 0.943 0.915 0.887 0.861 0.834 0.809 0.784 0.760 0.736 0.714 0.691 0.670 0.648 0.628

6.554 6.423 6.295 6.168 6.043 5.920 5.800 5.681 5.563 5.448 5.335 5.223 5.113 5.005 4.899 4.794 4.692 4.590 4.491 4.393 4.297 4.203 4.110 4.019 3.930 3.842 3.755 3.671 3.587 3.505 3.425 3.346 3.269 3.193 3.118 3.045 2.974 2.903 2.834 2.766 2.700 2.635 2.571 2.508 2.447 2.387 2.328 2.270 2.214 2.158 2.104 2.051 1.999 1.948 1.898 1.850 1.802 1.755 1.709 1.665 1.621 1.578 1.536 1.495 1.455 1.416 1.378 1.340 1.304 1.268 1.233 1.199 1.166 1.134 1.102 1.071 1.041 1.011 0.982 0.954 0.927 0.900 0.874 0.848 0.824 0.799 0.776 0.753 0.730 0.709 0.687

6.597 6.468 6.340 6.214 6.090 5.967 5.847 5.729 5.612 5.498 5.385 5.274 5.164 5.057 4.951 4.847 4.745 4.644 4.545 4.448 4.352 4.258 4.166 4.075 3.986 3.898 3.812 3.728 3.645 3.563 3.483 3.404 3.327 3.252 3.177 3.104 3.033 2.962 2.893 2.826 2.759 2.694 2.630 2.568 2.506 2.446 2.387 2.330 2.273 2.218 2.163 2.110 2.058 2.007 1.957 1.908 1.860 1.813 1.767 1.722 1.678 1.635 1.593 1.551 1.511 1.472 1.433 1.395 1.359 1.323 1.287 1.253 1.219 1.186 1.154 1.123 1.092 1.062 1.033 1.004 0.977 0.949 0.923 0.897 0.872 0.847 0.823 0.799 0.777 0.754 0.732

6.619 6.489 6.362 6.236 6.112 5.990 5.870 5.752 5.636 5.522 5.409 5.298 5.189 5.082 4.976 4.873 4.771 4.670 4.572 4.474 4.379 4.285 4.193 4.103 4.013 3.926 3.840 3.756 3.673 3.591 3.511 3.433 3.356 3.280 3.206 3.133 3.061 2.991 2.922 2.854 2.788 2.723 2.659 2.597 2.535 2.475 2.416 2.358 2.302 2.246 2.192 2.139 2.086 2.035 1.985 1.936 1.888 1.841 1.795 1.750 1.706 1.663 1.620 1.579 1.538 1.499 1.460 1.422 1.385 1.349 1.314 1.279 1.245 1.212 1.180 1.148 1.117 1.087 1.058 1.029 1.001 0.974 0.947 0.921 0.895 0.870 0.846 0.822 0.799 0.777 0.755

6.640 6.510 6.383 6.258 6.134 6.013 5.893 5.775 5.660 5.545 5.433 5.323 5.214 5.107 5.001 4.898 4.796 4.696 4.597 4.501 4.405 4.312 4.220 4.129 4.040 3.953 3.867 3.783 3.700 3.619 3.539 3.461 3.384 3.308 3.234 3.161 3.089 3.019 2.950 2.883 2.816 2.751 2.688 2.625 2.564 2.504 2.445 2.387 2.330 2.275 2.220 2.167 2.114 2.063 2.013 1.964 1.916 1.869 1.823 1.777 1.733 1.690 1.647 1.606 1.565 1.526 1.487 1.449 1.412 1.375 1.340 1.305 1.271 1.238 1.205 1.173 1.142 1.112 1.082 1.053 1.025 0.998 0.971 0.944 0.919 0.893 0.869 0.845 0.822 0.799 0.777

6.650 6.521 6.394 6.269 6.145 6.024 5.905 5.787 5.671 5.557 5.445 5.335 5.226 5.119 5.014 4.910 4.809 4.709 4.610 4.513 4.418 4.325 4.233 4.143 4.054 3.966 3.881 3.797 3.714 3.633 3.553 3.474 3.397 3.322 3.248 3.175 3.103 3.033 2.964 2.897 2.830 2.765 2.702 2.639 2.578 2.518 2.459 2.401 2.344 2.288 2.234 2.181 2.128 2.077 2.027 1.978 1.929 1.882 1.836 1.791 1.747 1.703 1.661 1.619 1.579 1.539 1.500 1.462 1.425 1.388 1.352 1.318 1.284 1.250 1.218 1.186 1.155 1.124 1.095 1.065 1.037 1.009 0.982 0.956 0.930 0.905 0.880 0.856 0.833 0.810 0.787

For sample size n and quality index Q, g(Q, n) 

Q n 1  and then the nonconforming fraction is 2(n - 1) 2

p  1  I  g (Q, n), n  w hereI ( g , n) 

1 n n  Β  1,  1 2 2 



g

0

n

2

t 2 ( 1  t)

n 2 2

dt

Γ(q)Γ(s) (beta function) Γ(q  s) This can be simplified for use on a spreadsheet, using the Β(q,s) 

familiar F distribution (a transformation of the beta distribution) 1 F(Q,n)  1 and degrees of freedom  n - 2 g(Q,n) Then the nonconforming fraction is the cumulative F distribution for F w ith n - 2, n - 2 degrees of freedom. EXAMPLE : Q  1.59, n  5 as previously calculated. g

1.59 5 1   0.9444 2(5 - 1) 2

I(0.9444,5)  0.9781 1 - 0.9781  0.0219

From the previous page, g(1.59,5)  0.9444 1 1 1   1  0.05885 g(Q,n) 0.9444 The nonconforming fraction is the cumulative F distribution for F  0.05885 and degrees of freedom n 1  n 2  n  2 w hichis 3 in this case. Then F 

In Microsoft Excel, FDIST(F,n1,n2) returns the upper tail for F with n1,n2 degrees of freedom. In this application, this is the conforming fraction so the nonconforming fraction is 1-FDIST(F,n-2,n-2) for any sample size. In this case, 1-FDIST(0.05885,3,3) equals 0.0219 as before.