By default, normal capability analysis for multiple variables estimates within-subgroup standard deviation and overall standard deviation using the methods described below. However, you can also choose to analyze between/within subgroup variation for this analysis. For information on the methods that are used to estimate between/'within standard deviation, go to Estimating standard deviation.
The method used to estimate σwithin depends on the subgroup size.
where:
If you change the default method and choose not to use the unbiasing constant, σwithin is estimated by Sp.
Term | Description |
---|---|
d | Degrees of freedom for Sp= Σ (ni- 1) |
Xij | jth observation in the ith subgroup |
X̅i | Mean of the ith subgroup |
ni | Number of observations in the ith subgroup |
C4(d+1) | Unbiasing constant |
Γ(.) | Gamma function |
where:
If n are all the same:
Term | Description |
---|---|
ri | Range of the ithsubgroup |
d2 (ni) | An unbiasing constant read from a table (for more information, see the section Unbiasing constants d2(), d3(), and d4() |
d3 (ni) | An unbiasing constant read from a table (for more information, see the section Unbiasing constants d2(), d3(), and d4() |
ni | Number of observations in the ith subgroup |
where:
If you change the default setting and do not use the unbiasing constant, σwithin is estimated by Σ Si / number of subgroups.
Term | Description |
---|---|
c4(ni) | Unbiasing constant (as defined for pooled standard deviation) |
Si | Standard deviation of subgroup i |
ni | Number of observations in the ith subgroup |
where:
Term | Description |
---|---|
Ri | The ith moving range |
w | The number of observations used in the moving range. The default is w = 2 |
d2(w) | An unbiasing constant read from a table (for more information, see the section Unbiasing constants d2(), d3(), and d4() |
where:
Term | Description |
---|---|
MRi | The ith moving range |
MRbar̅ | Median of the MRi |
w | The number of observations used in the moving range. The default is w = 2 |
d4(w) | An unbiasing constant read from a table (for more information, see the section Unbiasing constants d2(), d3(), and d4()) |
If you change the default setting and do not use the unbiasing constant, σwithin is estimated by
Term | Description |
---|---|
di | Successive differences |
c4(ni) | Unbiasing constant (as defined for the pooled standard deviation) |
c4(ni)' | Unbiasing constant ≈ c4(ni). For more information, see the section Unbiasing constant c4'() |
N | Total number of observations |
ni | Number of observations in the ith subgroup |
where:
By default, Minitab does not use the unbiasing constant when estimating σoverall. σoverall is estimated by S. If you want to estimate overall standard deviation using the unbiasing constant, change the default settings in the Estimate subdialog box.
Term | Description |
---|---|
Xij | The jth observation in the ith subgroup |
X̅ | Process mean |
ni | Number of observations in the ith subgroup |
C4 (N) | Unbiasing constant (as defined for the pooled standard deviation) |
N (or Σ ni) | Total number of observations |
The Box-Cox transformation estimates a lambda value, as shown in the following table, which minimizes the standard deviation of a standardized transformed variable. The resulting transformation is Yλ when λ ҂ 0 and ln Y when λ = 0.
The Box-Cox method searches through many types of transformations. The following table shows some common transformations where Y' is the transform of the data Y.
Lambda (λ) value | Transformation |
---|---|
d2(N) is the expected value of the range of N observations from a normal population with standard deviation = 1. Thus, if r is the range of a sample of N observations from a normal distribution with standard deviation = σ, then E(r) = d2(N)σ.
d3(N) is the standard deviation of the range of N observations from a normal population with σ = 1. Thus, if r is the range of a sample of N observations from a normal distribution with standard deviation = σ, then stdev(r) = d3(N)σ.
Use the following table to find an unbiasing constant for a given value, N. (To determine the value of N, consult the formula for the statistic of interest.)
N | d2(N) | d3(N) | d4(N) |
---|---|---|---|
2 | 1.128 | 0.8525 | 0.954 |
3 | 1.693 | 0.8884 | 1.588 |
4 | 2.059 | 0.8798 | 1.978 |
5 | 2.326 | 0.8641 | 2.257 |
6 | 2.534 | 0.8480 | 2.472 |
7 | 2.704 | 0.8332 | 2.645 |
8 | 2.847 | 0.8198 | 2.791 |
9 | 2.970 | 0.8078 | 2.915 |
10 | 3.078 | 0.7971 | 3.024 |
11 | 3.173 | 0.7873 | 3.121 |
12 | 3.258 | 0.7785 | 3.207 |
13 | 3.336 | 0.7704 | 3.285 |
14 | 3.407 | 0.7630 | 3.356 |
15 | 3.472 | 0.7562 | 3.422 |
16 | 3.532 | 0.7499 | 3.482 |
17 | 3.588 | 0.7441 | 3.538 |
18 | 3.640 | 0.7386 | 3.591 |
19 | 3.689 | 0.7335 | 3.640 |
20 | 3.735 | 0.7287 | 3.686 |
21 | 3.778 | 0.7242 | 3.730 |
22 | 3.819 | 0.7199 | 3.771 |
23 | 3.858 | 0.7159 | 3.811 |
24 | 3.895 | 0.7121 | 3.847 |
25 | 3.931 | 0.7084 | 3.883 |
N | d2(N) |
---|---|
26 | 3.964 |
27 | 3.997 |
28 | 4.027 |
29 | 4.057 |
30 | 4.086 |
31 | 4.113 |
32 | 4.139 |
33 | 4.165 |
34 | 4.189 |
35 | 4.213 |
36 | 4.236 |
37 | 4.259 |
38 | 4.280 |
39 | 4.301 |
40 | 4.322 |
41 | 4.341 |
42 | 4.361 |
43 | 4.379 |
44 | 4.398 |
45 | 4.415 |
46 | 4.433 |
47 | 4.450 |
48 | 4.466 |
49 | 4.482 |
50 | 4.498 |
Term | Description |
---|---|
Γ() | gamma function |
Use the following tables to find values for the unbiasing constant, c4'(), which is used in the formulas for the square root of MSSD method of estimating sigma.
N | c4'(N) | N | c4'(N) | N | c4'(N) |
---|---|---|---|---|---|
2 | 0.797850 | 41 | 0.990797 | 80 | 0.995215 |
3 | 0.871530 | 42 | 0.991013 | 81 | 0.995272 |
4 | 0.905763 | 43 | 0.991218 | 82 | 0.995328 |
5 | 0.925222 | 44 | 0.991415 | 83 | 0.995383 |
6 | 0.937892 | 45 | 0.991602 | 84 | 0.995436 |
7 | 0.946837 | 46 | 0.991782 | 85 | 0.995489 |
8 | 0.953503 | 47 | 0.991953 | 86 | 0.995539 |
9 | 0.958669 | 48 | 0.992118 | 87 | 0.995589 |
10 | 0.962793 | 49 | 0.992276 | 88 | 0.995638 |
11 | 0.966163 | 50 | 0.992427 | 89 | 0.995685 |
12 | 0.968968 | 51 | 0.992573 | 90 | 0.995732 |
13 | 0.971341 | 52 | 0.992713 | 91 | 0.995777 |
14 | 0.973375 | 53 | 0.992848 | 92 | 0.995822 |
15 | 0.975137 | 54 | 0.992978 | 93 | 0.995865 |
16 | 0.976679 | 55 | 0.993103 | 94 | 0.995908 |
17 | 0.978039 | 56 | 0.993224 | 95 | 0.995949 |
18 | 0.979249 | 57 | 0.993340 | 96 | 0.995990 |
19 | 0.980331 | 58 | 0.993452 | 97 | 0.996030 |
20 | 0.981305 | 59 | 0.993561 | 98 | 0.996069 |
21 | 0.982187 | 60 | 0.993666 | 99 | 0.996108 |
22 | 0.982988 | 61 | 0.993767 | 100 | 0.996145 |
23 | 0.983720 | 62 | 0.993866 | 101 | 0.996182 |
24 | 0.984391 | 63 | 0.993961 | 102 | 0.996218 |
25 | 0.985009 | 64 | 0.994053 | 103 | 0.996253 |
26 | 0.985579 | 65 | 0.994142 | 104 | 0.996288 |
27 | 0.986107 | 66 | 0.994229 | 105 | 0.996322 |
28 | 0.986597 | 67 | 0.994313 | 106 | 0.996356 |
29 | 0.987054 | 68 | 0.994395 | 107 | 0.996389 |
30 | 0.987480 | 69 | 0.994474 | 108 | 0.996421 |
31 | 0.987878 | 70 | 0.994551 | 109 | 0.996452 |
32 | 0.988252 | 71 | 0.994626 | 110 | 0.996483 |
33 | 0.988603 | 72 | 0.994699 | 111 | 0.996514 |
34 | 0.988934 | 73 | 0.994769 | 112 | 0.996544 |
35 | 0.989246 | 74 | 0.994838 | 113 | 0.996573 |
36 | 0.989540 | 75 | 0.994905 | 114 | 0.996602 |
37 | 0.989819 | 76 | 0.994970 | 115 | 0.996631 |
38 | 0.990083 | 77 | 0.995034 | 116 | 0.996658 |
39 | 0.990333 | 78 | 0.995096 | 117 | 0.996686 |
40 | 0.990571 | 79 | 0.995156 | 118 | 0.996713 |
N | c4'(N) | N | c4'(N) | N | c4'(N) |
---|---|---|---|---|---|
119 | 0.996739 | 160 | 0.997541 | 201 | 0.998016 |
120 | 0.996765 | 161 | 0.997555 | 202 | 0.998025 |
121 | 0.996791 | 162 | 0.997570 | 203 | 0.998034 |
122 | 0.996816 | 163 | 0.997584 | 204 | 0.998043 |
123 | 0.996841 | 164 | 0.997598 | 205 | 0.998052 |
124 | 0.996865 | 165 | 0.997612 | 206 | 0.998061 |
125 | 0.996889 | 166 | 0.997625 | 207 | 0.998070 |
126 | 0.996913 | 167 | 0.997639 | 208 | 0.998078 |
127 | 0.996936 | 168 | 0.997652 | 209 | 0.998087 |
128 | 0.996959 | 169 | 0.997665 | 210 | 0.998095 |
129 | 0.996982 | 170 | 0.997678 | 211 | 0.998104 |
130 | 0.997004 | 171 | 0.997691 | 212 | 0.998112 |
131 | 0.997026 | 172 | 0.997703 | 213 | 0.998120 |
132 | 0.997047 | 173 | 0.997716 | 214 | 0.998128 |
133 | 0.997069 | 174 | 0.997728 | 215 | 0.998137 |
134 | 0.997089 | 175 | 0.997741 | 216 | 0.998145 |
135 | 0.997110 | 176 | 0.997753 | 217 | 0.998152 |
136 | 0.997130 | 177 | 0.997765 | 218 | 0.998160 |
137 | 0.997150 | 178 | 0.997776 | 219 | 0.998168 |
138 | 0.997170 | 179 | 0.997788 | 220 | 0.998176 |
139 | 0.997189 | 180 | 0.997800 | 221 | 0.998184 |
140 | 0.997209 | 181 | 0.997811 | 222 | 0.998191 |
141 | 0.997227 | 182 | 0.997822 | 223 | 0.998199 |
142 | 0.997246 | 183 | 0.997834 | 224 | 0.998206 |
143 | 0.997264 | 184 | 0.997845 | 225 | 0.998214 |
144 | 0.997282 | 185 | 0.997856 | 226 | 0.998221 |
145 | 0.997300 | 186 | 0.997866 | 227 | 0.998228 |
146 | 0.997318 | 187 | 0.997877 | 228 | 0.998235 |
147 | 0.997335 | 188 | 0.997888 | 229 | 0.998242 |
148 | 0.997352 | 189 | 0.997898 | 230 | 0.998250 |
149 | 0.997369 | 190 | 0.997909 | 231 | 0.998257 |
150 | 0.997386 | 191 | 0.997919 | 232 | 0.998263 |
151 | 0.997402 | 192 | 0.997929 | 233 | 0.998270 |
152 | 0.997419 | 193 | 0.997939 | 234 | 0.998277 |
153 | 0.997435 | 194 | 0.997949 | 235 | 0.998284 |
154 | 0.997450 | 195 | 0.997959 | 236 | 0.998291 |
155 | 0.997466 | 196 | 0.997969 | 237 | 0.998297 |
156 | 0.997481 | 197 | 0.997978 | 238 | 0.998304 |
157 | 0.997497 | 198 | 0.997988 | 239 | 0.998311 |
158 | 0.997512 | 199 | 0.997997 | 240 | 0.998317 |
159 | 0.997526 | 200 | 0.998007 | 241 | 0.998323 |
N | c4'(N) | N | c4'(N) | N | c4'(N) |
---|---|---|---|---|---|
242 | 0.998330 | 283 | 0.998553 | 324 | 0.998720 |
243 | 0.998336 | 284 | 0.998558 | 325 | 0.998723 |
244 | 0.998342 | 285 | 0.998562 | 326 | 0.998727 |
245 | 0.998349 | 286 | 0.998567 | 327 | 0.998730 |
246 | 0.998355 | 287 | 0.998571 | 328 | 0.998734 |
247 | 0.998361 | 288 | 0.998576 | 329 | 0.998737 |
248 | 0.998367 | 289 | 0.998580 | 330 | 0.998740 |
249 | 0.998373 | 290 | 0.998585 | 331 | 0.998744 |
250 | 0.998379 | 291 | 0.998589 | 332 | 0.998747 |
251 | 0.998385 | 292 | 0.998593 | 333 | 0.998751 |
252 | 0.998391 | 293 | 0.998598 | 334 | 0.998754 |
253 | 0.998397 | 294 | 0.998602 | 335 | 0.998757 |
254 | 0.998403 | 295 | 0.998606 | 336 | 0.998761 |
255 | 0.998408 | 296 | 0.998611 | 337 | 0.998764 |
256 | 0.998414 | 297 | 0.998615 | 338 | 0.998767 |
257 | 0.998420 | 298 | 0.998619 | 339 | 0.998770 |
258 | 0.998425 | 299 | 0.998623 | 340 | 0.998774 |
259 | 0.998431 | 300 | 0.998627 | 341 | 0.998777 |
260 | 0.998436 | 301 | 0.998632 | 342 | 0.998780 |
261 | 0.998442 | 302 | 0.998636 | 343 | 0.998783 |
262 | 0.998447 | 303 | 0.998640 | 344 | 0.998786 |
263 | 0.998453 | 304 | 0.998644 | 345 | 0.998790 |
264 | 0.998458 | 305 | 0.998648 | 346 | 0.998793 |
265 | 0.998463 | 306 | 0.998652 | 347 | 0.998796 |
266 | 0.998469 | 307 | 0.998656 | 348 | 0.998799 |
267 | 0.998474 | 308 | 0.998660 | 349 | 0.998802 |
268 | 0.998479 | 309 | 0.998664 | 350 | 0.998805 |
269 | 0.998484 | 310 | 0.998668 | 351 | 0.998808 |
270 | 0.998489 | 311 | 0.998671 | 352 | 0.998811 |
271 | 0.998495 | 312 | 0.998675 | 353 | 0.998814 |
272 | 0.998500 | 313 | 0.998679 | 354 | 0.998817 |
273 | 0.998505 | 314 | 0.998683 | 355 | 0.998820 |
274 | 0.998510 | 315 | 0.998687 | 356 | 0.998823 |
275 | 0.998515 | 316 | 0.998690 | 357 | 0.998826 |
276 | 0.998519 | 317 | 0.998694 | 358 | 0.998829 |
277 | 0.998524 | 318 | 0.998698 | 359 | 0.998832 |
278 | 0.998529 | 319 | 0.998701 | 360 | 0.998835 |
279 | 0.998534 | 320 | 0.998705 | 361 | 0.998837 |
280 | 0.998539 | 321 | 0.998709 | 362 | 0.998840 |
281 | 0.998544 | 322 | 0.998712 | 363 | 0.998843 |
282 | 0.998548 | 323 | 0.998716 | 364 | 0.998846 |
k | c4'(k) | k | c4'(k) | k | c4'(k) |
---|---|---|---|---|---|
365 | 0.998849 | 411 | 0.998963 | 457 | 0.999054 |
366 | 0.998851 | 412 | 0.998965 | 458 | 0.999056 |
367 | 0.998854 | 413 | 0.998967 | 459 | 0.999058 |
368 | 0.998857 | 414 | 0.998970 | 460 | 0.999060 |
369 | 0.998860 | 415 | 0.998972 | 461 | 0.999061 |
370 | 0.998862 | 416 | 0.998974 | 462 | 0.999063 |
371 | 0.998865 | 417 | 0.998976 | 463 | 0.999065 |
372 | 0.998868 | 418 | 0.998978 | 464 | 0.999067 |
373 | 0.998871 | 419 | 0.998980 | 465 | 0.999068 |
374 | 0.998873 | 420 | 0.998982 | 466 | 0.999070 |
375 | 0.998876 | 421 | 0.998985 | 467 | 0.999072 |
376 | 0.998879 | 422 | 0.998987 | 468 | 0.999073 |
377 | 0.998881 | 423 | 0.998989 | 469 | 0.999075 |
378 | 0.998884 | 424 | 0.998991 | 470 | 0.999077 |
379 | 0.998886 | 425 | 0.998993 | 471 | 0.999078 |
380 | 0.998889 | 426 | 0.998995 | 472 | 0.999080 |
381 | 0.998892 | 427 | 0.998997 | 473 | 0.999082 |
382 | 0.998894 | 428 | 0.998999 | 474 | 0.999084 |
383 | 0.998897 | 429 | 0.999001 | 475 | 0.999085 |
384 | 0.998899 | 430 | 0.999003 | 476 | 0.999087 |
385 | 0.998902 | 431 | 0.999005 | 477 | 0.999088 |
386 | 0.998904 | 432 | 0.999007 | 478 | 0.999090 |
387 | 0.998907 | 433 | 0.999009 | 479 | 0.999092 |
388 | 0.998909 | 434 | 0.999011 | 480 | 0.999093 |
389 | 0.998912 | 435 | 0.999013 | 481 | 0.999095 |
390 | 0.998914 | 436 | 0.999015 | 482 | 0.999097 |
391 | 0.998917 | 437 | 0.999017 | 483 | 0.999098 |
392 | 0.998919 | 438 | 0.999019 | 484 | 0.999100 |
393 | 0.998921 | 439 | 0.999021 | 485 | 0.999101 |
394 | 0.998924 | 440 | 0.999023 | 486 | 0.999103 |
395 | 0.998926 | 441 | 0.999025 | 487 | 0.999104 |
396 | 0.998929 | 442 | 0.999027 | 488 | 0.999106 |
397 | 0.998931 | 443 | 0.999028 | 489 | 0.999108 |
398 | 0.998933 | 444 | 0.999030 | 490 | 0.999109 |
399 | 0.998936 | 445 | 0.999032 | 491 | 0.999111 |
400 | 0.998938 | 446 | 0.999034 | 492 | 0.999112 |
401 | 0.998940 | 447 | 0.999036 | 493 | 0.999114 |
402 | 0.998943 | 448 | 0.999038 | 494 | 0.999115 |
403 | 0.998945 | 449 | 0.999040 | 495 | 0.999117 |
404 | 0.998947 | 450 | 0.999042 | 496 | 0.999118 |
405 | 0.998950 | 451 | 0.999043 | 497 | 0.999120 |
406 | 0.998952 | 452 | 0.999045 | 498 | 0.999121 |
407 | 0.998954 | 453 | 0.999047 | 499 | 0.999123 |
408 | 0.998956 | 454 | 0.999049 | 500 | 0.999124 |
409 | 0.998959 | 455 | 0.999051 | ||
410 | 0.998961 | 456 | 0.999052 |
Use the table below to find a value for γN, 1 -α used in calculating the confidence interval for Z.Bench and then use the second equation to get the exact value of γN, 1 -α.
1 -α | |||||
N | 0.800 | 0.850 | 0.900 | 0.950 | 0.990 |
5 | 3.544 | 4.138 | 4.961 | 6.350 | 9.750 |
6 | 3.485 | 4.078 | 4.903 | 6.300 | 9.636 |
7 | 3.443 | 4.035 | 4.861 | 6.260 | 9.567 |
8 | 3.413 | 4.003 | 4.829 | 6.229 | 9.520 |
9 | 3.390 | 3.979 | 4.804 | 6.204 | 9.484 |
10 | 3.372 | 3.960 | 4.783 | 6.183 | 9.457 |
12 | 3.345 | 3.931 | 4.753 | 6.152 | 9.416 |
14 | 3.326 | 3.911 | 4.732 | 6.130 | 9.387 |
16 | 3.312 | 3.986 | 4.716 | 6.113 | 9.365 |
18 | 3.301 | 3.884 | 4.703 | 6.099 | 9.348 |
20 | 3.293 | 3.875 | 4.693 | 6.089 | 9.335 |
25 | 3.278 | 3.858 | 4.675 | 6.069 | 9.310 |
30 | 3.268 | 3.848 | 4.664 | 6.056 | 9.294 |
35 | 3.261 | 3.840 | 4.655 | 6.047 | 9.282 |
40 | 3.255 | 3.834 | 4.649 | 6.040 | 9.274 |
50 | 3.248 | 3.826 | 4.640 | 6.031 | 9.262 |
60 | 3.243 | 3.821 | 4.634 | 6.024 | 9.253 |
80 | 3.237 | 3.814 | 4.627 | 6.016 | 9.244 |
100 | 3.233 | 3.810 | 4.623 | 6.011 | 9.238 |
>100 | 3.219 | 3.794 | 4.605 | 5.991 | 9.210 |
When N and 1 - a are not listed in the table, use the extrapolation method to obtain the value for γN, 1 -α. For example,