Each data point, MR i , is the moving range of the x values in each group. MR i is not plotted for i < w because it is undefined.
Term | Description |
---|---|
MR | moving range |
w | Number of observations in the moving range. By default, w = 2. |
The center line is the unbiased estimate of the average of the moving range.
center line = MR * d2(w)
The LCL is the greater of the following:
or
Term | Description |
---|---|
d2() | a constant used to estimate the standard deviation |
w | number of observations in the moving range. By default, w = 2. |
σ | process standard deviation |
k | parameter for Test 1 (default is 3) |
d3() | A constant used to estimate LCL and UCL. |
The average moving range, , of length w is given by the following formula:
where MRi is the moving range for observation i, calculated as follows:
Minitab uses to calculate Smr, which is an unbiased estimate of σ:
Term | Description |
---|---|
n | number of observations |
w | length of the moving range. The default is 2. |
d2() | value of unbiasing constant d2 that corresponds to the value specified in parentheses. |
The median moving range, , of length w is given by the following formula:
where w is the number of observations used in the moving range and MRi is the moving range for observation i, calculated as follows:
Minitab uses to calculate Smr, which is an unbiased estimate of σ:
Term | Description |
---|---|
n | number of observations |
w | length of the moving range. The default is 2. |
d4() | value of unbiasing constant d4 that corresponds to the value specified in parentheses. |
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 |
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 |
If you use a Box-Cox transformation, Minitab transforms the original data values (Yi) according to the following formula:
where λ is the parameter for the transformation. Minitab then creates a control chart of the transformed data values (Wi). To learn how Minitab chooses the optimal value for λ, go to Methods and formulas for Box-Cox Transformation.
λ | Transformation |
---|---|
2 | |
0.5 | |
0 | |
−0.5 | |
−1 |