Normal capability analysis estimates within-subgroup standard deviation and
overall standard deviation.
Within-subgroup standard deviation
The method used to estimate σwithin depends on the subgroup
size.
When subgroup size > 1, Minitab estimates
σwithin using one of the following methods:
Pooled standard deviation:
where:
Note
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
Average of subgroup ranges
(Rbar):
where:
If n are all the same:
Term
Description
ri
Range of the
ith subgroup
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
Average of subgroup standard
deviations (Sbar):
where:
Note
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
When subgroup size = 1, Minitab estimates
σwithin using one of the following methods:
Average of moving range:
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())
Median of moving range:
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())
Square root of mean squared
successive differences (MSSD):
Note
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
Overall standard deviation
where:
Note
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, you
can change this option on the
Estimate
subdialog box when you perform the capability analysis. If you always want
Minitab to use the unbiasing constant by default, choose
File > Options > Control Charts and Quality Tools > Estimating Standard Deviation
and select the appropriate options.
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
Box-Cox transformation
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
Algorithm for Johnson transformation
The Johnson transformation optimally selects one of three families of distribution to transform the data to follow a normal distribution.
Johnson family
Transformation function
Range
SB
γ + η ln [(x – ε) / (λ + ε – x)]
η, λ > 0, –∞ < γ < ∞ , –∞ < ε < ∞, ε < x < ε + λ
SL
γ + η ln (x – ε)
η > 0, –∞ < γ < ∞, –∞ < ε < ∞, ε < x
SU
γ + η Sinh–1 [(x – ε) / λ] , where
Sinh–1(x) = ln [x + sqrt (1 + x2)]
η, λ > 0, –∞ < γ < ∞, –∞ < ε < ∞, –∞ < x < ∞
The algorithm uses the following procedure:
Considers almost all potential transformation functions from the Johnson system.
Estimates the parameters in the function using the method described in Chou, et al.1
Transforms the data using the transformation function.
Calculates Anderson-Darling statistics and the corresponding p-value for the transformed data.
Selects the transformation function that has the largest p-value that is greater than the p-value criterion (default is 0.10) that you specify in the Transform dialog box. Otherwise, no transformation is appropriate.
Notation
Term
Description
SB
The Johnson family distribution with the variable bounded (B)
SL
The Johnson family distribution with the variable lognormal (L)
SU
The Johnson family distribution with the variable unbounded (U)
For more information on the Johnson transformation, see Chou, et al.1 Minitab replaces the Shapiro-Wilks normality test used in that text with the Anderson-Darling test.
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.)
For values of N from 51 to 100, use the following approximation for d2(N):
For values of N from 26 to 100, use the following approximations for d3(N) and d4(N):
For more information on these constants, see the following:
D. J. Wheeler and D. S. Chambers. (1992). Understanding Statistical Process Control, Second Edition, SPC Press, Inc.
H. Leon Harter (1960). "Tables of Range and Studentized Range". The Annals of Mathematical Statistics, Vol. 31, No. 4, Institute of Mathematical Statistics, 1122−1147.
Table 1. Table of values
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
Unbiasing constants c4() and c5()
c4()
c5()
Notation
Term
Description
Γ()
gamma function
Unbiasing constant c4'()
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
1 Y. Chou, A.M. Polansky, and R.L. Mason (1998). "Transforming Nonnormal Data to Normality in Statistical Process Control", Journal of Quality Technology, 30, April, 133–141.
