FORMULARIO CERTAMEN 1
GESTIÓN Y CONTROL DE CALIDAD
Límites de Control 3 sigmas para Grafico 𝑋̅.
IMPORTANTE:
𝐿𝑆𝐶𝑋̅ = 𝜇 + 3𝜎𝑋̅
𝐿𝐶𝑋̅ = 𝜇
𝐿𝐼𝐶𝑋̅ = 𝜇 − 3𝜎𝑋̅
Se puede estimar 𝜇 y 𝜎:
𝑚
1
𝜇̂ = 𝑥̿ = ∑ 𝑥𝑖
𝑚
𝑖=1
̅̅̅
𝑅
𝜎̂ =
,
𝑑2
𝜎̂𝑋̅ =
𝜎̂
√𝑛
=
𝑅̅
𝑑2 ∙ √𝑛
𝑚
,
1
𝑑𝑜𝑛𝑑𝑒 𝑅̅ = ∑ 𝑅𝑖
𝑚
𝑖=1
Límites de Control 3 sigmas para Gráfico 𝑅̅.
𝐿𝑆𝐶𝑅̅ = 𝑅̅ + 3𝜎𝑅̅
𝐿𝐶𝑅̅ = 𝑅̅
𝐿𝐼𝐶𝑅̅ = 𝑅̅ − 3𝜎𝑅̅
Se puede estimar 𝜎𝑅̅ .
𝜎̂ 𝑅̅ = 𝑑3
̅̅̅
𝑅
𝑑2
Límites de Tolerancia Natural (SIEMPRE SON 3 SIGMAS).
𝐿𝑆𝑇𝑁 = 𝜇 + 3𝜎
𝐿𝐶 = 𝜇
𝐿𝐼𝑇𝑁 = 𝜇 − 3𝜎
Capacidad del Proceso (Sujeto a ESPECIFICACIONES).
𝑇=
𝐿𝑆𝐸 + 𝐿𝐼𝐸
2
• 𝒅𝟐 , 𝒅𝟑 y 𝒄𝟒 son factores
que dependen
del
tamaño de la muestra
(𝑛)
• 𝒎: Número de
muestras
• 𝑹: Rango
• 𝑻: Especificación
nominal
• 𝟏 − 𝒑: Probabilidad de
cumplir las
especificaciones.
̂ 𝒑 : Capacidad potencial
• 𝑪
̂ 𝒑𝒌: Capacidad real
• 𝑪
• 𝑨𝑹𝑳: Número promedio
de muestras necesarias
antes de detectar un
cambio en la media.
𝑵(𝝁,𝝈)
1 − 𝑝 = 𝑃(𝐿𝐼𝐸 ≤ 𝑥 ≤ 𝐿𝑆𝐸) ⇒ 𝑃 (
𝐶̂𝑝 =
𝐿𝐼𝐸 − 𝜇
𝐿𝑆𝐸 − 𝜇
)
≤𝑍≤
𝜎
𝜎
𝐿𝑆𝐸−𝐿𝐼𝐸
̂
6𝜎
𝐿𝑆𝐸 − 𝜇 𝜇 − 𝐿𝐼𝐸
,
}
̂
̂
3𝜎
3𝜎
𝐶̂𝑝𝑘 = 𝑀𝑖𝑛 {
•
•
•
̂𝒑 > 𝟏. 𝟑𝟑 ⇒ 𝑃𝑟𝑜𝑐𝑒𝑠𝑜 𝐶𝐴𝑃𝐴𝑍
𝑪
̂𝒑 ≤ 𝟏. 𝟑𝟑 ⇒ 𝑃𝑟𝑜𝑐𝑒𝑠𝑜 𝑠𝑢𝑗𝑒𝑡𝑜 𝑎 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑐𝑖ó𝑛.
𝟏. 𝟎𝟎 ≤ 𝑪
̂𝒑 < 𝟏. 𝟎𝟎 ⇒ 𝑃𝑟𝑜𝑐𝑒𝑠𝑜 𝑁𝑂 𝐶𝐴𝑃𝐴𝑍
𝑪
̂𝒑 ≥ 𝟏. 𝟎𝟎 ⇒ 𝑃𝑟𝑜𝑐𝑒𝑠𝑜 𝐶𝐴𝑃𝐴𝑍
•𝑪
̂𝒑 < 𝟏. 𝟎𝟎 ⇒ 𝑃𝑟𝑜𝑐𝑒𝑠𝑜 𝑁𝑂 𝐶𝐴𝑃𝐴𝑍
•𝑪
Curvas OC
Error tipo I:
𝛼 = 1−𝛽
1 − 𝛽 = 𝑃 (𝑋̅ > 𝐿𝑆𝐶/ 𝜇 = 𝜇 + 𝑘𝜎) = 𝑃 (𝑋̅ < 𝐿𝐼𝐶 / 𝜇 = 𝜇 − 𝑘𝜎)
𝜶: Probabilidad de asumir
proceso fuera de control
cuando en realidad si lo está.
⇒ 𝛽 = 𝑃 (𝐿𝐼𝐶 ≤ 𝑋̅ ≤ 𝐿𝑆𝐶/ 𝜇 = 𝜇 ± 𝑘𝜎)
Error tipo II:
𝐴𝑅𝐿 =
1
1−𝛽
𝜷: Probabilidad de asumir
proceso en control cuando
en realidad no lo está.
Límites de Control 3 sigmas para Gráfico 𝑆̅ (Recomendado para 𝒏 > 𝟏𝟎, o 𝒏 variable)
𝐿𝑆𝐶𝑆̅ = 𝜇𝑠 + 3𝜎𝑆
𝐿𝐶𝑆̅ = 𝜇𝑠
𝐿𝐼𝐶𝑆̅ = 𝜇𝑠 − 3𝜎𝑆
Donde se puede calcular 𝜇𝑠 𝑦 𝜎𝑆 , ya que la varianza (𝜎) es conocida:
𝜇𝑠 = 𝜎 ∙ 𝑐4
𝜎𝑆 = 𝜎 ∙ √1 − 𝑐4 2
Si la varianza (𝜎) es desconocida, se puede estimar:
𝑆̅
𝜎̂ = ,
𝑐4
𝑚
1
𝑑𝑜𝑛𝑑𝑒 𝑆̅ =
∙ ∑ 𝑆𝑖
𝑚
𝑖=1
Entonces, los límites de control para 𝑋̅ y 𝑆̅ quedan de la siguiente manera:
𝐿𝑆𝐶𝑆̅ = 𝑆̅ (1 + 3
𝐿𝐶𝑆̅ = 𝑆̅
1
√1 − 𝑐4 2 )
𝑐4
1
𝐿𝐼𝐶𝑆̅ = 𝑆̅ (1 − 3 √1 − 𝑐4 2 )
𝑐4
𝐿𝑆𝐶𝑋̅ = 𝑥̿ + 3
𝑆̅
𝑐4 √𝑛
𝐿𝐶𝑋̅ = 𝑥̿
𝐿𝐼𝐶𝑋̅ = 𝑥̿ − 3
𝑆̅
𝑐4 √𝑛
Para tamaño muestral (𝑛) variable:
𝑚
𝑚
𝑥̿ = ∑ 𝑛𝑖 𝑥̅𝑖 ⁄∑ 𝑛𝑖
𝑖=1
𝑖=1
*Se calculan límites de control para cada tamaño de muestra
𝑚
𝑛
𝑚
𝑆̅ = √∑(𝑛𝑖 − 1)𝑆𝑖 2 ⁄∑(𝑛𝑖 − 𝑚)
𝑖=1
𝑖=1
2
𝑆𝑖 = ∑(𝑥𝑖𝑗 − 𝑋̅𝑖 )2⁄(𝑛𝑖 − 1)
𝑗=1
Alternativas de cálculo de desviación estándar (𝑆):
•
•
•
•
Calcular los límites usando el tamaño muestral promedio. Esto es adecuado si los
tamaños muestrales no difieren mucho.
Como este tamaño promedio puede ser un número no entero, puede usarse el
tamaño muestral modal.
Estimación de 𝜎: Usar el tamaño muestral modal
Gráficos de Control estandarizados de la siguiente manera:
𝐿𝑆𝐶𝑍 = 3
𝐿𝐶𝑍 = 0
𝐿𝐼𝐶𝑍 = −3
𝑧𝑖 =
𝑥̅𝑖 − 𝑥̿
𝑆̅
𝑐4√𝑛𝑖
Límites de Control 3 sigmas para Gráficos 𝑆 2 .
𝑆̅ 2
𝐿𝑆𝐶𝑆 2 = (
) 𝜒 2 𝛼/2,𝑛−1
𝑛−1
𝐿𝑆𝐶𝑆 2 = 𝑆̅ 2
𝑆̅ 2
𝐿𝑆𝐶𝑆 2 = (
) 𝜒 21−(𝛼/2),𝑛−1
𝑛−1
*Luego de esto se verifica si 𝑧𝑖 se encuentra
en el intervalo de control [−3,3].
702
F o r n > 25.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2.121
1.732
1.500
1.342
1.225
1.134
1.061
1.000
0.949
0.905
0.866
0.832
0.802
0.775
0.750
0.728
0.707
0.688
0.671
0.655
0.640
0.626
0.612
0.600
1.880
1.023
0.729
0.577
0.483
0.419
0.373
0.337
0.308
0.285
0.266
0.249
0.235
0.223
0.212
0.203
0.194
0.187
0.180
0.173
0.167
0.162
0.157
0.153
2.659
1.954
1.628
1.427
1.287
1.182
1.099
1.032
0.975
0.927
0.886
0.850
0.817
0.789
0.763
0.739
0.718
0.698
0.680
0.663
0.647
0.633
0.619
0.606
Factors for
O bserv ations
Control
L im its
in
S am p le, n
A
A2
A3
0.7979
0.8862
0.9213
0.9400
0.9515
0.9594
0.9650
0.9693
0.9727
0.9754
0.9776
0.9794
0.9810
0.9823
0.9835
0.9845
0.9854
0.9862
0.9869
0.9876
0.9882
0.9887
0.9892
0.9896
1.2533
1.1284
1.0854
1.0638
1.0510
1.0423
1.0363
1.0317
1.0281
1.0252
1.0229
1.0210
1.0194
1.0180
1.0168
1.0157
1.0148
1.0140
1.0133
1.0126
1.0119
1.0114
1.0109
1.0105
3
A3 =
3
2( n − 1)
B5
B6 = c4 +
B4 = 1 +
B6
2( n − 1)
3
c4 2( n − 1)
3
4( n − 1)
4n − 3
2.606
2.276
2.088
1.964
1.874
1.806
1.751
1.707
1.669
1.637
1.610
1.585
1.563
1.544
1.526
1.511
1.496
1.483
1.470
1.459
1.448
1.438
1.429
1.420
c4 ≅
0
0
0
0
0.029
0.113
0.179
0.232
0.276
0.313
0.346
0.374
0.399
0.421
0.440
0.458
0.475
0.490
0.504
0.516
0.528
0.539
0.549
0.559
3
c4 n
3.267
2.568
2.266
2.089
1.970
1.882
1.815
1.761
1.716
1.679
1.646
1.618
1.594
1.572
1.552
1.534
1.518
1.503
1.490
1.477
1.466
1.455
1.445
1.435
c4 2( n − 1)
3
n
B5 = c4 −
B3 = 1 −
A=
0
0
0
0
0.030
0.118
0.185
0.239
0.284
0.321
0.354
0.382
0.406
0.428
0.448
0.466
0.482
0.497
0.510
0.523
0.534
0.545
0.555
0.565
B4
B3
c4
1/c4
Factors for Control L im its
Chart for S tand ard D ev iations
Factors for
Center L ine
Chart for A v erages
APPENDIX VI
Factors for Constructing Variables Control Charts
1.128
1.693
2.059
2.326
2.534
2.704
2.847
2.970
3.078
3.173
3.258
3.336
3.407
3.472
3.532
3.588
3.640
3.689
3.735
3.778
3.819
3.858
3.895
3.931
d2
0.8865
0.5907
0.4857
0.4299
0.3946
0.3698
0.3512
0.3367
0.3249
0.3152
0.3069
0.2998
0.2935
0.2880
0.2831
0.2787
0.2747
0.2711
0.2677
0.2647
0.2618
0.2592
0.2567
0.2544
1/d2
Factors for
Center L ine
0.853
0.888
0.880
0.864
0.848
0.833
0.820
0.808
0.797
0.787
0.778
0.770
0.763
0.756
0.750
0.744
0.739
0.734
0.729
0.724
0.720
0.716
0.712
0.708
d3
0
0
0
0
0
0.204
0.388
0.547
0.687
0.811
0.922
1.025
1.118
1.203
1.282
1.356
1.424
1.487
1.549
1.605
1.659
1.710
1.759
1.806
D1
3.686
4.358
4.698
4.918
5.078
5.204
5.306
5.393
5.469
5.535
5.594
5.647
5.696
5.741
5.782
5.820
5.856
5.891
5.921
5.951
5.979
6.006
6.031
6.056
D2
0
0
0
0
0
0.076
0.136
0.184
0.223
0.256
0.283
0.307
0.328
0.347
0.363
0.378
0.391
0.403
0.415
0.425
0.434
0.443
0.451
0.459
D3
D4
3.267
2.574
2.282
2.114
2.004
1.924
1.864
1.816
1.777
1.744
1.717
1.693
1.672
1.653
1.637
1.622
1.608
1.597
1.585
1.575
1.566
1.557
1.548
1.541
Factors for Control L im its
Chart for R anges
APPENDIX A
1z2 P1Z z2 z
1
22
e
1 u2
2 d
653
u
Φ (z)
z
T a b le I I
z
3 .9
3 .8
3 .7
3 .6
3 .5
3 .4
3 .3
3 .2
3 .1
3 .0
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
C u m u la tiv e S ta n d a rd No rm a l Distrib u tio n
0.09
0.00003 3
0.00005 0
0.00007 5
0.000112
0.00016 5
0.00024 2
0.0003 5 0
0.0005 01
0.0007 11
0.001001
0.0013 95
0.001926
0.0026 3 5
0.003 5 7 3
0.004 7 99
0.006 3 8 7
0.008 4 24
0.011011
0.014 26 2
0.018 3 09
0.023 295
0.0293 7 9
0.03 6 7 27
0.04 5 5 14
0.05 5 917
0.06 8 112
0.08 226 4
0.098 5 25
0.117 023
0.13 7 8 5 7
0.16 108 7
0.18 6 7 3 3
0.214 7 6 4
0.24 5 097
0.27 7 5 95
0.3 1206 7
0.3 4 8 26 8
0.3 8 5 908
0.4 24 6 5 5
0.4 6 4 14 4
0.08
0.00003 4
0.00005 2
0.00007 8
0.000117
0.00017 2
0.00025 1
0.0003 6 2
0.0005 19
0.0007 3 6
0.00103 5
0.0014 4 1
0.00198 8
0.0027 18
0.003 6 8 1
0.004 94 0
0.006 5 6 9
0.008 6 5 6
0.0113 04
0.014 6 29
0.018 7 6 3
0.023 8 5 2
0.03 005 4
0.03 7 5 3 8
0.04 6 4 7 9
0.05 7 05 3
0.06 94 3 7
0.08 3 7 93
0.10027 3
0.119000
0.14 007 1
0.16 3 5 4 3
0.18 94 3 0
0.217 6 95
0.24 8 25 2
0.28 095 7
0.3 15 6 14
0.3 5 197 3
0.3 8 97 3 9
0.4 28 5 7 6
0.4 6 8 119
0.07
0.06
0.05
0.04
0.00003 6
0.00005 4
0.00008 2
0.000121
0.00017 9
0.00026 0
0.0003 7 6
0.0005 3 8
0.0007 6 2
0.00107 0
0.0014 8 9
0.00205 2
0.0028 03
0.003 7 93
0.005 08 5
0.006 7 5 6
0.008 8 94
0.0116 04
0.015 003
0.019226
0.024 4 19
0.03 07 4 2
0.03 8 3 6 4
0.04 7 4 6 0
0.05 8 208
0.07 07 8 1
0.08 5 3 4 3
0.10204 2
0.121001
0.14 23 10
0.16 6 023
0.19215 0
0.2206 5 0
0.25 14 29
0.28 4 3 3 9
0.3 1917 8
0.3 5 5 6 91
0.3 93 5 8 0
0.4 3 25 05
0.4 7 2097
0.00003 7
0.00005 7
0.00008 5
0.000126
0.00018 5
0.00027 0
0.0003 90
0.0005 5 7
0.0007 8 9
0.001107
0.0015 3 8
0.002118
0.0028 90
0.003 907
0.005 23 4
0.006 94 7
0.00913 7
0.011911
0.015 3 8 6
0.0196 99
0.024 998
0.03 14 4 3
0.03 9204
0.04 8 4 5 7
0.05 93 8 0
0.07 214 5
0.08 6 915
0.103 8 3 5
0.123 024
0.14 4 5 7 2
0.16 8 5 28
0.194 8 94
0.223 6 27
0.25 4 6 27
0.28 7 7 4 0
0.3 227 5 8
0.3 5 94 24
0.3 97 4 3 2
0.4 3 6 4 4 1
0.4 7 6 07 8
0.00003 9
0.00005 9
0.00008 8
0.00013 1
0.000193
0.00028 0
0.0004 04
0.0005 7 7
0.0008 16
0.00114 4
0.0015 8 9
0.00218 6
0.00298 0
0.004 025
0.005 3 8 6
0.007 14 3
0.0093 8 7
0.012224
0.015 7 7 8
0.02018 2
0.025 5 8 8
0.03 215 7
0.04 005 9
0.04 94 7 1
0.06 05 7 1
0.07 3 5 29
0.08 8 5 08
0.105 6 5 0
0.125 07 2
0.14 6 8 5 9
0.17 105 6
0.197 6 6 2
0.226 6 27
0.25 7 8 4 6
0.29116 0
0.3 26 3 5 5
0.3 6 3 16 9
0.4 01294
0.4 4 03 8 2
0.4 8 006 1
0.00004 1
0.00006 2
0.000092
0.00013 6
0.000200
0.000291
0.0004 19
0.0005 98
0.0008 4 5
0.00118 3
0.0016 4 1
0.00225 6
0.003 07 2
0.004 14 5
0.005 5 4 3
0.007 3 4 4
0.0096 4 2
0.0125 4 5
0.016 17 7
0.0206 7 5
0.026 190
0.03 28 8 4
0.04 0929
0.05 05 03
0.06 17 8 0
0.07 4 93 4
0.090123
0.107 4 8 8
0.127 14 3
0.14 917 0
0.17 3 6 09
0.2004 5 4
0.2296 5 0
0.26 108 6
0.294 5 99
0.3 2996 9
0.3 6 6 928
0.4 05 16 5
0.4 4 4 3 3 0
0.4 8 4 04 7
0.03
0.00004 2
0.00006 4
0.000096
0.00014 2
0.000208
0.0003 02
0.0004 3 4
0.0006 19
0.0008 7 4
0.001223
0.0016 95
0.0023 27
0.003 16 7
0.004 26 9
0.005 7 03
0.007 5 4 9
0.009903
0.0128 7 4
0.016 5 8 6
0.02117 8
0.026 8 03
0.03 3 6 25
0.04 18 15
0.05 15 5 1
0.06 3 008
0.07 6 3 5 9
0.0917 5 9
0.1093 4 9
0.12923 8
0.15 15 05
0.17 6 18 5
0.203 26 9
0.23 26 95
0.26 4 3 4 7
0.298 05 6
0.3 3 3 5 98
0.3 7 07 00
0.4 0904 6
0.4 4 8 28 3
0.4 8 8 03 3
0.02
0.00004 4
0.00006 7
0.000100
0.00014 7
0.000216
0.0003 13
0.0004 5 0
0.0006 4 1
0.000904
0.00126 4
0.0017 5 0
0.0024 01
0.003 26 4
0.004 3 96
0.005 8 6 8
0.007 7 6 0
0.01017 0
0.013 209
0.017 003
0.0216 92
0.027 4 29
0.03 4 3 7 9
0.04 27 16
0.05 26 16
0.06 4 25 6
0.07 7 8 04
0.093 4 18
0.11123 3
0.13 13 5 7
0.15 3 8 6 4
0.17 8 7 8 6
0.206 108
0.23 5 7 6 2
0.26 7 6 29
0.3 015 3 2
0.3 3 7 24 3
0.3 7 4 4 8 4
0.4 1293 6
0.4 5 224 2
0.4 92022
0.01
0.00004 6
0.00006 9
0.000104
0.00015 3
0.000224
0.0003 25
0.0004 6 7
0.0006 6 4
0.00093 5
0.0013 06
0.0018 07
0.0024 7 7
0.003 3 6 4
0.004 5 27
0.006 03 7
0.007 97 6
0.0104 4 4
0.013 5 5 3
0.017 4 29
0.022216
0.028 06 7
0.03 5 14 8
0.04 3 6 3 3
0.05 3 6 99
0.06 5 5 22
0.07 927 0
0.095 098
0.113 14 0
0.13 3 5 00
0.15 6 24 8
0.18 14 11
0.208 97 0
0.23 8 8 5 2
0.27 093 1
0.3 05 026
0.3 4 0903
0.3 7 8 28 1
0.4 16 8 3 4
0.4 5 6 205
0.4 96 011
0.00
0.00004 8
0.00007 2
0.000108
0.00015 9
0.00023 3
0.0003 3 7
0.0004 8 3
0.0006 8 7
0.00096 8
0.0013 5 0
0.0018 6 6
0.0025 5 5
0.003 4 6 7
0.004 6 6 1
0.006 210
0.008 198
0.0107 24
0.013 903
0.017 8 6 4
0.0227 5 0
0.028 7 17
0.03 5 93 0
0.04 4 5 6 5
0.05 4 7 99
0.06 6 8 07
0.08 07 5 7
0.096 8 01
0.115 07 0
0.13 5 6 6 6
0.15 8 6 5 5
0.18 4 06 0
0.2118 5 5
0.24 196 4
0.27 4 25 3
0.3 08 5 3 8
0.3 4 4 5 7 8
0.3 8 208 9
0.4 207 4 0
0.4 6 017 2
0.5 00000
654
APPENDIX A STATISTICAL TABLES AND CHARTS
1z2 P1Z z2 z
1
22
e
1 u2
2
du
Φ (z)
z
0
Table II
Cumulative Standard Normal Distribution ( continued)
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.500000
0.539828
0.579260
0.617911
0.655422
0.691462
0.725747
0.758036
0.788145
0.815940
0.841345
0.864334
0.884930
0.903199
0.919243
0.933193
0.945201
0.955435
0.964070
0.971283
0.977250
0.982136
0.986097
0.989276
0.991802
0.993790
0.995339
0.996533
0.997445
0.998134
0.998650
0.999032
0.999313
0.999517
0.999663
0.999767
0.999841
0.999892
0.999928
0.999952
0.503989
0.543795
0.583166
0.621719
0.659097
0.694974
0.729069
0.761148
0.791030
0.818589
0.843752
0.866500
0.886860
0.904902
0.920730
0.934478
0.946301
0.956367
0.964852
0.971933
0.977784
0.982571
0.986447
0.989556
0.992024
0.993963
0.995473
0.996636
0.997523
0.998193
0.998694
0.999065
0.999336
0.999533
0.999675
0.999776
0.999847
0.999896
0.999931
0.999954
0.507978
0.547758
0.587064
0.625516
0.662757
0.698468
0.732371
0.764238
0.793892
0.821214
0.846136
0.868643
0.888767
0.906582
0.922196
0.935744
0.947384
0.957284
0.965621
0.972571
0.978308
0.982997
0.986791
0.989830
0.992240
0.994132
0.995604
0.996736
0.997599
0.998250
0.998736
0.999096
0.999359
0.999550
0.999687
0.999784
0.999853
0.999900
0.999933
0.999956
0.511967
0.551717
0.590954
0.629300
0.666402
0.701944
0.735653
0.767305
0.796731
0.823815
0.848495
0.870762
0.890651
0.908241
0.923641
0.936992
0.948449
0.958185
0.966375
0.973197
0.978822
0.983414
0.987126
0.990097
0.992451
0.994297
0.995731
0.996833
0.997673
0.998305
0.998777
0.999126
0.999381
0.999566
0.999698
0.999792
0.999858
0.999904
0.999936
0.999958
0.515953
0.555760
0.594835
0.633072
0.670031
0.705401
0.738914
0.770350
0.799546
0.826391
0.850830
0.872857
0.892512
0.909877
0.925066
0.938220
0.949497
0.959071
0.967116
0.973810
0.979325
0.983823
0.987455
0.990358
0.992656
0.994457
0.995855
0.996928
0.997744
0.998359
0.998817
0.999155
0.999402
0.999581
0.999709
0.999800
0.999864
0.999908
0.999938
0.999959
0.519939
0.559618
0.598706
0.636831
0.673645
0.708840
0.742154
0.773373
0.802338
0.828944
0.853141
0.874928
0.894350
0.911492
0.926471
0.939429
0.950529
0.959941
0.967843
0.974412
0.979818
0.984222
0.987776
0.990613
0.992857
0.994614
0.995975
0.997020
0.997814
0.998411
0.998856
0.999184
0.999423
0.999596
0.999720
0.999807
0.999869
0.999912
0.999941
0.999961
0.532922
0.563559
0.602568
0.640576
0.677242
0.712260
0.745373
0.776373
0.805106
0.831472
0.855428
0.876976
0.896165
0.913085
0.927855
0.940620
0.951543
0.960796
0.968557
0.975002
0.980301
0.984614
0.988089
0.990863
0.993053
0.994766
0.996093
0.997110
0.997882
0.998462
0.998893
0.999211
0.999443
0.999610
0.999730
0.999815
0.999874
0.999915
0.999943
0.999963
0.527903
0.567495
0.606420
0.644309
0.680822
0.715661
0.748571
0.779350
0.807850
0.833977
0.857690
0.878999
0.897958
0.914657
0.929219
0.941792
0.952540
0.961636
0.969258
0.975581
0.980774
0.984997
0.988396
0.991106
0.993244
0.994915
0.996207
0.997197
0.997948
0.998511
0.998930
0.999238
0.999462
0.999624
0.999740
0.999821
0.999879
0.999918
0.999946
0.999964
0.531881
0.571424
0.610261
0.648027
0.684386
0.719043
0.751748
0.782305
0.810570
0.836457
0.859929
0.881000
0.899727
0.916207
0.930563
0.942947
0.953521
0.962462
0.969946
0.976148
0.981237
0.985371
0.988696
0.991344
0.993431
0.995060
0.996319
0.997282
0.998012
0.998559
0.998965
0.999264
0.999481
0.999638
0.999749
0.999828
0.999883
0.999922
0.999948
0.999966
0.535856
0.575345
0.614092
0.651732
0.687933
0.722405
0.754903
0.785236
0.813267
0.838913
0.862143
0.882977
0.901475
0.917736
0.931888
0.944083
0.954486
0.963273
0.970621
0.976705
0.981691
0.985738
0.988989
0.991576
0.993613
0.995201
0.996427
0.997365
0.998074
0.998605
0.998999
0.999289
0.999499
0.999650
0.999758
0.999835
0.999888
0.999925
0.999950
0.999967
Appendix
APPENDIX III
P er c entag e P o ints o f th e χ2 D is tr ib u tio na
α
χ 2α,ν
a
v
0 .9 9 5
0 .9 9 0
0 .9 7 5
0 .9 5 0
0 .5 0 0
0 .0 5 0
0 .0 25
0 .0 1 0
0 .0 0 5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
25
30
40
50
60
70
80
90
100
0.00 +
0.01
0.07
0.2 1
0.4 1
0.6 8
0.9 9
1.3 4
1.7 3
2 .16
2 .6 0
3 .07
3 .5 7
4 .07
4 .6 0
5 .14
5 .7 0
6 .2 6
6 .8 8 4
7 .4 3
10.5 2
13 .7 9
2 0.7 1
2 7 .9 9
3 5 .5 3
4 3 .2 8
5 1.17
5 9 .2 0
6 7 .3 3
0.00 +
0.02
0.11
0.3 0
0.5 5
0.8 7
1.2 4
1.6 5
2 .09
2 .5 6
3 .05
3 .5 7
4 .11
4 .6 6
5 .2 3
5 .8 1
6 .4 1
7 .01
7 .6 3
8 .2 6
11.5 2
14 .9 5
2 2 .16
2 9 .7 1
3 7 .4 8
4 5 .4 4
5 3 .5 4
6 1.7 5
7 0.06
0.00 +
0.05
0.2 2
0.4 8
0.8 3
1.2 4
1.6 9
2 .18
2 .7 0
3 .2 5
3 .8 2
4 .4 0
5 .01
5 .6 3
6 .2 7
6 .9 1
7 .5 6
8 .2 3
8 .9 1
9 .5 9
13 .12
16 .7 9
2 4 .4 3
3 2 .3 6
4 0.4 8
4 8 .7 6
5 7 .15
6 5 .6 5
7 4 .2 2
0.00 +
0.10
0.3 5
0.7 1
1.15
1.6 4
2 .17
2 .7 3
3 .3 3
3 .9 4
4 .5 7
5 .2 3
5 .8 9
6 .5 7
7 .2 6
7 .9 6
8 .6 7
9 .3 9
10.12
10.8 5
14 .6 1
18 .4 9
2 6 .5 1
3 4 .7 6
4 3 .19
5 1.7 4
6 0.3 9
6 9 .13
7 7 .9 3
0.4 5
1.3 9
2 .3 7
3 .3 6
4 .3 5
5 .3 5
6 .3 5
7 .3 4
8 .3 4
9 .3 4
10.3 4
11.3 4
12 .3 4
13 .3 4
14 .3 4
15 .3 4
16 .3 4
17 .3 4
18 .3 4
19 .3 4
2 4 .3 4
2 9 .3 4
3 9 .3 4
4 9 .3 3
5 9 .3 3
6 9 .3 3
7 9 .3 3
8 9 .3 3
9 9 .3 3
3 .8 4
5 .9 9
7 .8 1
9 .4 9
11.07
12 .5 9
14 .07
15 .5 1
16 .9 2
18 .3 1
19 .6 8
2 1.03
2 2 .3 6
2 3 .6 8
2 5 .00
2 6 .3 0
2 7 .5 9
2 8 .8 7
3 0.14
3 1.4 1
3 7 .6 5
4 3 .7 7
5 5 .7 6
6 7 .5 0
7 9 .08
9 0.5 3
101.8 8
113 .14
12 4 .3 4
5 .02
7 .3 8
9 .3 5
11.14
12 .3 8
14 .4 5
16 .01
17 .5 3
19 .02
2 0.4 8
2 1.9 2
2 3 .3 4
2 4 .7 4
2 6 .12
2 7 .4 9
2 8 .8 5
3 0.19
3 1.5 3
3 2 .8 5
3 4 .17
4 0.6 5
4 6 .9 8
5 9 .3 4
7 1.4 2
8 3 .3 0
9 5 .02
106 .6 3
118 .14
12 9 .5 6
6 .6 3
9 .2 1
11.3 4
13 .2 8
15 .09
16 .8 1
18 .4 8
2 0.09
2 1.6 7
2 3 .2 1
2 4 .7 2
2 6 .2 2
2 7 .6 9
2 9 .14
3 0.5 8
3 2 .00
3 3 .4 1
3 4 .8 1
3 6 .19
3 7 .5 7
4 4 .3 1
5 0.8 9
6 3 .6 9
7 6 .15
8 8 .3 8
100.4 2
112 .3 3
12 4 .12
13 5 .8 1
7 .8 8
10.6 0
12 .8 4
14 .8 6
16 .7 5
18 .5 5
2 0.2 8
2 1.9 6
2 3 .5 9
2 5 .19
2 6 .7 6
2 8 .3 0
2 9 .8 2
3 1.3 2
3 2 .8 0
3 4 .2 7
3 5 .7 2
3 7 .16
3 8 .5 8
4 0.00
4 6 .9 3
5 3 .6 7
6 6 .7 7
7 9 .4 9
9 1.9 5
104 .2 2
116 .3 2
12 8 .3 0
14 0.17
v = d e g re e s o f fre e d o m .
a A d a p te d w ith p e rm is s io n fro m B io m e tr ik a T ab le s fo r S tatis tic ian s , V o l. 1, 3 rd e d ., b y E . S . P e a rs o n a n d H . O . H a rtle y ,
C a m b rid g e U n iv e rs ity P re s s , C a m b rid g e , 19 6 6 .
695