apéndice ii Áreas bajo la curva normal estándar,...
TRANSCRIPT
Apéndice II Áreas bajo
la curva normal
estándar,
desde 0 a z
z 0 1 2 3 4 5 6 7 8 9
0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0754
0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
0.6 .2258 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549
0.7 .2580 .2612 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
0.8 .2881 .2910 .2939 .2967 .2996 .3023 .3051 .3078 .3106 .3133
0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389
1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621
1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830
1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015
1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177
1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319
1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441
1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545
1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706
1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767
2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817
2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857
2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890
2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916
2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936
2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952
2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964
2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974
2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981
2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986
3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990
3.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .4993
3.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .4995
3.3 .4995 .4995 .4995 .4996 .4996 .4996 .4996 .4996 .4996 .4997
3.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4998
3.5 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998
3.6 .4998 .4998 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999
3.7 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999
3.8 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999
3.9 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000
0 z
Áreas bajo
la curva normal
estándar,
desde -∞ a z
z 0 1 2 3 4 5 6 7 8 9
0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5754
0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
0.6 .7258 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7518 .7549
0.7 .7580 .7612 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
0.8 .7881 .7910 .7939 .7967 .7996 .8023 .8051 .8078 .8106 .8133
0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.6 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993
3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995
3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997
3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998
3.5 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998
3.6 .9998 .9998 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999
3.7 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999
3.8 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
z
Apéndice III Valores percentiles (tp) para la distribución t
de Student, con v
grados de libertad
(área sombreada = p)
v t.995 t.99 t.975 t.95 t.90 t.80 t.75 t.70 t.60 t.55
1 63.66 31.82 12.71 6.31 3.08 1.376 1.000 .727 .325 .158
2 9.92 6.96 4.30 2.92 1.89 1.061 .816 .617 .289 .142
3 5.84 4.54 3.68 2.35 1.64 .978 .765 .584 .277 .137
4 4.60 3.75 2.78 2.63 1.53 .941 .741 .569 .271 .134
5 4.03 3.36 2.57 2.02 1.48 .920 .727 .559 .267 .132
6 3.71 3.64 2.45 1.94 1.44 .906 .718 .553 .265 .131
7 350 3.00 2.36 1.90 1.42 .896 .711 .549 .263 .130
8 3.36 2.90 2.31 1.86 1.40 .889 .706 .546 .262 .130
9 3.25 2.82 2.26 1.83 1.38 .883 .703 .543 .261 .129
10 3.67 2.76 2.23 1.81 1.37 .879 .700 .542 .260 .129
11 3.61 2.72 2.20 1.80 1.36 .876 .697 .540 .260 .129
12 3.06 2.68 2.18 1.78 1.36 .873 .695 .539 .259 .128
13 3.01 2.65 2.16 1.77 1.35 .870 .694 .538 .259 .128
14 2.98 2.62 2.14 1.76 1.34 .868 .692 .537 .258 .128
15 2.95 2.60 2.13 1.75 1.34 .866 .691 .536 .258 .128
16 2.92 2.58 2.12 1.75 1.34 .865 .690 .535 .258 .128
17 2.90 2.57 2.11 1.74 1.33 .863 .689 .534 .257 .128
18 2.88 2.55 2.10 1.73 1.33 .862 .688 .534 .257 .127
19 2.86 2.54 2.09 1.73 1.33 .861 .688 .533 .257 .127
20 2.84 2.53 2.09 1.72 1.32 .880 .687 .533 .257 .127
21 2.83 2.52 2.08 1.72 1.32 .859 .686 .532 .257 .127
22 2.82 2.51 2.07 1.72 1.32 .858 .686 .532 .256 .127
23 2.81 2.50 2.07 1.71 1.32 .858 .685 .532 .256 .127
24 2.80 2.49 2.06 1.71 1.32 .857 .685 .531 .256 .127
25 2.79 2.48 2.06 1.71 1.32 .856 .684 .531 .256 .127
26 2.78 2.48 2.06 1.71 1.32 .856 .684 .531 .256 .127
27 2.77 2.47 2.05 1.70 1.31 .855 .684 .531 .256 .127
28 2.76 2.47 2.05 1.70 1.31 .855 .683 .530 .256 .127
29 2.76 2.46 2.04 1.70 1.31 .854 .683 .530 .256 .127
30 2.75 2.46 2.04 1.70 1.31 .854 .683 .530 .256 .127
40 2.70 2.42 2.02 1.68 1.30 .851 .681 .529 .255 .126
60 2.66 2.39 2.00 1.67 1.30 .848 .679 .527 .254 .126
120 2.62 2.36 1.98 1.66 1.29 .845 .677 .526 .254 .126
∞ 2.58 2.33 1.96 1.645 1.28 .842 .674 .524 .253 .126
tp
Apéndice IV
Valores percentiles (2p)
para la distribución
chi cuadrada,
con v grados de libertad
(área sombreada = p)
2
.995 2.99
2.975
2.95
2.90
2.75
2.50
2.25
2.10
2.05
2.025
2.01
2.005
1 7.88 6.63 5.02 3.84 2.71 1.32 .455 .102 .0158 .0039 .0010 .0002 .0000
2 10.6 9.21 7.38 5.99 4.61 2.77 1.39 .575 .211 .103 .0506 .0201 .0100
3 12.8 11.3 9.35 7.81 6.25 4.11 2.37 1.21 .584 .352 .216 .115 .072
4 14.9 13.3 11.1 9.49 7.78 5.39 3.36 1.92 1.06 .711 .484 .297 .207
5 16.7 15.1 12.8 11.1 9.24 6.63 4.35 2.67 1.61 1.15 .831 .554 .412
6 18.5 16.8 14.4 12.6 10.6 7.84 5.35 3.45 2.20 1.64 1.24 .872 .676
7 20.3 18.5 16.0 14.1 12.0 9.04 6.35 4.25 2.83 2.17 1.69 1.24 .989
8 22.0 20.1 17.5 15.5 13.4 10.2 7.34 5.07 3.49 2.73 2.18 1.65 1.34
9 23.6 21.7 19.0 16.9 14.7 11.4 8.34 5.90 4.17 3.33 2.70 2.09 1.73
10 25.2 23.2 20.5 18.3 16.0 12.5 9.34 6.74 4.87 3.94 3.25 2.56 2.16
11 26.8 24.7 21.9 19.7 17.3 13.7 10.3 7.58 5.58 4.57 3.82 3.05 2.60
12 28.3 26.2 23.3 21.0 18.5 14.8 11.3 8.44 6.30 5.23 4.40 3.57 3.07
13 29.8 27.7 24.7 22.4 19.8 16.0 12.3 9.30 7.04 5.89 5.01 4.11 3.57
14 31.3 29.1 26.1 23.7 21.1 17.1 13.3 10.2 7.79 6.57 5.63 4.66 4.07
15 32.8 30.6 27.5 25.0 22.3 18.2 14.3 11.0 8.55 7.26 6.26 5.23 4.60
16 34.3 32.0 28.8 26.3 23.5 19.4 15.3 11.9 9.31 7.96 6.91 5.81 5.14
17 35.7 33.4 30.2 27.6 24.8 20.5 16.3 12.8 10.1 8.67 7.56 6.41 5.70
18 37.2 34.8 31.5 28.9 26.0 21.6 17.3 13.7 10.9 9.39 8.23 7.01 6.26
19 38.6 36.2 32.9 30.1 27.2 22.7 18.3 14.6 11.7 10.1 8.91 7.63 6.84
20 40.0 37.6 3'4.2 31.4 28.4 23.8 19.3 15.5 12.4 10.9 9.59 8.26 7.43
21 41.4 38.9 35.5 32.7 29.6 24.9 20.3 16.3 13.2 11.6 10.3 8.90 8.03
22 42.8 40.3 36.8 33.9 30.8 26.0 21.3 17.2 14.0 12.3 11.0 9.54 8.64
23 44.2 41.6 38.1 35.2 32.0 27.1 22.3 18.1 14.8 13.1 11.7 10.2 9.26
24 45.6 43.0 39.4 36.4 33.2 28.2 23.3 19.0 15.7 13.8 12.4 10.9 9.89
25 46.9 44.3 40.6 37.7 34.4 29.3 24.3 19.9 16.5 14.6 13.1 11.5 10.5
26 48.3 45.6 41.9 38.9 35.6 30.4 25.3 20.8 17.3 15.4 13.8 12.2 11.2
27 49.6 47.0 43.2 40.1 36.7· 31.5 26.3 21.7 18.1 16.2 14.6 12.9 11.8
28 51.0 48.3 44.5 41.3 37.9 32.6 27.3 22.7 18.9 16.9 15.3 13.6 12.5
29 52.3 49.6 45.7 42.6 39.1 33.7 28.3 23.6 19.8 17.7 16.0 14.3 13.1
30 53.7 50.9 47.0 43.8 40.3 34.8 29.3 24.5 20.6 18.5 16.8 15.0 13.8
40 66.8 63.7 59.3 55.8 51.8 46.6 39.3 33.7 29.1 26.5 24.4 22.2 20.7
50 79.5 76.2 71.4 67.5 63.2 56.3 49.3 42.9 37.7 34.8 32.4 29.7 28.0
60 92.0 88.4 83.3 79.1 74.4 67.0 59.3 52.3 46.5 43.2 40.5 37.5 35.5
70 104.2 100.4 95.0 90.5 85.5 77.6 69.3 61.7 55.3 51.7 48.8 45.4 43.3
80 116.3 112.3 106.6 101.9 96.6 88.1 79.3 71.1 64.3 60.4 57.2 53.5 51.2
90 128.3 124.1 118.1 113.1 107.6 98.6 89.3 80.6 73.3 69.1 65.6 61.8 59.2
100 140.2 135.8 129.6 124.3 118.5 109.1 99.3 90.1 82.4 77.9 74.2 70.1 67.3
Fuente: Calherine M. Thompson. Table of percentage points of the X2 distribution. Biometrika, Vol. 32 (1941). bajo permiso del autor y
el editor.
525
2
pχ
Apéndice V
Valores del percentil 95
para la distribución F
(v1 grados de libertad
en el numerador)
(v2 grados de libertad
en el denominador)
v1 v2 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞
1 161 200 216 225 230 234 237 239 241 242 244 246 248 249 250 251 252 253 254
2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.5 19.5 19.5 19.5 19.5 19.5
3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 3.66 8.64 8.62 8.59 8.57 8.55 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.37
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30
13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21
14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01
17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92
19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81
22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78
23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76
24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71
26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69
27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67
28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65
29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39
120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25
∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00
Fuente: E. S. Pearson y H. O. Hartley, Biometrika Tables for Statisticians. Vol. 2 (1972), tabla 5, página 178, bajo permiso del autor y el
editor.
526
F.95
0.05
0.95
Apéndice VI
Valores del percentil 99
para la distribución F
(v1 grados de libertad
en el numerador)
(v2 grados de libertad
en el denominador)
v 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞
1 4052 5000 5403 5625 5764 5859 5928 5981 6023 6056 6106 6157 6209 6235 6261 6287 6313 6339 6366
2 98.5 99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 99.4 99.4 99.4 99.4 99.5 99.5 99.5 99.5 99.5 99.5
3 34.1 30.8 29.5 28.7 28.2 27.9 27.7 27.5 27.3 27.2 27.1 26.9 26.7 26.6 26.5 26.4 26.3 26.2 26.1
4 21.2 18.0 16.7 16.0 15.5 15.2 15.0 14.8 14.7 14.5 14.4 14.2 14.0 13.9 13.8 13.7 13.7 13.6 13.5
5 16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.2 10.1 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02
6 13.7 10.9 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88
7 12.2 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65
8 11.3 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86
9 10.6 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31
10 10.0 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91
11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60
12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36
13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17
14 8.86 6.51 5.56 5.04 4.70 4.46 4.28 4.14 4.03 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00
15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87
16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75
17 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65
18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57
19 8.18 5.93 5.01 4.50 4.17 3.94 3.7
7 3.63 3.52 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49
20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42
21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36
22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31
23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.07 2.93 2.78 2.70 2.82 2.54 2.45 2.35 2.26
24 7.82 5.81 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21
25 4.2
4 5.57 4.68 4.18 3.86 3.63 3.46 3.32 3.22 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.17
26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18 3.09 2.96 2.82 2.66 2.58 2.50 2.42 2.33 2.23 2.13
27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.10
28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12 3.03 2.90 2.75 2.80 2.52 2.44 2.35 2.28 2.17 2.06
29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03
30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01
40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80
60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60
120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.86 2.56 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.65 1.53 1.38
∞ 8.63 4.61 3.78 3.32 3.02 2.80 2.84 2.51 2.41 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00
Fuente: E. S. Pearson y H. O. Hartley, Biometrika Tables for Statisticians, Vol. 2 (1972), tabla 5, página 180, bajo permiso de los autores y el
editor.
527
F.99 0.01
0.99
0 F
Los elementos de la tabla son valores Fa siendo el área o probabilidad en el extremo superior de la distribución F. Por ejemplo, con 12 grados de
libertad en el numerador y 15 en el denominador, el área de .05 en el extremo superior determina F.05 = 2.48.
Tabla de valores de F.05
Grados de
libertad en el Grados de libertad en el numerador
denominador 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞
1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 24 1.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30
13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21
14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01
17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92
19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81
22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78
23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76
24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71
26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69
27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1,67
28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65
29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39
120 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 1.91 1.83 1.75 1.66 J.61 1.55 1.50 1.43 1.35 1.25
∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00
Área de
probabilidad
TABLA E Percentiles de la distribución t
g.1. t.90 t.95 t.975 t.99 t.995 1 3.078 6.3138 12.706 31.821 63.6579 2 l.886 2.9200 4.3027 6.965 9.9248 3 l.638 2.3534 3.1825 4.541 5.8409 4 1.533 2.1318 2.7764 3.747 4.6041 5 1.476 2.0150 2.5706 3.365 4.0321 6 1.440 1.9432 2A469 3.143 3.7074 7 1.415 1.8946 2.3646 2.998 3.4995 8 1.397 1.8595 2.3060 2.896 3.3554 9 1.383 1.8331 2.2622 2.821 3.2498
10 1.372 1.8125 2.2281 2.764 3.1693 11 1.363 1.7959 2.2010 2.718 3.1058 12 1.356 1.7823 2.1768 2.681 3.0545 13 1.350 1.7709 2.1604 2.650 3.0123 14 1.345 1.76 I 3 2.1448 2.fi24 2.9768 15 1.341 1.7530 2.1315 2.602 2.9467 16 1.337 1.7459 2.1199 2.583 2.9208 17 1.333 1.7396 2.1098 2.567 2.8982 18 1.330 1.7341 2.1009 2.552 2.8784 19 1.328 1.7291 2.0930 2.539 2.8609 20 1.325 1.7247 2.0860 2.528 2.8453 21 1.323 1.7207 2.0796 2.518 2.8314 22 1.321 1.7171 2.0739 2.508 2.8188 23 1.319 1.7139 2.0687 2.500 2.8073 24 1.318 1.7109 2.0639 2.492 2.7969 25 1.316 1.7081 2.0595 2.485 2.7874 26 1.315 1.7056 2.0555 2.479 2.7787 27 1.314 1.7033 2.0518 2.473 2.7707 28 1.313 1.7011 2.0484 2.467 2.7633 29 1.311 1.6991 2.0452 2.462 2.7564 30 1.310 1.6973 2.0423 2.457 2.7500 35 1.3062 1.6896 2.0301 2.438 2.7239 40 1.3031 1.6839 2.0211 2.423 2.7045 45 1.3007 1.6794 2.0141 2.412 2.6896 50 1.2987 1.6759 2.0086 2.403 2.6778 60 1.2959 1.6707 2.0003 2.390 2.6603 70 1.2938 1.6669 1.9945 2.381 2.6480 80 1.2922 1.6641 1.9901 2.374 2.6388 90 1.2910 1.6620 1.9867 2.368 2.6316
100 1.2901 1.6602 1.9840 2.364 2.6260 120 1.2887 1.6577 1.9799 2.358 2.6175 140 1.2876 1.6558 1.9771 2.353 2.6114 160 1.2869 1.6545 1.9749 2.350 2.6070 180 1.2863 1.6534 1.9733 2.347 2.6035 200 1.2858 1.6525 1.9719 2.345 2.6006
∞ 1.262 1.645 1.96 2.326 2.576
TABLA F Percentiles de la distribución ji-cuadrada
0 31.410 2.20
P (2.20 ≤ 31.410) = .95
g.l. 2.005
2.025
2.05
2.09
2.95
2.975
2.99
2.995
1 .0000393 .000982 .00393 2.706 3.841 5.024 6.635 7.879 2 .0100 .0506 .103 4.605 5.991 7.378 9.210 10.597 3 .0717 .216 .352 6.251 7.815 9.348 11.345 12.838 4 .207 .484 .711 7.779 9.488 11.l43 13.277 14.860 5 .412 .831 1.145 9.236 11.070 12.832 15.086 16.750 6 .676 1.237 1.635 10.645 12.592 14.449 16.812 18.548 7 .989 1.690 2.167 12.017 14.067 16.013 18.475 20.278 8 1.344 2.180 2.733 13.362 15.507 17.535 20.090 21.955 9 1.735 2.700 3.325 14.684 16.919 19.023 21.666 23.589
10 2.156 3.247 3.940 15.987 18.307 20.483 23.209 25.188 11 2.603 3.816 4.575 17.275 19.675 21.920 24.725 26.757 12 3.074 4.404 5.226 18.549 21.026 23.336 26.217 28.300 13 3.565 5.009 5.892 19.812 22.362 24.736 27.688 29.819 14 4.075 5.629 6.571 21.064 23.685 26.119 29.141 31.319 15 4.601 6.262 7.261 22.307 24.996 27.488 30.578 32.801 16 5.142 6.908 7.962 23.542 26.296 28.845 32.000 34.267 17 5.697 7.564 8.672 24.769 27.587 30.191 33.409 35.718 18 6.265 8.231 9.390 25.989 28.869 31.526 34.805 37.156 19 6.814 8.907 10.117 27.204 30.144 32.852 36.191 38.582 20 7.434 9.591 10.851 28.412 31.410 34.170 37.566 39.997 21 8.034 10.283 11.591 29.615 32.671 35.479 38.932 41.401 22 8.643 10.982 12.338 30.813 33.924 36.781 40.289 42.796 23 9.260 11.688 13.091 32.007 35.172 38.076 41.638 44.181 24 9.836 12.401 13.848 33.196 36.415 39.364 42.980 45.558 25 10.520 13.120 14.611 34.382 37.652 40.646 44.314 46.928 26 11.160 13.844 15.379 35.563 38.885 41.923 45.612 48.290 27 11.808 14.573 16.151 36.741 40.113 43.194 46.963 49.645 28 12.461 15.308 16.928 37.916 41.337 44.461 48.278 50.993 29 13.121 16.047 17.708 39.087 42.557 45.722 49.588 52.336 30 13.787 16.791 18.493 40.256 43.773 46.979 50.892 53.672 35 17.192 20.569 22.465 46.059 49.802 53.203 57.342 60.275 40 20.707 24.433 26.509 51.805 55.758 59.342 63.691 66.766 45 24.311 28.366 30.612 57.505 61.656 65.410 69.957 73.166 50 27.991 32.357 34.764 63.167 67.505 71.420 76.154 79.490 60 35.535 40.482 43.188 74.397 79.082 83.298 88.379 91.952 70 43.275 48.758 51.739 85.527 90.531 95.023 100.425 104.215 80 51.172 57.153 60.391 96.578 101.879 106.629 112.329 116.321 90 59.196 65.647 69.126 107.565 113.145 118.136 124.116 128.299
100 67.328 74.222 77.929 118.498 124.342 129.561 135.807 140.169
.95
0 2.2281 t10
p (t10 < 2.2281) = .975
.975
654 Apéndice: Tablas estadísticas
Tabla A-5. Valores críticos de la distribución ji cuadrada
v .995 .99 .98 .975 .95 .90 .80 .75 .70 .50
1 .04393 .0
3157 .0
3628 .0
3982 .00393 .0158 .0642 .102 .148 .455
2 .0100 .0201 .0404 .0506 .103 .211 .446 .575 .713 1.386 3 .0717 .115 .185 .216 .352 .584 1.005 1.213 1.424 2.366 4 .207 .297 .429 .484 .711 1.064 1.649 1.923 2.195 3.357 5 .412 .554 .752 .831 1.145 1.610 2.343 2.675 3.000 4.351
6 .676 .872 1.134 1.237 1.635 2.204 3.070 3.455 3.828 5.348 7 .989 1.239 1.564 1.690 2.167 2.833 3.822 4.255 4.671 6.346 8 1.344 1.646 2.032 2.180 2.733 3.490 4.594 5.071 5.527 7.344 9 1.735 1.088 2.532 2.700 3.325 4.168 5.380 5.899 6.393 8.343
10 2.156 2.558 3.059 3.247 3.940 4.865 6.179 6.737 7.267 9.342
11 2.603 3.053 3.609 3.816 4.575 5.578 6.989 7.584 8.148 10.341 12 3.074 3.571 4.178 4.404 5.226 6.304 7.807 8.438 9.034 11.340 13 3.565 4.107 4.765 5.009 5.892 7.042 8.634 9.299 9.926 12.340 14 4.075 4.660 5.368 5.629 6.571 7.790 9.467 10.165 10.821 13.339 15 4.601 5.229 5.985 6.262 7.261 8.547 10.307 11.036 11.721 14.339
16 5.142 5.812 6.614 6.908 7.962 9.312 11.152 11.912 12.624 15.338 17 5.697 6.408 7.255 7.564 8.672 10.085 12.002 12.792 13.531 16.338 18 6.265 7.015 7.906 8.231 9.390 10.865 12.857 13.675 14.440 17.338 19 6.844 7.633 8.567 8.907 10.117 11.651 13.716 14.562 15.352 18.338 20 7.434 8.260 9.237 9.591 10.851 12.443 14.578 15.452 16.266 19.337
21 8.034 8.897 9.915 10.283 11.591 13.240 15.445 16.344 17.182 20.337 22 8.643 9.542 10.600 10.982 12.338 14.041 16.314 17.240 18.101 21.337 23 9.260 10.196 11.293 11.688 13.091 14.848 17.187 18.137 19.021 22.337 24 9.886 10.856 11.992 12.401 13.848 15.659 18.062 19.037 19.943 23.337 25 10.520 11.524 12.697 13.120 14.611 16.473 18.940 19.939 20.867 24.337
26 11.160 12.198 13.409 13.844 15.379 17.292 19.820 20.843 21.792 25.336 27 11.808 12.879 14.125 14.573 16.151 18.114 20.703 21.749 22.719 26.336 28 12.461 13.565 14.847 15.308 16.928 18.939 21.588 22.657 23.647 27.336 29 13.121 14.256 15.574 16.047 17.708 19.768 22.475 23.567 24.577 28.336 30 13.787 14.953 16.306 16.791 18.493 20.599 23.364 24.478 25.508 29.336
Apéndice: Tablas estadísticas 655
Tabla A-5. Valores críticos de la distribución ji cuadrada (continúa)
v .30 .25 .20 .10 .05 .025 .02 .01 .005 .001
1 1.074 1.323 1.642 2.706 3.841 5.024 5.412 6.635 7.879 10.827 2 2.408 2.773 3.219 4.605 5.991
1
7.378 7.824 9.210 10.597 13.815 3 3.665 4.108 4.642 6.251 7.815 9.348 9.837 11.345 12.838 16.268 4 4.878 5.385 5.989 7.779 9.488 11.143 11.668 13.271 14.860 18.465 5 6.064 6.626 7.289 9.236 11.070 12.832 13.388 15.086 16.750 20.517
6 7.211 7.841 8.558 10.645 12.592 14.449 15.033 16.812 18.548 21.457 7 8.383 9.037 9.803 12.017 14.067 16.013 16.622 18.475 20.278 24.322 8 9.524 10.219 11.030 13.362 15.507 17.535 18.168 20.090 21.955 26.125 9 10.656 11389 12.242 14.684 16.919 19.023 19.679 21.666 23.589 27.871
10 11.781 12.549 13.442 15.987 18.307 20.483 21.161 23.209 25.188 29.588
11 12.899 13.701 14.631 17.275 19.675 21.920 22.618 24.725 26.757 31.264 12 14.011 14.845 15.812 18.549 21.026 23.337 24.054 26.217 28.300 32.909 13 15.119 15.984 16.985 19.812 22.362 24.736 25.472 27.688 29.819 34.528 14 16.222 17.117 18.151 21.064 23.685 26.119 26.873 29.141 31319 36.123 15 17.322 18.245 19.311 22.307 24.996 27.488 28.259 30.578 32.801 37.697
16 18.418 19.369 20.465 23.542 26.296 28.845 29.633 32.000 34.267 39.252 17 19.511 20.489 21.615 24769 27.587 30.191 30.995 33.409 35.718 40.790 18 20.601 21.605 22.760 25.989 28.869 31.526 32.346 34.805 37.156 42.312 19 21.689 22.718 23.900 27.204 30.144 32.852 33.687 36.191 38.582 43.820 20 22.715 23.828 25.038 28.412 31.410 34.170 35.020 37.566 39.997 45.315
21 23.858 24.935 26.171 29.615 32.671 35.479 36.343 38.932 41.401 46.797 22 24.919 26.039 27.301 30.813 33.924 36.781 37.659 40.289 42.796 48.268 23 26.018 27.141 28.429 32.007 35.172 38.076 38.968 41.638 44.181 49.728 24 27.096 28.241 29.553 33.196 36.415 39.364 40.270 42.980 45.558 51.179 25 28.172 29.339 30.675 34.382 37.652 40.646 41.566 44.314 46.928 52.620
26 29.246 30.434 31.795 35.563 38.885 41.923 42.856 45.642 48.290 54.052 27 30.319 31.528 32.912 36.741 40.113 43.194 44.140 46.963 49.645 55.476 28 31.391 32.620 34.027 37.916 41.337 44.461 45.419 48.278 50.993 56.893 29 32.461 33.711 35.139 39.087 42.557 45.722 46.693 49.588 52.336 58.302 30 33.530 34.800 36.250 40.256 43.773 46.979 47.962 50.892 53.672 59.703
2
αχ0
656 Apéndice: Tablas estadísticas
Tabla A-6*. Valores críticos de la distribución F
f0.05 (ν1,ν2)
ν1
ν2 1 2 3 4 5 6 7 8 9
1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 17 4.45 3.59 ! 3.20 2.96 2.81 2.70 2.61 2.55 2.49 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04
120 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 ∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88
Apéndice: Tablas estadísticas 657
Tabla A-6. Valores críticos de la distribución F (continúa)
f0.05 (ν1,ν2)
ν1
ν2 10 12 15 20 24 30 40 60 120 ∞
1 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3 2 19.40 19.41 19.43 19.45 19.45 19.46 19.4 19.
4
19.49 19.50 3 8.79 8.74 8.70 8.66 8.64 8.62 8.5 8.5 8.5 8.53 4 5.96 5.91 5.86 5.80 5.77 5.75 5.7 5.6 5.6 5.63
5 4.74 4.68 4.62 4.56 4.53 4.50 4.4 4.4 4.4 4.36 6 4.06 4.00 3.94- 3.87 3.84 3.81 3.7 3.7 3.7 3.67 7 3.64 3.57 3.51 3.44 3.41 3.38 3.3 3.3 3.2 3.23 8 3.35 3.28 3.22 3.15 3.12 3.08 3.0 3.0 2.9 2.93 9 3.14 3.07 3.01 2.94 2.90 2.86 2.8 2.7 2.7 2.71
10 2.98 2.91 2.85 2.77 2.74 2.70 2.6 2.6 2.5 2.54 11 2.85 2.79 2.72 2.65 2.61 2.57 2.5 2.4 2.4 2.40 12 2.75 2.69 2.62 2.54 2.51 2.47 2.4 2.3 2.3 2.30 13 2.67 2.60 2.53 2.46 2.42 2.38 2.3 2.3 2.2 2.21 14 2.60 2.53 2.46 2.39 2.35 2.31 2.2 2.2 2.1 2.13
15 2.54 2.48 2.40 2.33 2.29 2.25 2.2 2.1 2.1 2.07 16 2.49 2.42 2.35 2.28 2.24 2.19 2.1 2.1 2.0 2.01 17 2.45 2.38 2.31 2.23 2.19 2.15 2.1 2.0 2.0 1.96 18 2.41 2.34 2.27 2.19 2.15 2.11 2.0 2.0 1.9 1.92 19 2.38 2.31 2.23 2.16 2.11 2.07 2.0 1.9 1.9 1.88
20 2.35 2.28 2.20 2.12 2.08 2.04 1.9 1.9 1.9 1.84 21 2.32 2.25 2.18 2.10 2.05 2.01 1.9 1.9 1.8 1.81 22 2.30 2.23 2.15 2.07 2.03 1.98 1.9 1.8 1.8 1.78 23 2.27 2.20 2.13 2.05 2.01 1.96 1.9 1.8 1.8 1.76 24 2.25 2.18 2.11 2.03 1.98 1.94 1.8 1.8 1.7 1.73
25 2.24 2.16 2.09 2.01 1.96 1.92 1.8 1.8 1.7 1.71 26 2.22 2.15 2.07 1.99 1.95 1.90 1.8 1.8 1.7 1.69 27 2.20 2.13 2.06 1.97 1.93 1.88 1.8 1.7 1.7 1.67 28 2.19 2.12 2.04 1.96 1.91 1.87 1.8 1.7 1.7 1.65 29 2.18 2.10 2.03 1.94 1.90 1.85 1.8 1.7 1.7 1.64
30 2.16 2.09 2.01 1.93 1.89 1.84 1.7 1.7 1.6 1.62 40 2.08 2.00 1.92 1.84 1.79 1.74 1.6 1.6 1.5 1.51 60 1.99 1.92 1.84 1.75 1.70 1.65 1.5 1.5 1.4 1.39
120 1.91 1.83 1.75 1.66 1.61 1.55 1.5 1.4 1.3 1.25 ∞ 1.83 1.75 1.67 1.57 1.52 1.46 1.3 1.3 1.2 1.00
0 f
658 Apéndice: Tablas estadísticas
Tabla A-6. Valores críticos de la distribución F (continúa)
f0.01 (ν1,ν2)
ν1
ν2 1 2 3 4 5 6 7 8 9
1 4052 4999.5 5403 5625 5764 5859 5928 5981 6022
2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66
5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16
6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 9 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35
10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94
11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03
15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89
16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52
20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46
21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26
25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22
26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18 27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15 28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12 29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09
30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07
40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72
120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 ∞ 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41
Apéndice: Tablas estadísticas 659
Tabla A-6. Valores críticos de la distribución F (continúa)
f0.01 (ν1,ν2)
ν1
ν2 10 12 15 20 24 30 40 60 120 ∞
1 6056 6106 6157 6209 6235 6261 6287 6313 6339 6366
2 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.50 3 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.13 4 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.46
5 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02
6 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88 7 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65 8 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86 9 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31
10 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91
11 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60 12 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36 13 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17 14 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00
15 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87
16 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75 17 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65 18 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57 19 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49
20 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42
21 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36 22 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31 23 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26 24 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21
25 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.17
26 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.13 27 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.10 28 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.06 29 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03
30 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01
40 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80 60 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60
120 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38 ∞ 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00
TABLA A.5
Percentiles de la distribución F.
gl
den
Grados de libertad en el numerador (gl)
1 2 3 4 5 6 7 8 12 24 ∞
2 0.050 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.41 19.45 19.50
0.025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.41 39.46 39.50 0.010 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.42 99.46 99.50
0.005 198.5 199.0 199.2 199.3 199.3 199.3 199.4 199.4 199.4 199.5 199.5
0.001 998.5 999.0 999.2 999.3 999.3 999.3 999.4 999.4 999.4 999.5 999.5
3 0.050 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.74 8.64 8.53
0.025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.34 14.12 13.90 0.010 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.05 26.60 26.13
0.005 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.39 42.62 41.83
0.001 167.0 148.5 141.1 137.1 134.6 132.9 131.6 130.6 128.3 125.9 123.5
4 0.050 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 5.91 5.77 5.63
0.025 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.75 8.51 8.26 0.010 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.37 13.93 13.46
0.005 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 20.70 20.03 19.32
0.001 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 47.41 45.77 44.05
5 0.050 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.68 4.53 4.36
0.025 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.52 6.28 6.02 0.010 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 9.89 9.47 9.02
0.005 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.38 12.78 12.14
0.001 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 26.42 25.13 23.79
6 0.050 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.00 3.84 3.67
0.025 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.37 5.12 4.85 0.010 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.72 7.31 6.88
0.005 18.63 14.54 12.92 12.03 1 1.46 11.07 10.79 10.57 10.03 9.47 8.88
0.001 35.51 27.00 23.70 21.92 20.80 20.03 19.46 19.03 17.99 16.90 15.75
7 0.050 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.57 3.41 3.23
0.025 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.67 4.41 4.14 0.010 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.47 6.07 5.65
0.005 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.18 7.64 7.08
0.001 29.25 21.69 18.77 17.20 16.21 15.52 15.02 14.63 13.71 12.73 11.70
8 0.050 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.28 3.12 2.93
0.025 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.20 3.95 3.67 0.010 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.67 5.28 4.86
0.005 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.01 6.50 5.95
0.001 25.41 18.49 15.83 14.39 13.48 12.86 12.40 12.05 11.19 10.3 9.33
9 0.050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.07 2.90 2.71
0.025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 3.87 3.61 3.33 0.010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.11 4.73 4.31
0.005 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.23 5.73 5.19
0.001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 9.57 8.72 7.81
10 0.050 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 2.91 2.74 2.54
0.025 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.62 3.37 3.08 0.010 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.71 4.33 3.91
0.005 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.66 5.17 4.64
0.001 21.04 14.91 12.55 11.28 10.48 9.93 9.52 9.20 8.45 7.64 6.76
12 0.100 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.15 2.04 1.90
0.050 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.69 2.51 2.30 0.025 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.28 3.02 2.72
0.010 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.16 3.78 3.36
0.005 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 4.91 4.43 3.90 0.001 18.64 12.97 10.80 9.63 8.89 8.38 8.00 7.71 7.00 6.25 5.42
TABLA A.5 Percentiles de la distribución F.
gl
den
Grados de libertad en el numerador (gl)
1 2 3 4 5 6 7 8 12 24 ∞
14 0.050 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.53 2.35 2.13
0.025 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.05 2.79 2.49 0.010 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 3.80 3.43 3.00 0.005 11.06 7.92 6.68 6.00 5.56 5.26 5.03 4.86 4.43 3.96 3.44 0.001 17.14 11.78 9.73 8.62 7.92 7.44 7.08 6.80 6.13 5.41 4.60
16 0.050 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.42 2.24 2.01
0.025 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 2.89 2.63 2.32 0.010 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.55 3.18 2.75 0.005 10.58 7.51 6.30 5.64 5.21 4.91 4.69 4.52 4.10 3.64 3.11 0.001 16.12 10.97 9.01 7.94 7.27 6.80 6.46 6.19 5.55 4.85 4.06
18 0.050 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.34 2.15 1.92
0.025 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.77 2.50 2.19 0.010 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.37 3.00 2.57 0.005 10.22 7.21 6.03 5.37 4.96 4.66 4.44 4.28 3.86 3.40 2.87 0.001 15.38 10.39 8.49 7.46 6.81 6.35 6.02 5.76 5.13 4.45 3.67
20 0.050 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.28 2.08 1.84
0.025 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.68 2.41 2.09 0.010 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.23 2.86 2.42 0.005 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.68 3.22 2.69 0.001 14.82 9.95 8.10 7.10 6.46 6.02 5.69 5.44 4.82 4.15 3.38
30 0.050 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.09 1.89 1.62
0.025 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.41 2.14 1.79 0.010 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 2.84 2.47 2.01 0.005 9.18 6.35 5.24 4.62 4.23 3.95 3.74 3.58 3.18 2.73 2.18 0.001 13.29 8.77 7.05 6.12 5.53 5.12 4.82 4.58 4.00 3.36 2.59
40 0.050 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.00 1.79 1.51
0.025 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.29 2.01 1.64 0.010 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.66 2.29 1.80 0.005 8.83 6.07 4.98 4.37 3.99 3.71 3.51 3.35 2.95 2.50 1.93 0.001 12.61 8.25 6.59 5.70 5.13 4.73 4.44 4.21 3.64 3.01 2.23
60 0.050 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 1.92 1.70 1.39
0.025 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.17 1.88 1.48
0.010 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.50 2.12 1.60 0.005 8.49 5.79 4.73 4.14 3.76 3.49 3.29 3.13 2.74 2.29 1.69
0.001 11.97 7.77 6.17 5.31 4.76 4.37 4.09 3.86 3.32 2.69 1.89
80 0.050 3.96 3.11 2.72 2.49 2.33 2.21 2.13 2.06 1.88 1.65 1.32
0.025 5.22 3.86 3.28 2.95 2.73 2.57 2.45 2.35 2.11 1.82 1.40
0.010 6.96 4.88 4.04 3.56 3.26 3.04 2.87 2.74 2.42 2.03 1.49 0.005 8.33 5.67 4.61 4.03 3.65 3.39 3.19 3.03 2.64 2.19 1.56
0.001 11.67 7.54 5.97 5.12 4.58 4.20 3.92 3.70 3.16 2.54 1.72
100 0.050 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.85 1.63 1.28
0.025 5.18 3.83 3.25 2.92 2.70 2.54 2.42 2.32 2.08 1.78 1.35
0.010 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.37 1.98 1.43 0.005 8.24 5.59 4.54 3.96 3.59 3.33 3.13 2.97 2.58 2.13 1.49
0.001 11.50 7.41 5.86 5.02 4.48 4.11 3.83 3.61 3.07 2.46 1.62
120 0.050 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.83 1.61 1.25
0.025 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.05 1.76 1.31
0.010 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.34 1.95 1.38 0.005 8.18 5.54 4.50 3.92 3.55 3.28 3.09 2.93 2.54 2.09 1.43
0.001 11.38 7.32 5.78 4.95 4.42 4.04 3.77 3.55 3.02 2.40 1.54
∞ 0.050 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.75 1.52 1.00
0.025 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 1.94 1.64 1.00
0.010 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.18 1.79 1.00 0.005 7.88 5.30 4.28 3.72 3.35 3.09 2.90 2.74 2.36 1.90 1.00
0.001 10.83 6.91 5.42 4.62 4.10 3.74 3.47 3.27 2.74 2.13 1.00
TABLA A.5 Percentiles de la distribución F.
gl en el
denominador
Área en la
sección
superior
Grados de libertad en el numerador (gl)
1 2 3 4 5 6 7 8 12 24 ∞
2 0.100 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.41 9.45 9.49
0.050 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.41 19.45 19.50 0.025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.41 39.46 39.50 0.010 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.42 99.46 99.50 0.005 198.5 199.0 199.2 199.3 199.3 199.3 199.4 199.4 199.4 199.5 199.5 0.001 998.5 999.0 999.2 999.3 999.3 999.3 999.4 999.4 999.4 999.5 999.5
3 0.100 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.22 5.18 5.13
0.050 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.74 8.64 8.53 0.025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.34 14.12 13.90 0.010 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.05 26.60 26.13 0.005 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.39 42.62 41.83 0.001 167.0 148.5 141.1 137.1 134.6 132.9 131.6 130.6 128.3 125.9 123.5
4 0.100 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.90 3.83 3.76
0.050 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 5.91 5.77 5.63 0.025 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.75 8.51 8.26 0.010 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.37 13.93 13.46 0.005 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 20.70 20.03 19.32 0.001 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 47.41 45.77 44.05
5 0.100 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.27 3.19 3.10
0.050 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.68 4.53 4.36 0.025 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.52 6.28 6.02 0.010 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 9.89 9.47 9.02 0.005 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.38 12.78 12.14 0.001 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 26.42 25.13 23.79
6 0.100 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.90 2.82 2.72
0.050 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.00 3.84 3.67 0.025 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.37 5.12 4.85 0.010 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.72 7.31 6.88 0.005 18.63 14.54 12.92 12.03 1 1.46 11.07 10.79 10.57 10.03 9.47 8.88 0.001 35.51 27.00 23.70 21.92 20.80 20.03 19.46 19.03 17.99 16.90 15.75 (continúa)
TABLA A.5 Percentiles de la distribución F.
gl en el
denominador
Área en la
sección
superior
Grados de libertad en el numerador (gl)
1 2 3 4 5 6 7 8 12 24 ∞
7 0.100 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.67 2.58 2.47
0.050 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.57 3.41 3.23 0.025 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.67 4.41 4.14 0.010 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.47 6.07 5.65 0.005 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.18 7.64 7.08 0.001 29.25 21.69 18.77 17.20 16.21 15.52 15.02 14.63 13.71 12.73 11.70
8 0.100 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.50 2.40 2.29
0.050 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.28 3.12 2.93 0.025 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.20 3.95 3.67 0.010 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.67 5.28 4.86 0.005 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.01 6.50 5.95 0.001 25.41 18.49 15.83 14.39 13.48 12.86 12.40 12.05 11.19 10.3 9.33
9 0.100 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.38 2.28 2.16
0.050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.07 2.90 2.71 0.025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 3.87 3.61 3.33 0.010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.11 4.73 4.31 0.005 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.23 5.73 5.19 0.001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 9.57 8.72 7.81
10 0.100 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.28 2.18 2.06
0.050 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 2.91 2.74 2.54 0.025 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.62 3.37 3.08 0.010 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.71 4.33 3.91 0.005 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.66 5.17 4.64 0.001 21.04 14.91 12.55 11.28 10.48 9.93 9.52 9.20 8.45 7.64 6.76
12 0.100 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.15 2.04 1.90
0.050 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.69 2.51 2.30 0.025 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.28 3.02 2.72 0.010 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.16 3.78 3.36 0.005 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 4.91 4.43 3.90 0.001 18.64 12.97 10.80 9.63 8.89 8.38 8.00 7.71 7.00 6.25 5.42 (continúa)
TABLA A.5 Percentiles de la distribución F.
gl en el
denominador
Área en la
sección
superior
Grados de libertad en el numerador (gl)
1 2 3 4 5 6 7 8 12 24 ∞
14 0.100 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.05 1.94 1.80
0.050 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.53 2.35 2.13 0.025 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.05 2.79 2.49 0.010 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 3.80 3.43 3.00 0.005 11.06 7.92 6.68 6.00 5.56 5.26 5.03 4.86 4.43 3.96 3.44 0.001 17.14 11.78 9.73 8.62 7.92 7.44 7.08 6.80 6.13 5.41 4.60
16 0.100 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 1.99 1.87 1.72
0.050 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.42 2.24 2.01 0.025 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 2.89 2.63 2.32 0.010 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.55 3.18 2.75 0.005 10.58 7.51 6.30 5.64 5.21 4.91 4.69 4.52 4.10 3.64 3.11 0.001 16.12 10.97 9.01 7.94 7.27 6.80 6.46 6.19 5.55 4.85 4.06
18 0.100 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 1.93 1.81 1.66
0.050 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.34 2.15 1.92 0.025 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.77 2.50 2.19 0.010 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.37 3.00 2.57 0.005 10.22 7.21 6.03 5.37 4.96 4.66 4.44 4.28 3.86 3.40 2.87 0.001 15.38 10.39 8.49 7.46 6.81 6.35 6.02 5.76 5.13 4.45 3.67
20 0.100 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.89 1.77 1.61
0.050 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.28 2.08 1.84 0.025 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.68 2.41 2.09 0.010 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.23 2.86 2.42 0.005 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.68 3.22 2.69 0.001 14.82 9.95 8.10 7.10 6.46 6.02 5.69 5.44 4.82 4.15 3.38
30 0.100 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.77 1.64 1.46
0.050 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.09 1.89 1.62 0.025 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.41 2.14 1.79 0.010 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 2.84 2.47 2.01 0.005 9.18 6.35 5.24 4.62 4.23 3.95 3.74 3.58 3.18 2.73 2.18 0.001 13.29 8.77 7.05 6.12 5.53 5.12 4.82 4.58 4.00 3.36 2.59
40 0.100 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.71 1.57 1.38
0.050 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.00 1.79 1.51 0.025 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.29 2.01 1.64 0.010 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.66 2.29 1.80 0.005 8.83 6.07 4.98 4.37 3.99 3.71 3.51 3.35 2.95 2.50 1.93 0.001 12.61 8.25 6.59 5.70 5.13 4.73 4.44 4.21 3.64 3.01 2.23 (continúa)
TABLA A.5 Percentiles de la
distribución F.
gl en el
denominador
Área en la
sección
superior
Grados de libertad en el numerador (gl)
1 2 3 4 5 6 7 8 12 24 ∞
60 0.100 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.66 1.51 1.29
0.050 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 1.92 1.70 1.39 0.025 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.17 1.88 1.48 0.010 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.50 2.12 1.60 0.005 8.49 5.79 4.73 4.14 3.76 3.49 3.29 3.13 2.74 2.29 1.69 0.001 11.97 7.77 6.17 5.31 4.76 4.37 4.09 3.86 3.32 2.69 1.89
80 0.100 2.77 2.37 2.15 2.02 1.92 1.85 1.79 1.75 1.63 1.48 1.24
0.050 3.96 3.11 2.72 2.49 2.33 2.21 2.13 2.06 1.88 1.65 1.32 0.025 5.22 3.86 3.28 2.95 2.73 2.57 2.45 2.35 2.11 1.82 1.40 0.010 6.96 4.88 4.04 3.56 3.26 3.04 2.87 2.74 2.42 2.03 1.49 0.005 8.33 5.67 4.61 4.03 3.65 3.39 3.19 3.03 2.64 2.19 1.56 0.001 11.67 7.54 5.97 5.12 4.58 4.20 3.92 3.70 3.16 2.54 1.72
100 0.100 2.76 2.36 2.14 2.00 1.91 1.83 1.78 1.73 1.61 1.46 1.21
0.050 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.85 1.63 1.28 0.025 5.18 3.83 3.25 2.92 2.70 2.54 2.42 2.32 2.08 1.78 1.35 0.010 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.37 1.98 1.43 0.005 8.24 5.59 4.54 3.96 3.59 3.33 3.13 2.97 2.58 2.13 1.49 0.001 11.50 7.41 5.86 5.02 4.48 4.11 3.83 3.61 3.07 2.46 1.62
120 0.100 2.75 2.35 2.13 1.99 1.90 1.82 1.77 1.72 1.60 1.45 1.19
0.050 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.83 1.61 1.25 0.025 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.05 1.76 1.31 0.010 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.34 1.95 1.38 0.005 8.18 5.54 4.50 3.92 3.55 3.28 3.09 2.93 2.54 2.09 1.43 0.001 11.38 7.32 5.78 4.95 4.42 4.04 3.77 3.55 3.02 2.40 1.54
∞ 0.100 2.71 2.30 2.08 1.94 1.85 1.77 1.72 1.67 1.55 1.38 1.00
0.050 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.75 1.52 1.00 0.025 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 1.94 1.64 1.00 0.010 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.18 1.79 1.00 0.005 7.88 5.30 4.28 3.72 3.35 3.09 2.90 2.74 2.36 1.90 1.00 0.001 10.83 6.91 5.42 4.62 4.10 3.74 3.47 3.27 2.74 2.13 1.00