statistical tables
DESCRIPTION
statistic & probabilityTRANSCRIPT
Table 1 Cumulative Binomial Probabilities - continued
v 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
n: LOO 1.0000.6340.2642.0794.0184
.0034
.0005
.0001
1.0000.8674.5967.3233.1410
.0508
.0155
.0041
.0009
.0002
1.0000.9993.9940.9'742.9256
.8368
.7086
.5557
.4012
.2660
.1620
.0908
.0469
.0224
.0099
.0041
.0016
.0006
.0002
.0001
r:0I2J
4
56789
1.0000 1.0000.9524 .9831.8054 .9128.5802 .7679.3528 .5705
.1821 .371r
.0808 .2t16
.0312 .t064
.0106 .0475
.0032 .0190
.0009 .0068
.0002 .0022.0007.0002
1.0000 1.0000.994r .9979.9629 .9848.8817 .9434.7422 .8570
.5640 .',t232
.3840 .s593
.2340 .3936
.1280 .2sr7
.0631 .1463
.0282 .0',175
.0115 .0376
.0043 .0168
.0015 .0069
.0005 .0026
.0001 .0009.0003.0001
1.0000 1.0000.9998 .9999.9977 .9991.9887 .99s2.9633 .9827
.9097 .9526
.8201 .8955
.6968 .8060
.s529 .6872
.4074 .5506
.2780 .4125
.1757 .2882
.1028 .1876
.0559 .1138
.0282 .064s
.0133 ". .0341
.0058 .0169
.0024 .0078
.0009 .0034
.0003 .0014
.0001 .0005.0002.0001
10111213t4
15T6t71819
202l22
n: 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
n=2
n=5
n: T0
n:20
1.0000.1900.0100
1.0000.4095.0815.0086.0005
1.0000.6513.2639.0702.0128
.0016
.0001
1.0000.8784.6083.3231.1330
.0432
.0113
.0024
.0004
.0001
r:0I2
r:0I2J
4
1.0000 1.0000.2775 .3600.0225 .0400
1.0000 1.0000.5563 .6723.1648 .2627.0266 .0519.0022 .0067
.0001 .0003
1.0000 1.0000.8031 .8926.4s57 .6242.1798 .3222.0s00 .t209
.0099 .0328
.0014 .0064
.0001 .0009.0001
1.0000 1.0000.96t2 .9885.8244 .9308.5951 .'7939.3523 .5886
.I't02 .3704
.0673 .1958
.0219 .0867
.0059 .0321
.0013 .0100
.0002 .0026.0006.0001
1.0000 1.0000.4375 .5100.0625 .0900
1.0000 1.0000."162',7 .8319.3612 .4',1r8.1035 .1631.0156 .0308
.0010 .0024
1.0000 1.0000.9437 .97t81560 .8507.4744 .6t',12.2241 .3504
.0781 .1503
.0197 .0473
.0035 .0106
.0004 .0016.0001
1.0000 1.0000.9968 .9992.9757 .9924.9087 .9645.7748 .8929
.5852 .7625
.3828 .5836
.2142 .3920
.1018 .22'17
.0409 .1133
.0139 .0480
.0039 .0171
.0009 .0051
.0002 .0013.0003
1.0000 1.0000.5775 .6400.t225 .1600
1.0000 1.0000.8840 .9222.5',1r6 .6630.2352 .3t74.0540 .0870
.0053 .0102
1.0000 1.0000.9865 .9940.9140 .9536.7384 .8327.4862 .6t77
.2485 .3669
.0949 .1662
.0260 .0548
.0048 .0123
.000s .0017
.0001
1.0000 1.0000.9998 1.0000.9979 .9995.9879 .9964.9556 .9840
.8818 .94903546 .8744.5834 .7500.3990 .5841.2376 .4044
.1218 .244',7
.0532 .r27s
.0196 .0565
.0060 .0210
.0015 .0065
.0003 .0016.0003
1.0000 1.0000.6975 .7500.2025 .2500
1.0000 1.0000.949',7 .9688.7438 .8125.4069 .5000.1312 .1875
.0185 .0313
1.0000 1.0000.9975 .9990.9767 .9893.9004 .94s3.7430 .8281
.4956 .6230
.2616 .3770
.1020 . t719
.0274 .0547
.0045 .0107
.0003 .0010
1.0000 1.00001.0000 1.0000.9999 1.0000.9991 .9998.9951 .9987
.9811 .9947
.9447 .9793
.8701 .9423
.7480 .8684
.5857 .7483
.4086 .5881
.2493 .4tt9
.1308 .2517
.0s80 .1316
.0214 .0577
.0064 .0207
.0015 .0059
.0003 .0013.0002
5
r:0I2J
4
56789
l0
r:012-l
4
56789
101lt213t4
1516l7l8
Table I Cumulative Binomial Probabilites - continued
0. l0 0. 15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
n=50 r=oI234
1.0000.9997.9971.9858.9544
.8879
.7806
.6387
.4812
.3319
.2089
.1 199
.0628
.0301
.0r32
.0053
.0019
.0007
.0002
.0001
1.00001.0000.9998.9987.9943
1.00001.00001.0000.9999.9995
1.00001.00001.00001.00001.0000
1.00001.00001.00001.00001.0000
1.0000.9999.9998.9992.9975
.9933
.9840
.9658
.9339
.8837
.8r22
.7r99
.6111
.4940
.3784
.2736
. l86l
. l 187
.0710
.0396
.0201
.0100
.0045
.0019
.0007
.0003
.0001
1.00001.00001.00001.00001.0000
1.00001.00001.0000.9999.9998
.9992
.9978
.9943
.9867
.9720
.9460
.9045
.8439
.7631
.6644
.5535
.4390
.3299
.2340
.1562
.0978
.0573
.0314
.0160
.00'76
.0034
.0014
.0005
.0002
.0001
1.0000.9948.9662.8883.7497
5 .56886 .38397 .22988 .t2219 .0s79
l0 .0245I I .0094t2 .003213 .001014 .0003
15 .0001l6t7l8l9
.9815 .9979 .9998
.9s20 .9930 .9993
.8966 .9806 .9975
.8096 .9547 .9927
.6927 .9084 .9817
.5563 .8363 .9598
.4t64 .7318 .92IT
.2893 .6184 .8610
.1861 .4890 .7711
.1106 .3630 .6721
.0607 .2519 .5532
.0308 .1631 .4308
.0144 .0983 .3161
.0063 .0551 .2178
.0025 .0287 .1406
.0009 .0139 .0848
.0003 .0063 .0478
.0001 .0026 .0251.0010 .0123.0004 .0056
.0001 .0024.0009.0003.0001
1.00001.00001.00001.00001.0000
1.00001.00001.00001.00001.0000
1.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.0000
.9999 1.0000
.9998 1.0000
.9994 " 1.0000
.9982 .9998
.9955 .9995
.9896 .9987
.9780 .9967
.9573 .9923
.9235 .9836
.8727 .9675
.8026 .9405
.7138 .8987
.6100 .8389
.4981 .7601.3856 .6641
.2840 .5561
.1966 ' .4439
.1279 .3359
.0780 .2399
.0444 .1611
.0235 .1013
.0116 .0595
.0053 .0325
.0022 .0164
.0009 .0077
.0003 .0033
.0001 .0013.0005.0002
202l222324
2526272829
3031
3334
3536JI
38
Table I gives binomial probabilities only for a limited range of values of n and p since, in practice, either the more compacttabulation of the Poisson distribution (Table 2) or that of the Nonnal distribution (Table 3) can usually be used to give anadequate approximation.
As a reasonable working rule:
(i) use the Poisson approximationrf p < 0.1, putting m= np
(it use the Normal approximation if 0.1 <p i 0.9 and np > S,putting p : np and o=(iii) use the Poisson approximation ifp > 0.9, putting m = n(l -p) and working in terms of 'failures'.
Note: For values of p > 0.5, work in terms of lfailures' which will have probability q (= | - p).
Exarnple: What is the probability that 40 ol more seeds will germinate out of 50 if the germination rate is 70%? Since theprobability of 'success' is greater than 0.5, the table can not be used directly; howwer, 40 or more successes is the same as l0 orfewer' fai lures' .Theprobabi l i tyof l0orfewer' fai lures'=l-probabi l i tyof l lormore' fai lures'=1-0.9211:0.0739.
"dr- p)
Table I Cumulative Binomial Probabilities - continued
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.s0
n=100 r=012J
4
5o
789
101112t3
l+
15to
17l8l9
t.00001.0000.9997.9981.9922
.9763
.9424
.8828
.7939
.6791
.5487
.4168
.2970
.1982
.1239
.0726
.0399
.0206
.0100
.0046
.0020
.0008
.0003
.0001
1.00001.00001.00001.0000.9999
.9996
.9984
.9953
.9878
.9725
.9449
.9006
.8365
.7527
.6526
.5428
.4317
.3275
.2367
.1628
.1 065
.0663
.0393
.0221
.01 19
.0061
.0030
.0014
.0006
.0003
.0001
1.00001.00001.00001.00001.0000
1.00001.0000.9999.9997.9991
.9977
.9943
.9874
.9747
.9531
.9196
.8715
.8077
.7288
.6379
.5398
.4405
.3460
.zot l
.1891
.1 314
.0875
.u))6
.0342
.0200
.0112
.0061
.0031
.0016
.0007
.0003
.0001
.0001
1.00001.00001.00001.00001.0000
1.00001.00001.00001.00001.0000
1.0000.9999.9996.9990.997s
.9946
.9889
.9789
.9624
.9370
.9005
.8512
.7886
.7136
.6289
.5383
.4465
.3583
.2776
.2075
.1495
.1038
.0693
.0446
.0276
.0164
.0094
.0052
.0027
.0014
.0007
.0003
.0001
.0001
1.00001.00001.00001.00001.0000
1.00001.00001.00001.00001.0000
1.00001.00001.00001.0000.9999
.9998
.9996
.9990
.9978
.9955
.9911
.9835
.97t2
.9521
.9245
.8864
.8369
.7756
.7036
.6232
.5377
.4509
.3669
.2893
.2207
.1629
.1161
.0799
.0530
.0340
.0210
.0125
.0072
.0040
.0021
.0011
.UUU)
.0003
.0001
.0001
1.00001.00001.00001.00001.0000
1.00001.00001.00001.00001.0000
1.00001.00001.00001.00001.0000
1.00001.00001.0000.9999.9999
.9997
.9992
.9983
.9966
.9934
.9879
.9789
.9649
.9442
.9r52
.8764
.8270
.7669
.6971
.6197
.5376
.4542
.3731
.2976
.2301
.1724
.1250
.0877
.0594
.0389
.0246
.0150
.0088
.0050
.0027
.0015
.0007
.0004
.0002
.0001
1.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.0000
1.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.0000
1.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.0000
1.0000 1.0000 1.00001.0000 1.0000* 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.0000
1.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.0000.9999 1.0000 1.0000.9997 1.0000 l 0000
.9994 1.0000 1.0000
.9988 l.0000 1.0000
.9976 .9999 r.0000
.9954 .9998 1.0000
.9916 .9996 1.0000
.9852 .9992 1.0000
.9752 .9985 1.0000
.9602 .9970 .9999
.9385 .9945 .9998
.9087 .9902 .9996
.8697 .9834 .9991
.820s .9728 .9982
.7614 .9571 .9967
.6932 .9349 .9940
.6178 .9049 .9895
.5379 .8657 .9824
.4567 .8169 .9716
.3775 .7585 .9557
.3033 .6913 .9334
.236s .6172 .9033
.1789 .s387 .8644
.1311 .4587 .8159
.0930 .3804 .7579
.0638 .3069 .6914
.0423 .2404 .6t78
.0271 .1827 .5398
.0168 .1346 .4602
.0100 .0960 .3822
.0058 .0662 .3086
.0032 .0441 .2421
.0017 .0284 .1841
.0009 .0176 .1356
.0004 .0106 .0967
.0002 .0061 .0666
.000r .0034 .0443
.0018 .0284
.0009 .0176
.0005 .0105
.0002 .0060
.0001 .0033
.0018
.0009
.0004
.0002
.0001
2021222324
25
272829
JU
31
JJ
34
3536JI
3839
40411Z
4344
4546474849
5051525354
5556575859
6061oz6364
6566676869
Table 2 Cumulative Poisson Probabilities
The table gives the probability tltat r or more random events are contained in an interval when the average number ofsuclr events per interval is m, i.e.
€ -^f f i '
* ' ' !
Where there is no entry for a particular pair of values of r and m, this indicates that the appropriate probability is lessthan 0.000 05. Similarly, except for the case," = 0 when the eniry is exact, a tabulated value of 1.0000 represents aprobability greater than 0.999 95.
1.00.90.80.70.60.50.40.30.20.1
r:0I234
567
L0000 1.0000.0952 .1813.0047 .0175.0002 .001 I
.0001
1.0000.2592.0369.0036.0003
1.0000 1.0000.3297 .3935.0616 .0902.0079 .0144.0008 .0018
.0001 .0002
1.0000 1.0000.4512 .5034.12t9 .1558.0231 .0341.0034 .0058
.0004 .0008.0001
1.0000.5507.t9t2.0474.0091
.0014
.0002.0023.0003
1.0000 1.0000.5934. .6321.2275 .2642.0629 .0803.0135 .0190
.0037
.0006
.0001
2.01.91.8r .71.61.5r .41.3t .2tn= l . l
r : 0 1.0000| .667 |2 .30103 .09964 .0257
1.0000 1.0000.6988 .727s.3374 .3732.1205 .t429.0338 .0431
.0077 .0107
.0015 .0022
.0003 .0004.0001
1.0000 1.0000.7s34 .7769.4082 .4422.1665 .19t2.0537 .0656
.0143 .0186
.0032 .0045
.0006 .0009
.0001 .0002
L0000 1.0000.7981 .8t73.47s1 .5068.2166 .2428.0788 .0932
1.0000 1.0000.8504 .8647.5663 .s940.7963 .3233.1253 .t429
.0441 .05270132 .0166
.0034 .0045
.0008 001I
.0002 .0002
1.0000.8347.5372.2694.1087
56789
.0054
.0010
.0001
.0237 .0296 .0364
.0060 .0080 .0104
.0013 .0019 .0026
.0003 .0004 .0006.0001 .0001
3.02.92.82.72.62.52.42.32.22.1
r = 0 1.0000| .87752 .62043 .3s044 .16t4
5 06216 .07047 .00598 .00l-59 .0003
1.0000 1.0000.8892 .8997.6454 .6691.3773 .4040. I tr06 .2007
0725 0838.0249 .0300.0075 .0094.0020 .0026.0005 .0006
.0001 .0001
t.0000 1.0000.9093 .9179.6916 .7 t27.4303 .4562.22t3 .2424
.0959 .1088
.0357 .0470
.01t6 .0t42
.0033 .0042
.0009 .001I
.0002 .0003.0001
1.0000 1.0000.9257 .9328.7326 .7513.4816 .5064.2640 .28s9
.1226 t37l
.0490 .0567
.0172 .0206
.0053 .0066
.0015 .0019
.0004 .0005
.0001 .0001l0 0001l lt2
1.0000.9392.7689.5305.3081
.1523
.0651
.0244
.0081
.0024
.0007
.0002
1.0000 1.0000.9450 .9502.7854 .8009.5540 .5768.3304 .3528
.1682 t847
.0742 .0839
.0287 .0335
.0099 .0119
.0031 .0038
.0009 .0011
.0002 .0003
.0001 .0001
Table 2 Cumulative Poisson Probabilities - continued
4.03.93.83.73.63.53.43.33.23.1
r:0I2J
4
)6789
1.0000.9550.8153.5988.3752
.2018
.0943
.0388
.0142
.0047
.0014
.0004
.0001
1.0000.9592.8288.6201.3975
.2194
.1054
.0446
.0168
.0057
.0018
.0005
.0001
1.0000.9631.8414.6406.4197
.2374
. l l7 l
.0510
.0198
.0069
.0022
.0006
.0002
1.0000.9666.8s32.6603.44t6
.2558
.1295
.0579
.023r
.0083
.0027
.0008
.0002
.0001
1.0000.9698.8641.6792.4634
.2746
.r424
.0653
.0267
.0099
.0033
.0010
.0003
.0001
1.0000.9721.8743.6973..4848
.2936
.1559
.0733
.0308
.0rt7
.0040
.0013
.0004
.0001
1.0000.9753.8838.7t46.5058
.3128
.1699
.0818
.0352
.0137
.0048
.0016
.0005
.0001
1.0000.9776.8926.73tt.s265
.3322
.1844
.0909
.0401
.0160
.0058
.0019
.0006
.0002
1.0000 1.0000.9198 .98t7.9008 .9084.7469 .76t9.s468 .5665
.3516 . t i , t tz
.1994 .2149
.1005 .tto7
.0454 .0511
.0185 .0214
.0069 .0081
.0023 .0028
.0007 .0009
.0002 .0003
.0001 .0001
l0l lt2l3t4
5.04.94.84.74.64.54.44.34.24.1
r=0I2J
4
56789
r.0000.9834.9155.7762.5858
.3907
.2307
.12T4
.0s73
.0245
.0095
.0034
.001I
.0003
.0001
1.0000.9850.9220.7898.6046
.4t02
.2469
.1325
.0639
.02'19
.0l l l
.0041
.0014
.0004
.0001
1.0000.9864.9281.8026.6228
.4296
.2633
.t442
.0710
.03t7
.0129
.0048
.0017
.0005
.0002
1.0000.9877.9337.8149.6406
.4488
.2801
.1564
.0786
.0358
.0149
.0057
.0020
.0007
.0002
.0001
1.0000.9889.9389.8264.6577
.4679
.2971
.1689
.0866
.0403
.0171
.0067
.0024
.0008
.0003
.0001
1.0000.9899.9437.8374.6743
.4868
.3t42
.1820
.0951
.04s 1
.0195
.0078
.0029
.0010
.0003
.0001
1.0000.9909,9492.8477.6903
.5054
.3316
.t954
.1040
.0503
.0222
.0090
.0034
.0012
.0004
.0001
1.0000.9918.9523.8575.7058
.s237
.3490
.2092
.1 133
.0558
.0251
.0104
.0040
.0014
.0005
.0001
1.0000 1.0000.9926 .9933.9s61 .9s96.8661 .8753.7207 .7350
.5418 .5595
.366s .3840
.2233 .2378
.t231 .1334
.0618 .0681
.0283 .0318
.0120 .0137
.0047 .0055
.00n .0020
.0006 .0007
.0002 .0002
.0001 .0001
l0l lT2l3T4
l5l6
7.06.86.66.46.26.05.85.65.45,2
r:0I234
l0l lt2l3t4
)6
89
l5l6t7l8t9
1.0000.9945.9658.8912.76t9
.5939
.4191
.2676
.155 I
.0819
.0397
.0t77
.0073
.(ru28
.00I()
(XX)3.0001
1.0000.9955.9711.9052.7867
.6261
.4539
.2983
.1783
.0974
.0488
.0225
.0096
.0038
.0014
.Ofi)5
.0002
.0001
1.0000.9963.9756.9r76.8094
.6579
.4881
.3297
.2030
.1 143
.0591
.0282
.0125
.0051
.0020
.0007
.0002
.0001
1.0000.9970.9794.9285.8300
.6873
.52t7
.3616
.2290
.1328
.0708
.0349
.0160
.0068
.(r027
.0010
.0004
.0001
1.0000.9975.9826.9380.8488
.7149
.5543
.393',7
.2560
.1528
.0839
.0426
.0201
.0088
.0036
.0014
.0005
.0002
.0001
1.0000.9980.9854.9464.8658
.7408
.5859
.4258
.2840
.t741
.0984
.0514
.0250
.0113
.0048
.0019
.0007
.0003
.0001
1.0000.9983.9877.9537.8811
.7649
.6t63
.4577
.3127
.t967
.t t42
.0614
.0307
.0143
.0063
.0026
.0010
.0004
.0001
1.0000.9986.9897.9600.8948
.7873
.6453
.4892
.3419
.2204
.13 14
.0726
.0373
.0t19
.0080
.0034
.0014
.0005
.0002
.0001
1.0000 1.0000.9989 .9991.9913 .9927.9656 .9704.9072 .9182
.8080 .8270
.6730 .6993
.5201 .5503
.37t5 .4013
.2452 .2709
.1498 .1695
.0849 .0985
.0448 .0534
.0221 .0270
.0t02 .0128
.0044 .0057
.0018 .0024
.0007 .0010
.0003 .0004
.0001 .0001
Table 2 Cumulative Poisson Probabilities - continued
9.08.88.68.48.28.07.87.67.47.2m:
r=OI234
202l2223
56789
l0l lt2l3l4
l5l6t7l8l9
1.0000.9993.9939.9745.9281
.8445
.7241
.5796
.431I
.2973
.1904
.1 133
.0629
.{t327
.0159
.00?3
.0031
.0013
.0005
.0002
.0001
1.0000.9994.9949.9781.9368
.8605
.7474
.6080
.4607
.3243
.2r23
.1293
.0735
.0391
.0195
.0092
.0041
.0017
.0007
.0003
.0001
1.0000.9995.9957.98t2.9446
.8751
.7693
.6354
.4900
.3518
.2351
.1465
.0852
.0464
.0238
.0114
.00s2
.0022
.0009
.0004
.0001
1.0000.9996.9964.9839.9515
.8883
.7897
.6616
.5188
.3796
.2589
.1648
.0980
.0546
.0286
.0141
.0066
.0029
.0012
.0005
.0002
.0001
1.0000.9997.9910.9862.9576
.9004
.8088
.6866
.5470
.4075
.2834
. l84l
. l l19
.0638
.0342
.0173
.0082
.0037
.0016
.0006
.0003
.0001
1.0000.9997.9975.9882.9630
.91l3
.8264
.7104
.5746
.4353
.3085
.2045
.1269
.0739
.0405
.0209
.0102
.0047
.0021
.0009
.0003
.0001
1.0000.9998.9979.9900.9677
.92t1
.8427
.7330
.6013
.463r
.3341
.2257
.1429
.0850
.0476
.02s1
.olzs
.0059
.0027
.0011
.0005
.0002
.0001
1.0000.9998.9982.99t4.9719
.9299
.8578
.7543
.62',72
.4906
.3600
.2478
.1600
.0971
.0555
.0299
.0t52
.0074
.0034
.0015
.0006
.0002
.0001
1.0000 1.0000.9998 .9999.9985 .9988.9927 .9938.97s6 .9788
.9379 .9450
.8716 .8843
.7744 .7932
.6522 .6161
.5177 .5443
.3863 .4126
.2706 .2940
.1780 .1970
.rto2 .1242
.0642 .0739
.0353 .0415
.0184 .0220
.0091 .0111
.0043 .0053
.0019 .0024
.0008 .0011
.0003 .0004
.0001 .0002.0001
m= 9.69.49.2 9.8 10.0 11.0 t2.0 13.0 14.0
r=Q1234
56789
l0nt2l3L4
l516^l7t8l9
1.0000.9999.9993.9962.9862
.9622
.9t62
.8426
.74t6
.6204
.491I
.3671
.2588
.t721
. l08l
.0643
.0362
.0194
.0098
.0048
.0022
.0010
.0004
.0002
.0001
1.00001.00001.0000.9998.9990
.9963
.9893
.9741
.9460
.9002
.8342
.7483
.6468
.5369
.4270
.3249
.2364
.t645
.1095
.0698
.0427
.0250
.0141
.0076
.0040
.0020
.0010
.000s
.0002
.0001
1.0000 1.0000.9999 .9999.9990 .999r.9947 .9955.9816 .9840
.9514 .957r
.8959 .9055
.8108 .8273
.6990 .7208
.5704 .5958
.4389 .4651
.3180 .3424
.2t68 .2374
.1393 .1552
.0844 .0958
.0483 .0559
.0262 .0309
.0135 .0162
.0066 .0081
.0031 .0038
.0014 .0017
.0006 .0008
.0002 .0003
.0001 .0001
1.0000 1.0000.9999 1.0000.9994 .9995.9967 .9972.9880 .9897
.9667 .9107
.9250 .9329
.8567 .8699
.7612 .7798
.6442 .6672
.5168 .5421
.3920 .4170
.2807 .3032
.1899 .2084
.12t4 .1355
.0735 .0835
.042r .0487
.0230 .0270
.0119 .0143
.0059 .0072
.0028 .0035
.0012 .0016
.0005 .0007
.0002 .0003
.0001 .0001
1.0000 1.00001.0000 1.0000.9998 .9999.9988 .9995.9951 .9977
.9849 .9924
.962s .9797
.9214 .9542
.8568 .9105
.7680 .84s0
.6595 .7576
.5401 .6528
.420'7 .5384
.3113 .4240
.2187 .3185
.1460 .2280
.0926 .1556
.0559 .1013
.0322 .0630
.0177 .0374
.0093 .0213
.004'7 .0116
.0023 .0061
.0010 .0030
.0005 .0015
.0002 .0007
.0001 .0003.0001.0001
1.0000 1.00001.0000 1.00001.0000 1.0000.9999 1.0000.9995 .9998
.9982 .9991
.9945 .9972
.9858 .9924
.9684 .9820
.9379 .9626
.8906 .9301
.8243 .8815
.7400 .8152
.6415 1324
.5356 .6368
.4296 .5343
.3306 .4319
.2441 .3359
.1728 .2511
.rt" l4 .1805
.076s .1248
.0479 .0830
.0288 .0531
.0167 .0327
.0093 .0195
.0050 .0122
.0026 .0062
.0013 .0033
.0006 .0017
.0003 .0009
.0001 .0004
.0001 .0002.0001
202l222324
2526272829
303l32
l0
Table 2 Cumulative Poisson Probabilities - continued
m= 16.0 t7.0 18.0 19.0 20.0 2t.0 22.0 23.0 24.0 25.0
r=0I234
567E9
1.0000 1.0000 1.0000I.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.0000.9999 1.0000 1.0000
.9996 .9998 .9999
.9986 .9993 .9997
.9960 .9979 .9990
.9900 .9946 .9971
.9780 .9874 .9929
.9567 .9739 ,9846
.9226 .9509 .9696
.8730 .9153 .9451
.8069 .8650 .9083
.7255 .7991 ,8574
.6325
.5333
.4340
.3407
.2577
.1 878
.13 18
.0892
.0582
.0367
.0223
.0131
.0075
.0041
.0022
.001I
.0006
.0003
.0001
.0001
.7192 .7919
.6285 .7t33
.5323 .6249
.4360 .53t4
.3450 .4378
.2637 .349t
.194s .2693
.1385 .2009
.0953 .t449
.0633 .1011
.0406 ,0683
.0252 .0446
.0ts2 .0282
.0088 .0t73
.0050 .0103
1.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.0000
1.0000 1.0000.9998 .9999.9995 .9997.9985 .9992.9961 .9979
.9911 .9950
.9817 .9892
.9653 .9786
.9394 .9610
.9016 .9339
.8503 .8951
.7852 .8435
.7080 .7789
.62t6 .t030
.5305 .5186
.4394 .5297
.3528 .4409
.2745 .3553
.2069 .2794
.1510 .212s
.1067 .1568
.0731 .tt22
.0486 .0779
.0313 .0525
.0195 .0343
1.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.00001.0000 1.0000 1.0000
1.0000 1.0000 1.0000r.0000 1.0000 1.0000.9999 .9999 1.0000.9996 .9998 .9999.9989 .9994 .9997
9972 .9985 .9992.9937 .9965 .9980.9871 .9924 .9956.9755 .9849 .9909.9s66 .9722 .9826
.9284 .9523 .9689
.8889 .923t .9480
.8371 .8830 .9179
.7730 .8310 .8772
.6983 .7675 .82s2
.6157 .6940 .7623
.s290 .6131 .6899
.4423 .5284 .6106
.3595 .4436 .5277
.2840 .3626 .4449
.2178 .2883 .3654
.1623 .2229 .2923
.1t74 .1676 .2277
.0825 .t225 .1726
.0564 .0871 .t274
.0374
.0242
.0152
.0093
.0055
.0032
.0018
.0010
.0005
.0003
.0001
.0001
.0602
.0405
.0265
.0169
.0105
.0064
.0038
.0022
.0012
.0007
.0004
.0002
.0001
.0915
.0640
.0436
.0289
.0187
.0118
.0073
.0044
.0026
.0015
.0008
.0004
.0002
.0001
.0001
1.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.0000
1.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.0000.9998 .9999
.9996 .9998
.9989 .9994
.9975 .9986
.9946 .9969
.9893 .9935
.9802 .9876
.9656 .9777
.9437 .9623
.9129 .9395
.87t7 .9080
.8t97 .8664
.7574 .8145
.6861 .7527
.6083 .6825
.5272 .6061
.4460 .5266
.3681 .4471
.2962 .3706
.2323 .2998
.1775 .2366
.1321 .t821
.0958 .1367
.0678 .1001
.0467 .07t5
.0314 .0498
.0206 .0338
.0t32 .0225
.0082 .0t46
.0050 .0092
.0030 .0057
.0017 .0034
.0010 .0020
.0005 .0012
.0003 .0007
.0002 .0004
.0001 .0002.0001
l0l lT2l3t4
15t6t7l8l9
202l222324
2526272829
303l323334s
3536373839
404l424344
4546
.0027
.0014
.0007
.0004
.0002
.0001
.0059
.0033
.0018
.0010
.0005
.0002
.0001
.0001
.01l8
.0070
.0040
.0022
.0012
.0006
.0003
.0002
.0001
.0218
.0135
.0081
.0047
.0027
.0015
.0008
.0004
.0002
.0001
.0001
l l
Table 2 Cumulative Poisson Probabilities - continued
m= 26.0 27.0 28.0 29.O 30.0 32.0 34.0 36.0 38.0 40.0
r = 9 1.0000l0 .999911 .9991t2 .999213 .9982t4 .9962
.9924
.9858
.9752
.9580
.9354
.9032
.8613
.8095
.1483
.6791
.6041
.5261
.4481
.3730
.3033
.2407
.1866
.1411
.1042
.0751
.0528
.0363'.0244
.0160
.0103
.0064
.0039
.0024
.0014
.0008
.0004
.0002
.0001
.0001
.99s4
.9912
.9840
.9726
.9555
.9313
.8985
.8564
.8048
.7441
.6158
.602t
.5256
.449r
.3753
.3065
.2447
.1908
.1454
.1082
.0787
.0559
.0388
.0263
.0r75
.0113
.0072
.0045
.0027
.0016
.0009
.0005
.0003
.0002
.0001
1.00001.0000.9999.9998.9994.9987
.9973
.9946
.9899
.982r
.9700
.9522
.9273
.8940
.8517
.8002
.7401
.6728
.6003
.5251
.4500
.3774
.3097
.2485
.1949
.t495
.tt2l
.0822
.0589
.0413
.0283
.0190
.0t25
.0080
.0050
.0031
.0019
.0011
.0006
.0004
.0002
.0001
.0001
1.00001.00001.0000.9999.9997.9993
.9984
.9967
.9937
.9885
.9801
.9614
.9489
.9233
.8896
.8471
.7958
.7363
.6699
.5986
.5247
.4508
.3794
.3126
.2521
.1989
.1535
.1 159
.0856
.0619
.0438
.0303
.0205
.0136
.0089
.0056
.0035
.0022
.0013
.0008
.0004
.0002
.0001
.0001
1.00001.00001.0000.9999.9998.9996
.9991
.9981
.9961
.9927
.9871
.9781
.9647
.9456
.9t94
.88s4
.8428
.7916
.7327
.6671
.s969
.5243
.4516
.3814
.3155
.2556
.2027
.t574
.1196
.0890
.0648
.0463
.0323
.0221
.0148
.0097
.0063
.0040
.0025
.0015
.0009
.0005
.0003
.0002
.0001
.0001
1.00001.00001.00001.00001.0000.9999
.9997
.9993
.9986
.9912
.9948
.9901
.9841
.9740
.9s94
.9390
.9119
.8772
.8344
.7838
.1259
.6620
.5939
.5235
.4532
.3850
.3208
.262r
.2099
.1648
.1268
.0956
.0707
.0512
.0364
.0253
.0173
.0116
.0076
.0049
.0031
.0019
.0012
.0007
.0004
.0002
.0001
.0001
1.00001.00001.00001.00001.00001.0000
.9999
.9998
.9995
.9990
.9980
.9963
.9932
.9884
.9809
.9698
.9540
.9326
.9047
.8694
.8267
.7765
.7196
.6573
.591I
.5228
.4546
.3883
.3256
.2681
.2r66
.t717
.r336
.1019
.0763
.0561
.0404
.0286
.0199
.0136
.0091
.0060
.0039
.0024
.0015
.0009
.0006
.0003
.0002
.0001
.0001
1.00001.00001.00001.00001.00001.0000
1.0000.9999.9998.9997.9993
.9986
.9913
.9951
.991s
.9859
.9176
.9655
.9487
.9264
.8977
.8621
.8194
.7697
.7139
.6530
.5885
.5222
.4558
.3913
.3301
.2737
.2229
.1783
.1401
.1081
.0819
.0609
.0445
.0320
.0225
.0156
.0106
.0071
.0047
.0030
.0019
.0012
.0007
.0005
.0003
.0002
.0001
.0001
1.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.00001.0000 1.0000
1.0000 1.00001.0000 1.00001.0000 1.0000.9999 1.0000.9998 .9999
.9995 .9998
.9990 .9996
.9981 .9993
.996s .9986
.9938 .9974
.9897 .9955
.9834 .9924
.9141 .9871
.9611 .9807
.9435 .9706
.9204 .9568
.8911 .9383
.8552 .9t45
.8125 .8847
.1635 .8486
.7086 .8061
.6490 .7516
.5862 .7037
.52t6 .6453
.4s70 .5840
.394r .szr0
.3343 .4581
.2789 .3967
.2288 .3382
.1845 .2838
.1462 .2343
.1139 .1903
.0872 .rszr
.06s7 .1196
.0486 .0925
.0353 .0703
.0253 .0526
.0178 .0387
.0123 .0281
.0084 .0200
.0056 .0140
.0037 .0097
.0024 .0066
.0015 .0044
.0010 .0029
.0006 .0019
.0004 .00t2
.0002 .0008
.0001 .0005
.0001 .0003
.0002
.0001
.0001
l5t6t7t819
202T222324
2526272829
303132JJ
34
3536JI
3839
404T424344
4546474849
5051{ t
5354
))56JI
5859
606l626364
656667
For values of m greater than 30, use the table of areas under the Normal curve (Table 3) to obtain approximate Poisson
probabilities, putting p= m and o=Ji .
tz
Table 3 Areas in upper TaiI of the Normal Distribution
The function tabulated its l - <D(z) wtlerc <D(z) is the cumulative distribution function of a standardised Normal variable, z.
rhus 1-o(z)=#1"
varueorz(=+)
- .2 tcis the probabilitythat a standardised Normal variate selected at random will be greater than a
o (")
.09.08.07.06.05.04.03.02.01x- p .00o
0.00.10.20.30.4
0.50.60.70.80.9
1.0t . lt .21.31.4
1.5t .6r .71.81.9
2.02.12.22.32.4
2.52.62.72.82.9
3.03.13.23.33.4
3.53.63.7
.5000
.4602
.4207
.3821
.3446
.3085
.2743
.2420
.2119
.1841
.1s87
.t35'7
.1151
.0968
.0808
.0668
.0548
.0446
.0359
.0287
.02275
.01786
.01390
.0t072
.00820
.00621
.00466
.00347
.00256
.00187
.00135
.00097
.00069
.00048
.00034
.00023
.00016
.000108
.000072
.000048
.000032
.4960
.4562
.4168
.3783
.3409
.3050
.2709
.2389
.2090
.18 14
.1562
.1335
.1 13l
.0951
.0793
.0655
.0537
.0436
.0351
.0281
.02222
.0r743
.01355
.01044
.00798
.00604
.00453
.00336
.00248
.00181
.00131
.00094
.00066.
.00047
.00032
.00022
.00015
.000104
.000069
.000046
.4920
.4522
.4t29
.3745
.3372
.3015
.2676
.2358
.2061
.1788
.1539
.13t4
.ttt2
.0934
.0778
.0643
.0526
.0427
.0344
.0274
.02169
.01700
.01321
.01017
.00776
.00587
.00440
.00326
.00240
.00175
.00126
.00090
.00064
.00045
.00031
.00022
.00015
.000100
.000067
.000044
.4880
.4483
.4090
.3707
.3336
.2981
.2643
.2327
.2033
.t762
.15 l5
.1292
.1093
.0918
.0764
.0630
.0516
.0418
.0336
.0268
.021l8
.01659
.01287
.00990
.00755
.00570
.00427
.00317
.00233
.00169
.00122
.00087
.00062
.00043
.00030
.00021
.00014
.000096
.000064
.000042
.4840
.4443
.4052
.3669
.3300
.2946
.26t1
.2296
.2005
.t736
.r492
.1271
.t075
.0901
.0749
.0618
.0505
.0409
.0329
.0262
.02068
.01618
.01255
.00964
.00734
.00554
.00415
.00307
.00226
.00164
.00118
.00084
.00060
.40042
.00029
.00020
.00014
.000092
.000062
.000041
.4801
.4404
.4013
.3632
.3264
.2912
.2578
.2266
.r977
.t7lL
.1469
.1251
.1056
.0885
.0735
.0606
.4495
.0401
.0322
.0256
.02018
.01578
.01222
.00939
.00714
.00539
.00402
.00298
.00219
.00159
.00114
.00082
.000s8
.00040
.00028
.00019
.00013
.000088
.000059
.000039
.4761
.4364
.3974
.3594
.3228
.2877
.2546
.2236
.t949
.1685
.t446
.1230
.1038
.0869
.0721
.0594
.0485
.0392
.0314
.0250
.01970
.01539
.01l9l
.00914
.00695
.00523
.00391
.00289
.00212
.00154
.001I I
.00079
.000s6
.00039
.00027
.00019
.00013
.000085
.000057
.000037
.4721
.4325
.3936
.3557
.3192
.2843
.2514
.2206
.1922
.1660
.1423
.t210
.1020
.0853
.0708
.0582
.0475
.0384
.0307
.0244
.01923
.01500
.01160
.00889
.00676
.00508
.00379
.00280
.00205
.00149
.00107
.00076
.00054
.00038
.00026
.00018
.00012
.000082
.000054
.000036
.4681
.4286
.3897
.3520
.3156
.28t0
.2483
.2177
.1894
.1635
.1401
.1 190
.1003
.0838
.0694
.0571
.0465
.0375
.0301
.0239
.01876
.01463
.01130
.00866
.00657
.00494
.00368
.00272
.00199
.00144
.00104
.00074
.00052
.00036
.00025
.00017
.00012
.000078
.000052
.000034
.4641
.4247
.3859
.3483
.3t21
.2776
.2451
.2148
.1867
.161 I
.r379
.1t70
.0985
.0823
.0681
.0559
.0455
.0367
.0294
.0233
.01831
.01426
.01101
.00842
.00639
.00480
.00357
.00264
.00193
.00139
.00100
.00071
.00050
.00035
.00024
.00017
.0001I
.000075
.000050
.000033
3.83.9
4.0
5.0 -+ 0.000 000 286 7
l3
6.0 -+ 0.000 000 001 05.5 -+ 0.000 000 019 0
Table 4 Percentage Points of the Normal Distribution
The table gives the l00a percentage points, zo of a standardised normal distribution whsre
Thus zo is the value of a standardised normal variate which has probability a of being exceeded.
o = Ll* ,-,2124, ..lZt Jzo
zad. zad zaq zaq q za zdq
.50 0.0000
.45 0.t257
.40 0.2533
.35 0.3853
.30 0.5244
.25
.20
.15
.10
.05
0.67450.84161.03641.28161.6449
.050 1.6449
.048 1.6646
.046 t.6849
.044 1.7060
.042 13279
.040 1.7507
.038 t.7744
.036 t.7991
.034 1.8250
.032 1.8522
.030 1.8808
.029 1.8957
.028 1.9910
.027 1.9268,026 t.9431
1.9600t.97741.99542.0t412.0335
02502402302202r
.020 2.0537
.019 2.0749
.018 2.0969
.017 2.1201
.016 2.1444
.015 2.r70r
.0t4 2.t973
.013 2.2262
.012 2.2571
.0l l 2.2904
.010 2.3263
.009 2.3656
.008 2.4089
.007 2.4573
.006 2.5t21
005004003002001
2.57582.65212.74782.87823.0902
.050 1.6449
.010 2.3263
.001 3.0902
.000 1 3.7190
.000 0l 4.2649
.025 1.9600
.005 2.5758
.000 5 3.2905
.000 05 3.8905
.000 005 4.4t72
Table 5 Ordinates of the Normal Distribution
The table gsves SQ) for values of the standardised normat variate, z, inthe interval 0.0 (0.1) 4.0 where
obl=L"-t,,.l2tt
.8.7.6.5.4.J.2. t.0
0.0 .39891.0 .24202.0 .05403.0 .00444.0 0001
.3970
.2179
.0440
.0033
.3910
.r942
.0355
.0024
.3814
.17t4
.0283
.0017
.3683
.t497
.0224
.0012
.352t
.r295
.017s
.0009
.3332
.1 109
.0136
.0006
.3123
.0940
.0104
.0004
.2897 .2661
.0790 .0656
.0079 .0060
.0003 .0002
I4
Table 7 Percentage Points of the t Distribution
The table gives the value of lo u - the l00a percentage point of the r distribution for v degrees of freedom.
The values of r are obtained by solution of the equation:
a -- f[%(v+ t)]Ir(%v)l-' (ro)-'o
Note: The tabulation is for one tail only, that is, for positive values of l.For i r I the column headings for o should be doubled.
t a,v
I?t . x' / vy-<'*')rz *
0.10 0.05 0.025 0.01 0.005 0.001 0.0005
v: l2J
45
3.0781.8861.6381.533t.476
t.4401.4151.3971.383r.372
1.3631.3561.3501.3451.341
1.3371.3331.3301.328r.325
1.3231.3271.3191.3 l81.316
1.3 151.3141.3131.3111.310
1.303t.296L2891.282
6.3t42.9202.3532.1322.0t5
r.9431.8951.8601.8331.812
r.796t.782r.7711.761t.753
r.7461.740r.7341.729r.725
1.72rt .717t.714t.7lr1.708
t.7061.703|.7011.699t.697
1.6841.6711.6581.645
12j064.3033.1822.7762.57r
2.4472.3652.3062.2622.228
2.2012.1792.1602.r452.131
2.t202.IIO2. l0 l2.0932.086
2.0802.0742.0692.0642.060
2.0562.0522.0482.0452.042
2.02r2.0001.9801.960
3r.8216.9654.5413,7473.365
3.t432.9982.8962.8212.764
2.7t82.6812.6502.6242.602
2.5832.5672.5522.5392.528
2.5182.5082.5002.4922.485
2.4792.4732.4672.4622.457
2.4232.3902.3582.326
63.6579.9255.8414.6044.032
3.7073.4993.3553.2503.t69
3.1063.0553.0t22.9772.947
2.9212.8982.8782.86r2.845
2.8312.8192.8072.7972.787
2.7792.7712.7632.7562.750
2.7042.6602.6172.576
318.3122.326r0.2137.t735.893
5.2084.7854.5014.2974.144
4.0253.9303.8523.7873.733
3.6863.6463.6103.5793.552
3.5273.5053.4853.4673.450
3.4353.4213.4083.3963.3 85
3.3073 2323.1603.090
636.6231.598t2.9248.6106.869
5.9595.4085.0414.7814.587
4.4374.3184.22r4.1404.073
4.0153.9653.9223.8833.850
3.8193.7923.7673.7453.72s
3.7073.6903.6743.6593.646
3.55 I3.4603.3733.291
6789
l0
ilT2l3T4l5
t6t7l81920
2l22232425
2627282930
4060
r20
This table is taken from Table III of Fisher & Yates: Statistical Tables for Biological, Agicultural and Medical Research,reprinted by permission of Addison Wesley Longman Ltd. Also from Table 72 of Biometrika Tables for Statisticians.Volume I , by permission of Oxford University Press and the Biomehika Trustees.
l7
Table 8 Percentage Points of the f Distribution
Table of f o, - the l0Oapercentage point ofthe fdisribution for vdegrees of freedom.
r'o.,
a= .995 .99 .98
v: | .04393 .o3ls7 .036282 .0100 .020t .04043 .0717 .115 .1854 .201 .291 .4295 .4t2 .554 .752
.00393 .0158 .0642 .r02 .148
.103 .ztt .446 .s75 .7r3
.352 .584 1.005 t.2t3 t.424
.7ll l.064 1.649 1.923 2.19s1.145 1.610 2.343 2.675 3.000
1.635 2.204 3.070 3.455 3.8282.t67 2.833 3.822 4.255 4.6712333 3.490 4.594 5.071 5.5273.325 4.168 5.380 5.899 6.3933.940 4.865 6.179 6.731 7.267
.975 .95 .70.75.90
.03982
.0506
.216
.484
.831
.4551.3862.3663.3574.351
5.3486.3467.3448.3439.342
6 .676 .872 1.134 1.2377 .989 t.239 r.564 1.6908 1.344 1.646 2.032 2.1809 1.735 2.088 2.s32 2.700
l0 2.156 2.558 3.059 3.241
11 2.60312 3.07413 3.56514 4.01515 4.601
16 5.142t7 5.69118 6.26519 6.84420 7.434
3.053 3.609 3.8163.571 4.t78 4.4044.101 4.76s 5.0094.660 5.368 5.6295.229 5.985 6.262
5.812 6.614 6.9086.408 7.255 1.s647.Ots 1.906 8.2317.633 8.561 8.9078.260 9.237 9.591
4.515 5.578 6.989 7.584 8.148 10.3415.226 6.304 7.807 8.438 9.034 11.3405.892 1.042 8.634 9.299 9.926 12.3&6.s7r 7 .790 9.467 10.16s 10.821 13.3397.26t 8.541 10.307 11.036 tI.72t 14.339
2t 8.034 8.897 9.9t5 10.28322 8.643 9.542 10.600 10.98223 9.260 10.196 tt.293 11.68824 9.886 10.856 L1.992 t2.40125 10.s20 1t.524 t2.697 13.120
ll.59l t3.240 t5.44s 16.344 I7.tS2 20.33712.338 t4.041 t6.314 t7.240 18.101 2t.33713.091 14.848 17.t87 18.137 t9.O2t 22.33713.848 15.659 18.062 19.037 t9.943 23.337t4.6tt t6.473 18.940 19.939 20.867 24.337
7.962 9.3128.612 10.08s9.390 10.865
10.117 11.65110.85r t2.443
15.379 t7.292 19.82016.151 18.114 20.703t6.928 18.939 21.588t7.708 19.768 22.47518.493 20.599 23.364
ll.l52 Lt.9t2 t2.624 15.33Er2.oo2 t2192 13.531 16.33812.857 t3.675 t4.440 17.33813.716 14.s62 ts.352 18.338r4.s78 15.452 t6.266 19.337
26 11.160 12.19827 11.808 12.87928 t2.461 13.56529 13.t2r t4.2s630 13.787 t4.953
13.409 t3.84414.125 14.573t4.847 15.30815.514 16.04716.306 t6.791
65.64614.222
20.843 2t.79221.749 22.71922.657 23.64723.567 24.57724.478 25.508
25.33626.33627.33628.33629.336
40 20.706 22.t64 23.838 24.433s0 27.991 29.107 31.664 32.35760 3553s 37.485 39.699 40.48270 43.27s 45.442 47.893 48.75880 5t.t7l 53.539 56.213 51.L53
26.509 29.05t 32.345 33.660 34.872 39.33534.764 31.689 41.449 42.942 44.313 49.33s43.188 46.459 50.641 52.294 53.809 59.3355t.739 55.329 59.898 61.698 63.346 69.33460.391 64.278 69.207 1I.t45 72.9t5 79.334
69.126 73.291 78.558 80.625 82.51I 89.33477.929 82.358 87.945 90.133 92.t29 99.334
90 59.196 6t.754 64.634100 67.327 70.065 73.142
normal deviate cutting off the corresponding tails of a normal distribution. f It taken at the 0.02 level. so that 0.01o
normal distribution is in each tail, the expressionyields f atthe 0.99 and 0.01 points.
For very large values of v, it is sufficiently accurate to compute l*, the distribution of which is approximately
around a mean of ,,!2v - I and with a standard deviation of l.Tabfe taken liom Table IV of Fisher and Yates: Statistical Tables for Biological, Agricultural and Medical Research, reprintod by permission of AddisonLnngman Ltd, and from Table tl of Biometrika Tables for Statisticians,VoL l, by permission of Oxfbrd Uuiversity Pre,ss and the Biometrika Trustees.
l8
Table 8 Percentage Points of the f Distribufion - continued
001.01.05 .025 .02.10.20.25
I -r-.r t.323I 4rl8 2.773_: 665 4.108I 878 5.385r 064 6.626
:.231 7.8418. i83 9.0379 524 10.219
I ,656 11.389. l ?81 12.549
It 899 13..701r+ 0l l 14.845r. i l19 15.984\6 222 t7.t t7'-- 322 18.245
r 5.418 19.369-9., i I I 20.489:, 601 21.605: r 589 22.718:t --5 23.828
: i 858 24.935:.r 939 26.039:5 1l 18 27 . t4lt- ':196 28.241:s 1-2 29.339
:i :{6 30.434: -, -i 19 31.528- ' i l9 l 32.620-"1+bi 33.71I-'-: 5-: r 34.800
r-+ tr6_< 45.616-;-r -tl 56.334;, i l : - 66.981-_r 689 77.577...t i5 l : { ) 88.130
;r -51-t 98.650I . 'b 9f6 109.141
1.6423.2194.6425.9897.289
8.5589.803
l1.030r2.242t3.442
14.63115.81216.98518. l5 I19.311
20.46s2t.6ts22.76023.9002s.038
26.t7127.30128.42929.55330.675
3t.79532.91234.02735.13936.250
47.26958.16468.97279.7 t590.405
101.054Ilt.667
2.7064.6056.2517.7799.236
10.64s12.0t713.36214.68415.987
17.27518.54919.8122t.06422.307
23.54224.76925.98927.20428.412
29.6t530.81332.00733.19634.382
35.56336.74137.91639.08740.256
51.80563.16774.39785.52796.578
107.565118.498
3.8415.9917.8159.488
t 1.070
12.59214.06715.50716.91918.307
t9.67s2r.02622.36223.68s24.996
26.29627.s8728.86930.t4431.410
32.6'7133.92435.17236.41537.652
38.88540. I 1341.33742.55743.773
ss.75967.s0s79.08290.531
101.880
I 13.145r24.342
5.0247.3789.348
l 1.143t2.832
t4.44916.01317.s3519.02320.483
21.92023.33724.73626.1I927.488
28.84530. t9 l31.52632.8s234.170
35.47936.78138.07639.36440.646
41.92343.19444.46I45.72246.979
s9.3427r.42083.29895.023
t06.629
I 18.136t29.561
5.4127.8249.837
l1.66813.388
15.03316.62218.16819.6792t. t6I
22.61824.05425.47226.87328.2s9
29.63330.99532.34633.68735.020
36.34337.65938.96840.27041.566
42.85644.14045.4t946.69347.962
60.43672.6t384.58096.388
108.069
119.648t3r.I42
6.6359.210
r1.34513.27715.086
16.812t8.47520.09021.66623.209
24.72526.2r727.68829.14130.578
32.00033.40934.80536.1 9 137.566
38.93240.28941.63842.98044.314
45.64246.96348.27849.58850.892
63.69176.15488.379
t00.425112.329
124.176135.807
7.87910.59712.83814.86016.750
18.54820.2782t.95s23.58925.188
26.75728.30029.81931.31932.801
34.26735.71837.15638.58239.997
41.40142.79644. l8 l45.55846.928
48.29049.64550.99352.33653.672
66.76679.49091.952
t04.2t5lt6.32L
128.299140.170
10.827 v: I13.815 216.268 318.465 420.5t7 5
22.457 624.322 726.125 827.877 929.588 10
31.264 1l32.909 t234.528 1336.123 1437.697 15
39.2s2 1640.790 t742.312 1843.820 1945.315 20
46.79748.26849.7285L17952.620
s4.052s5.47656.89358.302s9.703
73.40286.66199.607
rt2.3t7124.839
2l22232425
2627282930
4050607080
137.208 90149.449 100
t9
Table 9 Percentage Points of the F Distribution
The table gives the vatu€s d Fo;\,r, tle l00a p€rD€trfiagE poid d the F ,tiarihtioo having yi degrees of freedom in the
num€f,ator and n degrees dfreedm ia the d.ryrnirr*r. Fc "-+r trir dvatu€s 6 vlnul v2, F=a;v1,"2is tabulated for a = 0.05,
O.O25,0.01, 0.001, the 0.025 valpes brng hactsEtThe low percemags Poiffi d6e fi$rilrrirrr 6sy be ffifromthe relaion:
Fr-orur,u, =l / F.,;rr,t,
Exaryle: F.cs,r2,s = t/F.*,r,r, :r lz15 = 0..3,5.L Fotrr,r,
t61.4 199.5 2t5.7(648) (800) (864)4052 5000 54p340531 5000* 5405*
224.6 230.2(e00) (n2)5625 57U5625* 57U*
t9.2 19.3(3e.2) (3e.3)99.2 99.3
999.2 999.3
9.t2 9.01(15.1) (14.e)28.7 28.2
t37.1 t34.6
6.39 6.26(e.60) (e.36)16.0 15.553.44 51.71
5.19 5.05(7.3e) (7.15)11.39 10.9731.09 29.75
4.53 4.39(6.23) (s.ee)9.15 8.7s
2r.92 20.80
4.12 3.97(s.sz) (s.2e)7.85 7.46
17.20 t6.21
3.84 3.69(s.05) (4.82)7.0r 6.63
14.39 13.48
3.63 3.48(4.72) (4.48)6.42 6.06
t2.56 tl.1l
3.48 3.33(4.47) (4.24)5.99 5.U
11.28 10.48
3.36 3.20(4.28) (4.04)5.67 5.32
10.35 9.58
3.26 3. l l(4.r2) (3.8e)5.41 5.069.63 8.89
2y.0 236.8(e37) (%8)5859 5y)85859* 5929'
19.3 19.4(3e.3) (3e.4)99.3 99.4
999.3 999.4
8.94 8.89(14.7) (14.6)27.9 27.7
132.8 l3 l .s
6.16 6.09(e.20) (e.07)15.2 ls.050.53 49.66
4.95 4.88(6.e8) (6.85)t0.67 10.4628.83 28.t6
4.28 4.21(s.82) (s.70)8.47 8.26
20.03 19.46
3.87 3.79(s.rz) (4.ee)7.19 6.99
15.52 t5.O2
3.58 3.50(4.6s) (4.s3)6.37 6.18
12.86 12.40
3.37 3.29(4.32) (4.20)5.80 5.61
I l . l3 10.69
3.22 3.14(4.07) (3.es)5.39 5.209.93 9.52
3.09 3.01(3.88) (3.76)5.07 4.899.05 8.66
3.00 2.91(3.73) (3.61)4.82 4.648.38 8.00
241.9 243.9(%e) Qm6056 61066056* 6107.
24e.0 254.3(ee7) (1018)6235 63666235* 6366*
19.5 19.5(3e.s) (3e.5)99.5 99.5
999.5 999.5
8.U 8.53(14.1) (13.e)26.6 26.1
125.9 123.5
5.77 5.63(8.51) (8.26)13.9 13.545.77 44.0s
4.53 4.36(6.28) (6.02)9.47 9.02
25.14 23.79
3.84 3.67(s.12) (4.85)7.3t 6.88
16.90 15.75
3.41 3.23(4.42) (4 14)6.07 5.65
12.73 1t.70
3.12 2.93(3.e5) (3.67)5.28 4.86
10.30 9.34
2.90 2.71(3.61) (3.33)4.73 4.318.72 7.81
2.74 2.54(3.37) (3.08)4.33 3.917.64 6.76
2.61 2.40(3.17) (2.88)4.02 3.606.85 6.00
2.51 2.30(3.02) (2.72)3.78 3.366.25 5.42
23t.9(es7)59815981.
18.5(38.5)98.5
998.5
10.13 9.55 9.28(r7.4) (16.0) (15.4)34.1 30.8 29.5
167.0 148.5 l4l.l
7.71 6.94 6.5902.n) (10.6s) (e.e8)21.2 18.0 t6.774.14 61.25 56.18
6.61 5.79 5.41(r0.0r) (8.43) (7.76)16.26 13.27 12.0647.18 37.12 33.20
s.99 5.14 4.76(8.8r) (7.26) (6.60)13.74 rO.92 9.7835.51 27.00 23.70
19.0 t9.2(3e.0) (3e.2)99.0 99.2
999.0 999.2
5.59(8.0712.2529.2s
4.74 4.35(6.54) (5.8e)9.55 8.45
2t.69 t8.77
5.32 4.46 4.07(7.s7) (6.06) (s.42)11.26 8.65 7.5925.42 18.49 15.83
5.12 4.26 3.86(7.2r) (5.71) (5.08)10.56 8.02 6.9922.86 16.39 13.90
4.96 4.10 3.71(6.e4) (s.46) (4.83)10.04 7.56 6.552t.04 14.91 r2.s5
4.84 3.98 3.59(6.72) (s.26) (4.63)9.65 7.21 6.22
19.69 13.81 I 1.56
4.75 3.89 3.49(6.s5) (s.10) (4.47)9.33 6.93 5.95
18.64 t2.97 10.80
t9.4 r9.4 r9.4(3e.4) (3e.4) (3e.4)99.4 99.4 99.4
999.4 999.4 999.4
8.85 8.79 8.74(14.s) (r4.4) (14.3)27.5 27.2 27.1
130.6 129.2 128.3
6.04 5.96 5.91(8,e8) (8.84) (8.75)14.8 t4.s t4.449.00 48.05 47.4r
4.82 4.74(6.76) (6.62)10.29 10.0527.65 26.92
4.15 4.06(s.60) (s.46)8.10 7.87
19.03 18.41
3.73 3.64(4.e0) (4.76)6.W 6.62
14.63 14.08
4.68(6.s2)9.89
26.42
4.00(s.37)7.72
17.99
3.57(4.67)6.47
13.7r
3.44 3.35 3.28(4.43) (4.30) (4.20)6.03 5.81 s.67
12.05 11.54 l l . l9
3.23 3.14 3.07(4.10) (3.e6) (3.87)5.47 5.26 s. l l
10.37 9.87 9.57
3.07 2.98 2.91(3.ss) (3.72) (3.62)5.06 4.85 4.719.20 8.74 8.44
2.95 2.85 2.79(3.66) (3.s3) (3.43)4.74 4.54 4.408.35 7.92 7.63
2.85 2.75 2.69(3.5r) (3.37) (3.28)4.50 4.30 4.167.71 7.29 7.00
l0
l l
t2
* Entries marked thus must be multiplied by 100
Table 9 Percentage points of the F Distribution - continued
l0
22", ,: :1. ,'^':!, .l ::. .: g:. 2s? 2 83 2.77 2.67 2 60(6 4t) (4 e7) (.1sr (4.00) a.77) (3 60) ri.iit rl rh ,iir, ,3 f3, ,3.17, ,121,9.07 6-70 5.74 5.21 4.86 '4.62' '4.44
4.30 '4.10'
3.96 3.59 3.r717-82 12.31 10.21 s.07 8.35 7.86 7.4g 7.21 6 80 6.52 5.78 4.s714 4.60 3.74 3.34 3. l l 2.96 2.85 2.76 2.70 2.60 2.53 2.35 2.13(6'30) (4.86) (4.24) (3.8e) (1gg (3.50) ri.:sl e.?e) ri. i ir (3.05) (z..,s) (2.4s)8'86 6.51 5.56 5.04 4.70 4.46 'q.zs '4.r4' '3.g4'.
3.80 3.43 3.00r7'r4 1l '78 9.73 8.62 7.92 7.44 7.08 6.80 6.40 6.13 5.4r 4.6016 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.5g 2.4g 2.42 2.24 2.01(6'12) (4'6e) (4.08) (3.73) (1
:g) Q.34) 1i.zz1 a lrl ri.gs) (2.8s) (2.63) Q.32)8'53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 ' i .os' , 3.55 3.r8 2.7516'12 to-97 9.01 7.94 7.27 6.80 a.qa 6.19 5.8r 5.s5 4.85 4.0618 4.4r 3.55 3.16 2.g3 2.77 2.66 2.58 2.5r 2.41 2.34 2.1s 1.g2(5'e8) (4.s6) (3.es) (3.6r) (l:g) G.22) rJ.ro; (191) Q.; (2.77) (2.50) Q.1s)8'29 6'01 5.09 4.58 4.25 4.0r '3.84
3.71 ':.sr', 3.37 3.00 2.5715'38 10.39 8-49 7.46 6.81 6.35 o.oz 5.76 s,s9 5.13 4.45 3.67
4'35 3 '49 3 '10 2.87 2.71 2.60 2.51 2.4s 2.35 2.28 2.08 1.84(s'87) (4.46) (3 g6) (3.5r) (3.2s) (1 11) (3.01) (?s-l) 1z.tt7 ri.z'el (2.41) (2.0e)8'10 5'85 4'94 4.43 4.10 3.87 3.70 3.56 3.37 3.23 2.86 2.4214'82 9.95 8.r0 7.10 6.46 6.02 s.6g 5.44 5.08 4.82 4.r5 3.384'30 3.44 3-05 2.82 2.66 2.5s 2.46 2.40 2.30 2.23 2.03 1.78(s.7e) (4.38) (3.78) (3.44) (3.22) (:9:) es3) (?\1) eto1 ri.Zol (2.33) (2.00)7'95 s'72 4-82 4'31 3.99 3.76 3.5g 3.45 3.26 3.12 2.7s 2.3114'38 9'61 7.80 6.81 6.19 5.76 5.44 5.19 4.83 4.58 3.s2 3.154'26 3.40 3.01 2.78 2.62 2.5r 2.42 2.36 2.2s 2.18 1.98 1.73(s.72) (4.32) (3 7? (3.38) (3.15) (?22) Q.87) q7l) Q.64) ri. i i t Q.27) (1.s4)7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.17 3.03 2.66 2.2114'03 9.34 7-s5 6.59 5.98 5.55 5.23 4.9g 4.64 4.3g 3.74 2.g74'23 3.37 2.98 2.74 2.5g 2.47 2.3g 2.32 2.22 2.r5 1.95 1.69(5.65) (4.27) (3.67) (3.33) (3.10) (?3-1) 1z.zz1 q71) QssT Q.4s) (2.22) (1.88)7'72 5.53 4.64 4.r4 3.82 3.5g 3.42 '3.2g
3.09 2.s6 2.s8 2.1313'74 9'r2 7-36 6.41 5.80 5.3S 5.07 4.83 4.48 4.24 3.59 2.8228 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.2g 2.19 2.12 l .9r 1.65(5'61) (4'22) (3'63) (3.2s) (:99) (2.e0) (?.7^g) 1z.ey ri ' \ l (2.4s) (2.17) (r.s3)7.64 s-4s 4.s7 4.07 3.7s 3.s3 3.36 3.23 ',.0t
2.90 2.s2 2.0613.50 8-93 7.r9 6.25 5.66 5.24 4.g3 4.69 4.35 4.11 3.46 2.6930 4'17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.16 2.0g 1.S9 r.62(5'57) (4'18) (3'5e) (3.2s) (: 91) Q.87) (?71) Qss) rz.si l Q 41) (2.14) (1.7e)7.s6 5.3e 4.s1 4.02 3.70 3.47 3.30 3.17 ' t . t i 2.84 2.47 2.0r13'29 8.77 7.05 6.12 5.53 5.12 4.82 4.s8 q.zq 4.00 3.36 2.sg40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.08 2.00 1.7g 1.51(s'42)" (4'05) (3.46) (3.13) (?2y Q.74)
'1g?, 1z.sz7 <i.igl (2.2s) (2.01) (r.64)7 -31 s. r 8 4-31 3.83 3.51 3.2s 3.12 2.s9 ' t . i '
2.66 2.2e r.8012-61 8.25 6.59 5-70 5.13 4.73 4.44 4.21 z.sl 3.64 3.01 2.2360 4.00 3.r5 2.76 2.53 2.37 2.25 2.r7 2.10 1.99 r.g2 1.70 1.39(s.2e) (3 e3) (3.34) (3.01) (r^7nt e.63) (?:l) r?jy r).zil (2.17) (1.88) (r.48)7.08 4.98 4.13 3.65 3.34 3.r2 2.95 2.82 2.63 2.50 2.12 1.60t1.97 7.77 6.17 5.31 4.76 4.37 4.Og 3.86 z.sq 332 2.69 1.89120 3 92 3.07 2.68 2.45 2.2g 2.18 2.0g 2.02 1.91 1.83 1.61 t .zs(s.rs) (3.80) (3.23) (2.8e) (?g? Q.s2) (212) qg ri. i i> (2.05) (1.76) (r.31)6.8s 4.79 3.9s 3.48 3.r7 2.s6 2.7g 2.66 'z qi 2.34 1.95 1.38
' .38 7.32 5.78 4.95 4.42 4.04 3.77 3.55 z.zq 3.02 2.40 r .54
@ 3.84 3.00 2.60 2.37 2.21 2. tO z.Ot t .g4 1.83 1.75 t .SZ 1.00(5'02) (3'6e) (3'r2) (2.7s) (?17) (2.4r) r,?n) {ltss rz.o\l (1 e4) (1.64) (r.00)6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 '2 32' 2.18 1.79 r .0010.83 6.91 5.42 4.62 4.10 3.74 3.47 3.27 2.96 2.74 2.13 1.00
This table i" t*""
Sffil ifi":lf$$:::,Ttrtff"t":l#ff-:tf#*1* ti"" i"ti" ti'or'iio^"t,tro rabti for statisticians,Vorume r, by permission or
2l
raulc I t_urlulartve rrlnomral frobabilit ies
p = probability of success in a single tial; n: number of trials. The table gives the probability of obtaining r or moresuccesses in n independent trials. That is
When there is no entry for a particular pair of values of r ffid p, this indicates that tlre appropriate probability is lessthan 0.000 05. Similarly, except for the case i'= 0, when the entry is exact, a tabulated value of t.bOOO represents aprobability greater than 0.999 95.
E(,)o'o-P)'-'
p: 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
n=2
n=5
n= l0
n=20
n= 50
1.0000.0591.0009
1.0000.14t3.0085.0003
1.0000.2626.0345.0028.0001
1.0000.4562.1 198.0210.0027
.0003
1.0000.7819.4447.1892.0628
.0168
.0037
.0007
.0001
r:0I2
r=0I2J
4
r=0I2aJ
4
56
r=0I2
4
5678
r:0I2a
4
56789
1.0000 1.0000.0199 .0396.0001 .0004
1.0000 1.0000.0490 .0961.0010 .0038
.0001
1.0000 1.0000.0956 .1829.0043 .0t62.0001 .0009
1.0000 1.0000 1.0000.0784 .0975 .t164.0016 .0025 .0036
1.0000 1.0000 1.0000.1846 .2262 .2661.0148 .0226 .0319.0006 .0012 .0020
,0001
1.0000 1.0000 1.0000.3352 .40t3 .46t4.0582 .0861 .1176.0062 .0115 .0188.0004 .0010 .0020
.0001 .0002
r.0000 1.0000 1.0000.5580 .6415 .7099.t897 .2642 .3395.0439 .0755 .1150.0074 .0159 .0290
.0010 .0026 .0056
.0001 .0003 .00090001
1.0000 1.0000 1.0000.8701 .9231 .9547.5995 .7206 .8100.3233 .4595 .5838.139t .2396 .3527
.0490 .1036 .1794
.0t44 .0378 .0776
.0036 .0118 .0289
.0008 .0032 .0094
.0001 .0008 .0027
.0002 .0007.0002
1.0000 1.0000 1.0000.1351 .1536 .1719.0049 .0064 .0081.
1.0000 1.0000 1.0000.3043 .3409 .3760.042s .0s44 .0674.0031 .0045 .0063.0001 .0002 .0003
1.0000 1.0000 1.0000.5160 .5656 .6106.1517 .1879 .2254.0283 .0401 .0540.0036 .0058 .0088
.0003 .0006 .0010.0001
1.0000 1.0000 1.0000.7658 .81 13 .8484.4131 .4831 .5484.1610 .212t .2666.0471 .0706' .0993
.0107 .0183 .0290
.0019 .0038 .0068
.0003 .0006 .0013.0001 .0002
1.0000 1.0000 1.0000.9734 .9845 .9910.8735 .9t73 .9468.6892 .t740 .8395.4673 .5747 .6697
.27t0 .3710 .4723
.13s0 .2081 .2928
.0583 .1019 1596
.0220 .0438 .0768
.0073 .0t67 .0328
.0022 .0056 .0125
.0006 .0017 .0043
.0001 .0005 .0013.0001 .0004
.0001
1.0000 1.0000.t821 .3324.0169 .0599.0010 .0071
.0006
1.0000 1.0000.3950 .6358.0894 .2642.0138 .0784.0016 .0178
.0001 .0032.0005.0001
10l lT2l3t4