NEW CAMBRIDGE
STATISTICAL
TABLES
D. V. LINDLEY
&
W. E SCOTT
Second Edition
CAMBRIDGE
UNIVERSITY PRESS
CONTENTS
PREFACES
page 3
TABLES:
1
The Binomial Distribution Function
2
The Poisson Distribution Function
3
Binomial Coefficients
4
The Normal Distribution Function
5
Percentage Points of the Normal Distribution
6
Logarithms of Factorials
7
The )(2-Distribution Function
8
Percentage Points of the )(2-Distribution
9
The t-Distribution Function
10
Percentage Points of the t-Distribution
11
Percentage Points of Behrens' Distribution
12
Percentage Points of the F-Distribution
13
Percentage Points of the Correlation Coefficient r when p = 0
14
Percentage Points of Spearman's S
15
Percentage Points of Kendall's K
16
The z-Transformation of the Correlation Coefficient
17
The Inverse of the z-Transformation
Percentage Points of the Distribution of the Number of Runs
18
19
Upper Percentage Points of the Two-Sample Kolmogorov—Smirnov
Distribution
20
Percentage Points of Wilcoxon's Signed-Rank Distribution
21
Percentage Points of the Mann—Whitney Distribution
22A Expected Values of Normal Order Statistics (Normal Scores)
22B Sums of Squares of Normal Scores
23
Upper Percentage Points of the One-Sample Kolmogorov—Smirnov
Distribution
24
Upper Percentage Points of Friedman's Distribution
25
Upper Percentage Points of the Kruskal—Wallis Distribution
26
Hypergeometric Probabilities
27
Random Sampling Numbers
28
Random Normal Deviates
29
Bayesian Confidence Limits for a Binomial Parameter
30
Bayesian Confidence Limits for a Poisson Mean
Bayesian Confidence Limits for the Square of a Multiple Correlation
31
Coefficient
A NOTE ON INTERPOLATION
CONSTANTS
4
24
33
34
35
36
37
40
42
45
46
50
56
57
57
58
59
60
62
65
66
68
70
70
71
72
74
78
79
80
88
89
96
96
CONVENTION. To prevent the tables becoming too dense with figures, the convention has been adopted of omitting the leading figure when this does not change
too often, only including it at the beginning of a set of five entries, or when it
changes. (Table 23 provides an example.)
PREFACE TO THE FIRST EDITION
The raison d'etre of this set of tables is the same as that of the set
it replaces, the Cambridge Elementary Statistical Tables (Lindley
and Miller, 1953), and is described in the first paragraph of their
preface.
This set of tables is concerned only with the commoner and
more familiar and elementary of the many statistical functions and tests of significance now available. It is hoped that
the values provided will meet the majority of the needs of
many users of statistical methods in scientific research, technology and industry in a compact and handy form, and that
the collection will provide a convenient set of tables for the
teaching and study of statistics in schools and universities.
The concept of what constitutes a familiar or elementary
statistical procedure has changed in 30 years and, as a result,
many statistical tables not in the earlier set have been included,
together with tables of the binomial, hypergeometric and
Poisson distributions. A large part of the earlier set of tables
consisted of functions of the integers. These are now readily
available elsewhere, or can be found using even the simplest of
pocket calculators, and have therefore been omitted.
The binomial, Poisson, hypergeometric, normal, X2 and t
distributions have been fully tabulated so that all values within
the ranges of the arguments chosen can be found. Linear, and
in some cases quadratic or harmonic, interpolation will sometimes be necessary and a note on this has been provided. Most
of the other tables give only the percentage points of distributions, sufficient to carry out significance tests at the usual 5 per
cent and i per cent levels, both one- and two-sided, and there
are also some io per cent, 2.5 per cent and 0•1 per cent points.
Limitation of space has forced the number of levels to be
reduced in some cases. Besides distributions, there are tables
of binomial coefficients, random sampling numbers, random
normal deviates and logarithms of factorials.
Each table is accompanied by a brief description of what is
tabulated and, where the table is for a specific usage, a
description of that is given. With the exception of Table 26,
no attempt has been made to provide accounts of other statistical procedures that use the tables or to illustate their use with
numerical examples, it being felt that these are more appropriate
in an accompanying text or otherwise provided by the teacher.
The choice of which tables to include has been influenced
by the student's need to follow prescribed syllabuses and to
pass the associated examinations. The inclusion of a table does
not therefore imply the authors' endorsement of the technique
associated with it. This is true of some significance tests, which
could be more informatively replaced by robust estimates of the
parameter being tested, together with a standard error.
All significance tests are dubious because the interpretation to be placed on the phrase 'significant at 5%' depends on
the sample size: it is more indicative of the falsity of the null
hypothesis with a small sample than with a large one. In
addition, any test of the hypothesis that a parameter takes a
specified value is dubious because significance at a prescribed
level can generally be achieved by taking a large enough
sample (cf. M. H. DeGroot, Probability and Statistics (1975),
Addison-Wesley, p. 421).
All the values here are exact to the number of places given,
except that in Table 14 the values for n > 17 were calculated by
an Edgeworth series approximation described in 'Critical values
of the coefficient of rank correlation for testing the hypothesis
of independence' by G. J. Glasser and R. F. Winter, Biometrika
48 (1961), pp. 444-8.
Nearly all the tables have been newly computed for this
publication and compared with existing compilations: the
exceptions, in which we have used material from other sources,
are listed below:
Table 14, n = 12 to 16, is taken from 'The null distribution
of Spearman's S when n = 13(1)16', by A. Otten, Statistica Neerlandica, 27 (1973), pp. 19-20, by permission of the
editor.
Table 24, k = 6, n = 5 and 6, is taken from 'Extended
tables of the distribution of Friedman's S-statistic in the
two-way layout', by Robert E. Odeh, Commun. Statist. —
Simula Computa., B6 (I), 29-48 (1977), by permission of
Marcel Dekker, Inc., and from Table 39 of The Pocket Book
of Statistical Tables, by Robert E. Odeh, Donald B. Owen,
Z. W. Birnbaum and Lloyd Fisher, Marcel Dekker (1977),
by permission of Marcel Dekker, Inc.
Table 25, k = 3, 4, 5, is partly taken from 'Exact probability levels for the Kruskal—Wallis test', by Ronald L. Iman,
Dana Quade and Douglas A. Alexander, Selected Tables in
Mathematical Statistics, Vol. 3 (1975), by permission of the
American Mathematical Society; k = 3 is also partly taken
from the MS thesis of Douglas A. Alexander, University
of North Carolina at Chapel Hill (1968), by permission of
Douglas A. Alexander.
We should like to thank the staff of the University Press for
their helpful advice and co-operation during the printing of the
tables. We should also like to thank the staff of Heriot-Watt
University's Computer Centre and Mr Ian Sweeney for help
with some computing aspects.
10
January 1984
PREFACE TO THE SECOND EDITION
The only change from the first edition is the inclusion of
tables of Bayesian confidence intervals for the binomial and
Poisson distributions and for the square of a multiple correlation coefficient.
D. V Lindley
Periton Lane,
Minehead
Somerset,
TA24 8AQ, U.K.
2
W. F Scott
Department of Actuarial Mathematics
and Statistics,
Heriot-Watt University
Riccarton, Edinburgh
EHI4 4AS, U.K.
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n= 2
r =o
1
n= 3 r =o
p=
i
P = 0.01 0.9801
0.9999
0 '9997
*9996
0•01
•02
0.9703
'9604
*9412
'9988
'9409
•9216
.9991
'9984
.03
'04
.9127
.8847
'9974
.9953
'02
'03
'04
0'05
0.9025
•8836
0 '9975
0'05
0'8574
•8306
0.9928
•9896
•8044
'9860
'7787
'7536
•9818
'9772
'07
'8649
•9964
'9951
.68
'8464
'993 6
•06
•07
.08
•09
•8281
•9919
'09
0. zo
0•8roo
•I I
•I2
'7921
0'10 0'7290 0'9720
' II '7050 '9664
'7744
'13
•14
'7569
.7396
0.9900
'9879
'9856
'9831
.9804
'12
'13
•x4
•6815
'6585
•6361
0'15
0'7225
•7056
0.6141
.5927
•6889
0.9775
'9744
'9711
0'15
•16
•17
-6724
•9676
'17
•18
•6561
•9639
'19
0'20
•21
'22
'23
o'6400
•6241
0.9600
'9559
0'5120
'4930
'6084
'5929
'9516
'24
'5776
'9471
'9424
0'20
'21
'22
'23
'24
0'25
•5625
0 '9375
•26
'5476
'5329
•9324
•9271
0'25
'26
•5184
•9216
'5041
'9159
0'30
0.4900
0.3430
'4761
0.9100
'9039
0'30
'31
.31
•3285
'32
'4624
'8976
'32
'3144
'33
.34
'4489
•8911
.8844
'33
.3008
'4356
'34
'2875
•66
•18
•19
z
The function tabulated is
0.9999
.9998
.9997
'9995
'9993
0.9990
'9987
.9983
'9603
'9537
'9978
.9467
.9973
0.9393
.9314
'9231
.9145
'9054
0.9966
'9959
.9951
.9942
'9931
0'8960
0'9920
'4390
•8862
•8761
•8656
'8548
.9907
'9894
•9878
•9862
0'4219
0.8438
0 '9844
'8324
'8207
'9824
'9803
•28
'4052
'3890
'3732
'29
'3579
•8087
'7965
'9780
'9756
0.7840
'7713
'7583
.7452
.7318
0 '9730
•x 6
'5718
•5514
'5314
F(r1n, p) = E (n) pt(i_p)n-t
0.9999
Pr {X < r} = F(rIn, p).
Note that
Pr {X 3 r} = 1 -Pr {X < r= I F(r iln, p).
F(nin, p) = I, and the values for p > 0.5 may be
-
•27
'28
'29
6.35
•36
'37
'38
.39
0.4225
•4096
.3969
'3844
.3721
'43
•44
0'2746
0.7182
•36
•2621
'7045
•37
.38
•2500
•6966
.9493
.2383
.6765
. 9451
'39
'2270
•662,3
'9407
0.8400
•8319
'3364
•82 36
•8151
'3249
•3136
•8064
0'40
0•2160
•41
'2054
0.6480
.6335
0.9360
.9311
'42
'1951
'6190
'9259
'43
•44
•1852
•1756
•6043
•9205
'9148
6.1664
'1575
'1489
0'5748
•1406
.1327
'5300
'5150
0.3025
0 '7975
0.45
•46
•2916
'2809
•7884
.7791
•46
'7696
•49
.2704
•26or
0'50
0'2500
.48
0.9571
'9533
0'35
•8704
•8631
.8556
.8479
0 '45
.47
'9702
'9672
.9641
•9607
0'8775
040 0.3600
.41
.3481
'42
•27
'7599
'47
'48
'49
0'7500
0'50
•5896
.5599
'5449
0.9089
.9027
.8962
•8894
•8824
F(rIn, p) =
0'5000
0'8750
4
-
F(n r
-
-
'In, x
-
p).
The probability of exactly r occurrences, Pr {X = r}, is
equal to
-
F(r
-
1ln, p ) = (nr) Pr(I
Linear interpolation in p is satisfactory over much of the
table but there are places where quadratic interpolation
is necessary for high accuracy. When r = o, x or n-1 a
direct calculation is to be preferred:
F(o1n, p) = (i - p)n ,
F(1 In, p) = (1 -p)"-1[1 + (n- 1)p]
F(n- iln, p) = 1 -p".
and
For n > 20 the number of occurrences X is approximately normally distributed with mean np and variance
np(1 p); hence, including for continuity, we have
-
F(rin, p) * (1)(s)
npand 0(s) is the normal distribution
A/np(i p)
where s = r+
-
-
function (see Table 4). The approximation can usually be
improved by using the formula
F(rin, p) * 0(s)
where y
I -
Y e-is (s2 - x)
6 A/27/
2P
Vnp(i -p)
An alternative approximation for n > zo when p is
small and np is of moderate size is to use the Poisson
distribution:
F(rIn, p) * F(rI#)
where # = np and F(r1,a) is the Poisson distribution
function (see Table 2). If 1 p is small and n(i p) is
of moderate size a similar approximation gives
-
F(rin, p) *
-
-
F(n r
-
-
11,a)
where u = n(i p).
Omitted entries to the left and right of tabulated
values are o and I respectively, to four decimal places.
-
0'1250
-
found using the result
F(r1n, p)
'4746
'4565
t
t
for r = o, r,
n- 1, n < 20 and p 5 O. 5 ; n is sometimes referred to as the index and p as the parameter
of the distribution. F(rin, p) is the probability that X, the
number of occurrences in n independent trials of an
event with probability p of occurrence in each trial, is
less than or equal to r; that is,
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n =4
p
= o•ox
'02
.03
.04
0.05
7= 0
0.9606
-9224
.8853
•8493
I
I
0•0I
'02
0'9510
0'9990
'9039
.8587
'9998
.81 54
.9962
.9915
'9852
0 '9999
-03
•04
0 '9995
0'05
0 '7738
0 '9774
'9992
'9987
•9981
'9973
-o6
•o8
'7339
'6957
'6 591
.9681
'9575
'9456
0.9988
•998o
•9969
'9955
0 '9999
.09
'6240
.9326
'9937
'9999
'9998
'9997
0.9963
'9951
'9937
.9921
'9902
0'9999
'9999
'9998
'9997
'9996
0•10
'II
0.5905
0.9185
'9035
-8875
.8708
.8533
0 '9914
0 '9995
.9888
.9857
'982,
.9780
'9993
'9991
'9987
'9983
0'9995
(Yr5
•i6
0'4437
0 '9734
0 '9999
'17
0.9978
'9971
.9964
'9955
'9945
0 '9933
0 '9997
'9919
•9903
•9886
•9866
'9996
'9995
'9994
'9992
p
.07
•o8
.7164
'09
•6857
0.9860
•9801
'9733
'9656
'9570
o•io
0.6561
0 '9477
'II
•12
•6z74
'5997
•13
'14
.5729
'5470
'9376
'9268
'9153
'9032
0'15
0.5220
•i6
'4979
•17
'4746
0.8905
'8772
.8634
•18
•19
.4521
'4305
'8344
0.9880
.9856
.9829
'9798
.9765
0'20
•21
0.4096
0.8192
0.9728
'3895
'22
•3702
•3515
'8037
.7878
'7715
'7550
•9688
'9644
'9597
'9547
•8491
.23
.24
'3336
0'25
0 '3164
•26
'2999
0.7383
.7213
•27
•2840
*7041
•28
'29
•2687
'2541
•6868
'6693
0.9492
'9434
'9372
•9306
'9237
0'30
0.2401
0.6517
0.9163
'31
'32
-2267
•2138
'6340
'6163
'9085
•9004
'33
'34
•2015
'1897
•5985
•5807
0.35
•36
'37
0.1785
•1678
'1 575
0.5630
•38
'1478
'39
•1385
0'40
0•1296
'41
•1212
•1132
•1056
'42
'43
'44
0'45
-46
'47
'48
•49
o.5o
'5453
.5276
-5100
'4925
0.4752
.4580
'4410
'4241
=
0 '9999
0.8145
•7807
'7481
•o6
3
0 '9994
'9977
'9948
•9909
n =5
7' = 0
2
'07
•12
•x3
'14
'5584
•5277
'4984
'4704
0.9984
0'20
0.3277
0 '7373
•9981
'9977
'9972
.9967
•21
'7167
'6959
.23
'3077
•2887
*2707
'24
.2536
.6539
0.9421
'9341
.9256
'9164
.9067
0 '9961
0'25
0 '2373
0.6328
0.8965
0 '9844
0 '9990
'9954
'9947
'9939
'9929
•26
•2219
•8857
*9819
'27
*2073
•6117
•5907
'5697
'5489
'8743
•8624
'8499
'9792
'9762
'9728
.9988
•9986
'9983
'9979
0•5282
'5077
'4875
'4675
'4478
0•8369
•8234
'8095
'7950
•7801
0.9692
'9653
.9610
'9564
'9514
0 '9976
0.7648
'7491
'7330
'7165
'6997
0.9460
.9402
'9340
'9274
'9204
0 '9947
0.6826
•665,
'6475
•6295
•6114
0.9130
•8967
•8879
'8786
0.9898
•9884
•9869
'9853
'9835
0 '5931
0.9815
'9794
'9771
'9745
'22
'6749
0'30
*9908
'31
0•1681
'1564
'32
'1454
•8918
'8829
'9895
•9881
•9866
'33
'34
'1350
-1252
0.8735
-8638
'8536
•8431
.8321
0.9850
-9832
'9813
'9791
'9769
0.35
0.1160
•36
•38
•1074
'0992
•09,6
'39
.0845
0.4284
'4094
'3907
'3724
'3545
0•82.08
•8091
'7970
•7845
'7717
0 '9744
0.40
0.0778
0'3370
'9717
•9689
•9658
•9625
'41
'42
'43
'44
.0715
•o6o2
•0551
'3199
'3033
•2871
0 '9590
0'45
0.0503
'9552
'9512
'9469
'9424
'46
'47
'48
'49
'0459
.0731
'3431
•0677
•3276
0.0625
0.3125
0.6875
0'9999
'3487
0'9919
0.7585
'7450
.7311
'7169
•7023
0 '9999
•i8
'1935
0.3910
'3748
'3588
'9997
'9994
'19
•1804
0.0915
•0850
'0789
4
'3939
'3707
-4182
.28
'4074
3
0.8352
•8165
'7973
'7776
'7576
'9993
'9992
'9990
.9987
'29
'0983
2
'37
-0656
'2714
•9051
'9999
'9999
'9998
'9998
'9971
•9966
- 9961
'9955
'9940
'9931
'9921
'9910
-2272
'5561
•2135
•2002
'5375
0.8688
'8585
'8478
.8365
'5187
'8248
'9718
0•50
0'0313
0'1875
0'9375
See page 4 for explanation of the use of this table.
0'5000
0.8125
0.9688
5
'0418
'0380
'0345
0•2562
'2415
•9682
'9625
'9563
'9495
'5747
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n =6
r
= 0
p = 0•01
0.9415
'02
'8858
•8330
.03
•04
0'05
•06
.07
•o8
.09
0•25
•26
.27
-28
•29
n =7
r=o
I
2
3
p = 0•01
0'9321
•o2
-03
.04
•8681
-8080
0'9980
•9921
•9829
•9706
0.9997
'9991
•9980
0.9999
0.9556
•9382
•9187
•8974
•8745
0.9962
'9937
•9903
.9860
•9807
0.9998
'9996
'9993
'9988
•9982
0 '9743
0'9973
'3773
'3479
0.8503
•825o
.7988
.7719
'7444
•9669
•9584
'9487
•9380
•9961
'9946
.9928
•9906
•z6
0.3206
•2951
0.7166
•6885
0.9262
.9134
.8995
.8846
•8687
0.9879
.9847
•9811
.9769
•9721
'4702
0•8520
.8343
•8159
'7967
*7769
0'9667
•9606
'9539
'9464
.9383
0'4449
0'7564
0'9294
•4204
•3965
'3734
•3510
'7354
.7139
•6919
•6696
•9198
.9095
.8984
•8866
0'6471
•6243
•6013
•5783
'5553
0.8740
•8606
•8466
•8318
.8163
0.9978
.9962
'9942
.9915
•9882
0.9999
'9998
'9997
'9995
'9992
0'9842
'9794
'9739
.9676
.9605
0.9987
.9982
'9975
-9966
'9955
0'9999
'9999
'9999
'9998
'9997
0:::
'4046
0'8857
.8655
.8444
.8224
'7997
0.3771
.3513
•3269
•3040
•2824
0-7765
.7528
•7287
•7044
.6799
0'9527
.9440
'9345
'9241
.9130
0.9941
.9925
.9906
•9884
.9859
0.9996
'9995
'9993
'9990
'9987
0'15
0'2621
•2431
0'9011
•8885
•8750
•8609
.8461
0•9830
'9798
*9761
.9720
.9674
0 '9984
'2252
'2084
'1927
0'6554
•6308
'6063
•5820
'5578
.9980
'9975
.9969
'9962
0.1780
•1642
0.5339
•5104
•1513
•1393
•1281
'4872
'4420
0.8306
•8144
'7977
.7804
•7626
o•9624
.9569
.9508
'9443
'9372
0'7443
•7256
'7064
.687o
•6672
0.7351
-6899
. 6470
-6064
'5679
'4970
•23
.24
5
4
0.9672
'9541
'9392
•9227
*9048
•7828
0.5314
0'20
'21
'22
3
0'9998
'9995
-9988
'II
-12
'13
'14
•i6
•17
•18
•19
2
0.9985
'9943
.9875
.9784
010
0'15
I
'4644
'4336
'4644
0'05
•06
.07
•o8
•09
*7514
0.6983
.6485
•6017
.5578
•5168
0:4783
4423
*4087
'12
•13
'14
•17
'2714
'6604
•18
•19
'2493
•2288
.6323
•6044
0'9999
'9999
'9999
'9999
'9998
0'20
0'2097
•2X
.22
•23
•24
•1920
'1757
'1605
*1465
0'5767
'5494
•5225
*4960
0'9954
'9944
'9933
.9921
•9907
0.9998
'9997
'9996
'9995
'9994
0.25
•26
•27
•28
.29
0'1335
'1215
'1105
0.9295
•9213
•9125
.9031
•8931
0.9891
.9873
'9852
'9830
•9805
0.9993
'9991
.9989
.9987
.9985
0'30
0'0824
.31
•32
•33
•34
•0745
•0606
.0546
0.3294
'3086
•2887
•2696
.2513
'1003
•0910
0.30
•31
-32
.33
•34
0.1176
.1079
.0905
•0827
0.4202
•3988
'3780
'3578
•3381
0.35
•36
•37
•38
-39
0.0754
•0687
•0625
0568
.0515
0.3191
•3006
•2828
•2657
•2492
0.6471
•6268
-6063
.5857
*5650
0.8826
-8714
.8596
.8473
'8343
0.9777
'9746
'9712
.9675
'9635
0.9982
'9978
'9974
'9970
'9965
0.35
•36
'37
•38
•39
0'0490
•0440
'0394
•0352
-0314
0'2338
'2172
•2013
•1863
'1721
0'5323
'5094
'4866
'4641
'4419
0'8002
.7833
.7659
'7479
•7293
0'40
0-0467
0.8208
'8067
•7920
.7768
•7610
0.9590
•9542
'9490
'9434
'9373
0'0280
0'1586
0'4199
'9952
'9945
'9937
'9927
•41
-42
.43
•44
.0249
'0343
•0308
0.5443
•5236
•5029
.4823
•4618
0'40
'0422
'0381
0'2333
•2181
-2035
•1895
•1762
0'9959
'41
•42
'43
•44
'0195
-0173
.1459
'1340
•1228
•1123
'3983
'3771
'3564
•3362
0'7102
-6906
-6706
•6502
•6294
'46
•47
•48
•49
0.0277
•0248
'0222
0198
•0176
0.1636
.1515
•1401
•1293
•1190
0.4415
'4214
•4015
•3820
'3627
0.7447
•7279
'7107
•6930
*6748
0.9308
•9238
•9163
•9083
'8997
0.9917
•9905
'9892
•9878
*9862
0.45
-46
'47
•48
'49
0'0152
'0134
•0117
•0103
•0090
0'1024
'0932
•o847
•0767
•o693
0'3164
•2973
.2787
•2607
'2433
0'6083
'5869
. 5654
'5437
'5219
0'50
0.0156
0.1094
0'3438
0.6562
0.8906
0.9844
0'50
0.0078
0.0625
0•2266
0•5000
0'45
'0989
'0672
•0221
See page 4 for explanation of the use of this table.
6
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n= 7
p
r=4
5
6
n= 8
r
=o
1
2
3
4
5
= O'OI
p = 0'01
0.9227
0'9973
0 '9999
•02
•03
'02
'8508
'03
•04
'7837
'9897
'9777
'9996
'9987
0'9999
'7214
'9619
.9969
'9998
0.6634
.6096
.5596
•5132
- 4703
0.9428
•9208
.8965
•8702
'8423
0.9942
.9904
'9853
.9789
'9711
0 '9996
.9993
.9987
.9978
'9966
0.9999
.9999
'9997
0.9619
0'9950
'9929
'9513
.9903
.9392
.9871
9257
9109 •9832
0.9996
'9993
.9990
'9985
*9979
0.9999
.9999
'9998
'04
0.05
•66
•o7
•68
'09
cros
•o6
'07
6
0.9999
'9999
•38
•09
0.10
'II
•I2
0.9998
'9997
'II
0.4305
'3937
'9996
•12
'3596
•X3
•14
'9994
'9991
•13
'14
•3282
•2992
0.8131
'7829
.7520
•7206
•6889
0'15
0.9988
.9983
'9978
-9971
'9963
0.9999
.9999
'9999
.9998
'9997
0'/5
0.2725
.2479
-2252
•2044
.1853
0.6572
•6256
'5943
. 5634
'5330
0.8948
.8774
•8588
'8392
•8185
0'9786
0.9971
0 '9998
.16
'17
.18
'19
'9733
-9672
•9603
.9524
'9962
-9950
'9935
.9917
'9997
.9995
'9993
-9991
0 '9999
0'20
0.9953
'9942
0.9996
'9995
'22
'23
'9928
'9912
'9994
0'20
'21
'22
'24
'9893
.9992
'9989
.23
•24
0'1678
'1517
'1370
'1236
'1113
0.5033
'4743
'4462
'4189
.3925
0'7969
•21
0.9437
'9341
*9235
'9120
.8996
0.9896
.9871
'9842
'9809
'9770
0.9988
'9984
'9979
'9973
.9966
0.9999
'9999
'9998
.9998
.9997
0'25
0.9871
*9847
'981 9
*9787
0•6785
- 6535
.6282
'6027
0.9727
'9678
'9623
.9562
'9495
0'9958
*9948
'9936
'9922
'9906
0.9996
'9995
'9994
.9992
'9990
.26
'27
'28
•29
'9752
•10
•i6
•17
•19
'7745
'7514
'7276
-7033
'0722
0.3671
'3427
. 3193
'2969
•29
•0646
'2756
'5772
0•8862
*8719
•8567
'8466
'8237
0'9998
'9997
'9997
'9996
'9995
0'30
0.0576
.0514
0'2553
•0406
-0360
'2360
•2178
•2006
•1844
0.5518
.5264
•5013
*4764
.4519
0.8059
.7874
•7681
*7481
.7276
0.9420
-9339
'9250
*9154
.9051
0.9887
'9866
.9841
'9813
-9782
0.9987
.9984
•9980
'9976
•9970
0'9910
'9895
'9877
•9858
.9836
0 '9994
0.35
•36
'37
•38
.39
0.0319
-0281
'0248
-0218
-0192
0.1691
'1548
'1414
•1289
'1172
0.4278
'4042
•3811
•3585
'3366
0.7064
'6847
-6626
'6401
'6172
0.8939
'8820
'8693
'8557
•8414
0'9747
0'9964
'9992
'9991
.9989
'9986
'9707
'9664
'9615
'9561
-9957
'9949
'9939
'9928
0.8263
'8105
'7938
'7765
'7584
0.9502
'9437
'9366
•9289
•9206
0.9915
.9900
•9883
'9864
.9843
0'9115
•9018
•8914
-8802
•868z
0'9819
.9792
•9761
.9728
•9690
0.8555
0.9648
0 '9987
0 '9999
0'25
0' IOOI
'9983
'9979
'9974
.9969
'9999
'9999
'9999
.9998
•26
•0899
•0806
'27
•28
•31
•32
0'9712
- 9668
•9620
0'9962
'9954
'33
'9566
'34
'9508
'9935
'9923
0.35
'36
'37
-38
-39
0 '9444
0'40
'41
'42
- 43
'44
0 '9037
0.9812
'9784
'9754
'9721
'9684
0'9984
0'40
'8937
.8831
'8718
'8598
.9981
0.0168
'0147
-0128
'011 1
'0097
0.1064
*0963
•0870
'0784
'0705
0 '3154
.2948
0 '5941
. 5708
'9977
'9973
•9968
'41
'42
'43
•44
' 2750
'2560
'2376
'5473
'5238
•5004
0'45
'46
'47
.48
'49
0'8471
'8337
'8197
'8049
'7895
0'9643
'9598
'9549
'9496
'9438
0'9963
.9956
'9949
'994'
'9932
0'45
•46
'47
•48
'49
0.0084
•0072
-0062
-0053
'0046
0.0632
•0565
•0504
.0448
'0398
0.2201
.2034
'1875
.1724
-1581
0.4770
-4537
'4306
'3854
0.7396
'7202
•7001
'6795
•6584
0•50
0.7734
0'9375
0.9922
0•50
0.0039
0.0352
0.1445
0.3633
0.6367
0'30
'9375
'9299
•9218
.9131
'9945
'31
.32
'33
-34
'0457
'4078
See page 4 for explanation of the use of this table.
7
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n =8
r= 7
n= 9
r
=o
I
2
0.9966
•9869
•9718
0.9999
'9994
•9980
'9522
'9955
0.9916
•9862
'9791
3
4
5
6
7
P = o•ox
p = O'OI
0.9135
•02
•04
'02
•03
'04
•8337
'7602
*6925
0'05
0'05
•o6
•o6
0.6302
'573 0
.07
'07
'5204
0.9288
•9022
•8729
•o8
•o8
•4722
•8417
'9702
•09
'09
•4279
•8088
'9595
o•xo
o•xo
0.3874
0 '7748
0 '9470
'II
•I2
'3504
'3165
'7401
•x3
'14
-x3
'14
•2855
'2573
•6696
•6343
.9328
.9167
.8991
•8798
0.9917
'9883
•9842
'9791
'9731
0.9991
.9986
'9979
'9970
'9959
0 '9999
'II
•I2
0'15
0'15
0.2316
0.5995
0.8591
0.9661
•x6
•17
•x8
•2082
•1869
•1676
'5652
•8371
•9580
.19
.19
'1501
•4988
'4670
.8139
.7895
•7643
'9488
'9385
•9270
'9991
.9987
.9983
'9977
0 '9999
'531 5
0.9944
•9925
.9902
.9875
'9842
0 '9994
•x6
•17
•x8
0'20
'21
'22
0'1342
0'4362
0'7382
0'9144
0'9804
0 '9969
0 '9997
'1199
•1069
'0952
'o846
•4066
•3782
'24
0'20
•2I
'22
'23
'24
•7115
•6842
•6566
•6287
•9006
•8856
•8696
'8525
-9760
'9709
•9650
.9584
.9960
'9949
'9935
'9919
'9996
'9994
'9992
'9990
0'25
•26
'27
0'25
'26
•27
0.075x
0.3003
•2770
•2548
0.8343
.8151
'7950
'7740
.7522
0.9511
'9429
'9338
•9238
'9130
0.9900
•9878
'9851
•9821
.9787
0.9987
.9983
'9978
'9972
.9965
0 '9999
'0665
'0589
•03
.23
•7049
'3509
•3250
8
0.9999
'9997
0.9994
'9987
'9977
'9963
'9943
0 '9999
'9998
'9997
'9995
'9999
'9998
'9997
'9996
'9999
'9998
'9998
0 '9999
'9999
-28
•0520
•2340
•29
0'9999
'29
'0458
'2144
0.6007
'5727
. 5448
•5171
'4898
0'30
0 '9999
0'30
0'0404
0'1960
0'4628
0'7297
'31
.0355
•4364
'9999
'32
'0311
'1788
•1628
'33
'34
'9999
'9998
.33
'0272
'1478
'4106
•3854
'34
.0238
'1339
•3610
•7065
•6827
.6585
•6338
0.9747
'9702
.9652
'9596
'9533
0 '9996
'9999
0.9012
•8885
'8748
•86oz
'8447
0 '9957
'31
.32
'9947
'9936
'9922
.9906
'9994
'9993
'9991
'9989
0 '9999
0'35
0 '9998
0'35
0.0207
.36
'9997
•36
'37
'38
'9996
.9996
•0135
0. 1 zi 1
•1092
•0983
•0882
0.3373
'3144
•2924
'2 713
'39
'9995
•37
-38
.39
•0i8o
•0156
'0117
'0790
'2511
0.6089
•5837
•5584
'5331
'5078
0.8283
.8110
'7928
'7738
'7540
0.9464
•9388
.9304
•9213
'9114
0.9888
.9867
.9843
•9816
•9785
0.9986
•9983
'9979
'9974
'9969
0.9999
'9999
'9999
'9998
'9998
0'40
0'9993
'9992
'9990
'9988
0'40
0.0101
0'0705
0'2318
0'4826
'41
•0087
•0628
•2134
'4576
0.7334
-7122
•6903
•6678
'6449
0.9006
•8891
•8767
•8634
'8492
0.9750
•9710
•9666
'9617
'9563
0.9962
'9954
'9945
'9935
'9923
0.9997
'9997
'9996
'9995
'9994
0.6214
'5976
'5735
'5491
0.8342
•8183
'80,5
'7839
•7654
0.9909
'9893
'9875
'9855
'9831
0 '9992
'5246
0.9502
'9436
'9363
•9283
'9196
0.5000
0.7461
0.9102
0'9805
0.9980
•28
'41
'42
'43
•44
'42
'0074
•0558
'1961
.4330
•oo64
'0495
•9986
'43
•44
•0054
'0437
•1796
.1641
•4087
'3848
0'45
0.9983
0.45
0.0046
0.0385
0.1495
'46
'47
•9980
•0039
'0338
'1358
•0033
•0296
•1231
0.3614
'3386
'3164
•0028
'49
'9976
'9972
•9967
•46
'47
'48
'49
'0023
'0259
'0225
'III,
•I00I
'2948
'2740
0'50
0.9961
0•50
0'0020
0'0195
0.0898
0.2539
'48
See page 4 for explanation of the use of this table.
8
'9999
'9998
'9997
'9997
'999!
•9989
'9986
'9984
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n
p
I0
r =0
1
2
= 0•0i
'02
'03
0.9044
0 '9957
0 '9999
'8171
•9838
9655
'9418
'9991
'9972
'9938
0.9885
'9812
'9717
'9599
'9460
0 '9990
0 '9999
'9980
'9964
'9942
•9912
'9998
'9997
'9994
'9990
0 '9999
0.9298
•9116
'8913
•8692
'8455
0'9872
•9822
'9761
•9687
'9600
0 '9984
0 '9999
'9975
•9963
'9947
•9927
'9997
'9996
'9994
'9990
0.8202
'7936
0.9500
'9386
0.9901
.9870
0.9986
.9980
7659
9259
983 2
9973
9997
. 7372
•7078
.9117
•8961
.9787
'9734
.9963
'9951
'9996
'9994
0 '9999
'7374
•6648
-04
3
4
5
6
7
8
9
0 '9999
'9996
•o6
.07
•o8
0.5987
.5386
'4840
'4344
'09
- 3894
0.9139
•8824
.8483
.81z1
'7746
0.10
0.3487
•3118
' 2785
•2484
0.7361
•6972
'6583
•6196
'2213
'5816
0.15
•16
'17
•18
•19
0.1969
•1749
0 '5443
:1135752
'1374
•12.16
.5080
'4730
'4392
-4068
0'20
•21
•22
0'1074
0'3758
.23
•24
'0733
•0643
0.8791
•8609
'8413
•8206
.7988
0'9672
'9601
.9521
'9431
'9330
0'9936
•9918
.9896
'9870
'9839
0 '9999
*3464
-3185
•2921
.2673
0.6778
'6474
. 6169
•5863
•5558
0 '9991
'0947
•o834
.9988
'9984
'9979
'9973
'9999
*9998
'9998
'9997
0'25
0'0563
0'2440
0.5256
'2222
'4958
'28
'29
'0374
•0326
•2019
.1830
'4665
'4378
'4099
'7274
'7021
•6761
0.9219
.9096
.8963
•8819
•8663
0.9803
.9761
'9713
'9658
'9596
0 '9996
'0492
'0430
0 '7759
. 7521
0 '9965
•26
•27
'9955
'9944
'9930
'9913
'9994
'9993
'9990
'9988
0'6496
•6228
o'8497
•8321
4
0'9894
0 '9984
0 '9999
'9871
'595 6
5684
4
'8133
'7936
'7730
0'9527
'9449
'9363
'9268
•9164
.9815
'9780
•9980
'9975
.9968
'9961
.9998
'9997
'9997
'9996
0.5138
'4868
'4600
'4336
'4077
0.7515
'7292
•7061
•6823
.658o
0.9051
•8928
'8795
•8652
•8500
0.9740
'9695
'9644
'9587
'9523
0'9952
'9941
'9929
'9914
'9897
0 '9995
0.6331
•6078
'5822
'5564
'5304
0.8338
•8166
'7984
'7793
'7593
0'9452
'9374
'9288
'9194
'9092
0'9877
'9854
'9828
'9798
'9764
0'9983
'9979
'9975
'9969
'9963
0 '9999
0.5044
'4784
.4526
0'7384
0.8980
•8859
'8729
'9726
'9683
'9634
0 '9955
0'9997
'9996
'9995
'9994
'9992
0 '9990
0'05
•II
•I2
•13
'14
0'30
'1655
0.1493
'1344
0'9999
'9999
0 '9999
'9998
0 '9999
'9999
.9999
'0211
•Iz(36
•0182.
•1o8o
•0157
'0965
0.3828
'3566
'3313
•3070
.2838
0.35
•36
'37
-38
'39
0.0135
•0115
•0098
-0084
0.0860
'0764
'0677
•0598
0.2616
•2405
•2206
'2017
•0071
'0527
'1840
0.40
'41
0.0060
'005!
- 0043
•0036
•0030
0.0464
•0406
'0355
•0309
0.1673
'1517
'1372
•1236
'0269
'III!
0.3823
'3575
'3335
•3102
•2877
0'45
•46
'47
-48
'49
0.0025
0.0233
'0021
•0017
•0014
'0012
'0201
'0173
•0148
'0126
0.0996
•0889
0.2660
'2453
'0791
•0702
'0621
'2255
•2067
'4270
'6712
'8590
'9580
•1888
•4018
'6474
'8440
'9520
'9946
'9935
'9923
'9909
0'50
0.0ozo
0.0107
0'0547
O'1719
0'3770
0.6230
0.8281
0 '9453
0 '9893
'31
•32
'33
'34
•42
'43
'44
0
4
85
2
•4
'7168
'6943
See page 4 for explanation of the use of this table.
9
'9993
'9991
'9989
'9986
0 '9999
'9999
'9999
'9998
'9998
'9997
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n = xx
p = 0•01
6
8
I
2
.03
•04
•8007
-7153
0'9948
•9805
.9587
0.9998
•9988
.9963
'6382
'9308
'9917
0.9998
.9993
0.05
•o6
0.5688
•5063
'4501
•o8
'09
.3996
'3544
0.9848
.9752
•963o
•9481
'9305
0.9984
'9970
'9947
'9915
.9871
0'9999
'07
0.8981
•8618
•8228
•7819
'7399
0'10
0.3138
*2775
0.6974
'6548
0'9104
'2451
'2161
•1903
.5311
'8985
0.9815
'9744
.9659
'9558
'9440
0.9972
'9958
'9939
•9913
•9881
0.9997
'9995
'9992
.9988
•9982
0 '9999
•5714
•888o
•8634
•8368
o•xs
•16
.17
-18
•19
0.1673
- 1469
-1288
•1127
.0985
0'4922
'4547
-4189
•3849
-3526
0'7788
'7479
•7161
•6836
•6506
0.9306
'9154
.8987
•8803
•8603
0.9841
'9793
'9734
.9666
-9587
0'9973
0'9997
'9963
'9949
.9932
-9910
'9995
'9993
'9990
.9986
0'20
0.0859
0748
.0650
•0489
0.3221
'2935
•2667
•2418
-2186
0.6174
'5842
'5512
.5186
•4866
0.8389
•8160
'7919
•7667
.7404
0.9496
'9393
'9277
. 9149
•9008
0.9883
'9852
'9814
'9769
'9717
0.9980
'9973
'9965
'9954
'9941
0.9998
'9997
'9995
'9993
'9991
0'25
0'0422
0'1971
0'4552
'1773
-4247
'27
'0314
'1590
'395 1
•28
•29
•0270
- 1423
•3665
0'9657
'9588
- 9510
. 9423
0'9924
'9905
-9881
.9854
'9984
'9979
'9973
'0231
•1270
'3390
'6570
-6281
-5989
0'8854
*8687
'8507
-8315
•8112
0 '9999
'0364
0'7133
.6854
0 '9988
•26
'9326
'9821
'9966
'9998
.9998
'9997
.9996
0.1130
•1003
.0888
0.3127
•2877
. 2639
'0784
'0690
'2413
'2201
0.5696
'5402
.5110
'4821
'4536
0.7897
'7672
'7437
'7193
•6941
0.9218
'9099
.8969
'8829
-8676
0.9784
'9740
.9691
'9634
.9570
0.9957
'9946
'33
'34
0.0198
•0169
-0144
•0122
•0104
0'35
0'0088
0'0606
0'2001
•0074
'0062
-0052
'0044
.0530
•0463
.0403
•0350
•1814
"1640
*1478
•1328
0.4256
*3981
. 3714
'3455
•3204
0.6683
'6419
'6150
'5878
'5603
0.8513
'8339
'8153
'7957
'7751
0 '9499
•36
'37
-38
'39
0.40
41
•42
'43
'44
0.0036
.0030
-0025
•0021
•0017
0.0302
•0261
-0224
0.1189
•./062
0.2963
-2731
0.5328
-5052
0 '7535
2510
'4777
0. 45
46
'47
'48
'49
0'50
'02
•II
•I2
•13
•14.
.21
-22
•23
-24
0'30
-31
-32
.
-
r
o
0'8953
-
'0564
'6127
3
5
4
'9997
'9995
'9990
.9983
7
9
0 '9999
.9998
.9999
.9998
0 '9999
'9999
.9998
0 '9999
'9999
0 '9994
0 '9999
'9918
'9899
'9992
'9990
'9987
'9984
'9419
'9330
'9232
'9124
0'9878
'9852
'9823
. 9790
'9751
0.9980
'9974
- 9968
'9961
'9952
0.9998
'9997
- 9996
'9995
'9994
0.9006
-8879
.8740
-8592
- 8432
0.9707
.9657
•9601
'9539
- 9468
0.9941
-9928
*9913
'9896
'9875
0 '9993
.7310
'7076
.6834
.6586
.9991
- 9988
'9986
'9982
0.9999
'9999
'9999
'9999
0.9998
'9998
'9998
.9997
.9996
0'9995
'9933
'9999
'9999
. 9998
'0945
.
'0192
'0838
'2300
'4505
•0164
. 0740
•2100
.4236
0'0014
0'0139
0'0652
0'1911
0'3971
•0011
•0009
•0008
•0006
•0118
.0100
-0084
•007o
-0572
•0501
.0436
•0378
'1734
•1567
.1412
•1267
'3712
'3459
. 3213
'2974
0.6331
.6071
•5807
. 5540
'5271
0•8262
-8081
-7890
•7688
'7477
0.9390
'9304
'9209
.9105
-8991
0.9852
'9825
'9794
.9759
'9718
0.9978
'9973
'9967
.9960
.9951
0.0005
0.0059
0.0327
0'1133
0 '2744
0.5000
0.7256
0.8867
0.9673
0.9941
See page 4 for explanation of the use of this table.
I0
xo
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n = 12
r
p=
(•oi
0•8864
•02
'03
•04
'7847
0.05
•o6
=o
I
•6938
'6127
'9191
0.9998
'9985
'9952
•9893
0'5404
0'8816
0'9804
0 '9978
'4759
'9684
-9532
.9348
'9134
'9957
'9925
-988o
.9820
0.8891
•8623
.8333
•8923
-7697
0 '9744
- 9649
0.4435
'4055
•3696
'3359
-3043
0'7358
'47
'49
0'9957
0'9995
0 '9999
0'50
0.9998
.9536
'9403
-9250
'9935
.9905
'9867
•9819
'9991
.9986
'9978
'9967
'9999
'9998
'9997
.9996
0.9078
•8886
•8676
.8448
'8205
0.9761
-9690
•9607
•9511
'9400
0.9954
.9935
.9912
-9884
'9849
0 '9993
0'9999
.7010
•6656
*6298
'5940
'9990
'9999
'9998
'9997
0.2749
-2476
•2224
'1991
0 '5583
0 '7946
•1778
'4222
*6795
0.9806
'9755
'9696
•9626
'9547
0.9961
'9948
'9932
•9911
'9887
0 '9999
-7674
- 7390
•7096
0'9274
'9134
'8979
•8808
'8623
0 '9994
.5232
•4886
-4550
'9992
'9989
'9984
'9979
'9999
'9999
'9998
'9997
0'9456
0 ' 9857
0'9972
0'9996
'9995
0 '9999
:9
9324
504
'9953
'9940
'9924
'9993
'9990
'9987
'9999
'9999
'9998
0'9983
- 9978
0'9998
'9997
.9996
'9995
'9993
0'9999
0.9999
'9999
'9999
-9998
'9998
0.2824
0.6590
•II
•I2
'2470
*6133
•5686
-0924
.0798
0'20
'2I
'22
'23
0.0687
•0591
'0507
•5252
.4834
'46
'48
'9985
.9979
'9971
.9996
.24
'0434
'0371
0'25
0.0317
0.1584
0'3907
0.6488
0.8424
•26
-27
•28
'0270
•0229
'1406
.1245
'3603
•6176
•8210
•0194
. 1 Ioo
'29
'0164
'0968
'3313
•3037
' 2 775
.5863
'5548
•5235
'7984
'7746
'7496
'9113
'8974
.•
'9733
.9678
0'30
0'0138
'oi 16
'0098
0'0850
0'2528
.2296
0'4925
0.7237
-6968
.6692
•6410
•6124
0.8822
•8657
'8479
•8289
•8087
0.9614
'9542
'9460
'9368
•9266
0 '9905
0.5833
'5541
'5249
'4957
•4668
0.7873
.7648
'7412
-7167
•6913
0'9154
0.9745
.9030
•8894
-8747
'8589
-9696
0'4382
0.6652
•6384
•2078
•1876
•1687
-4619
.4319
'4027
.3742
0.1513
•1352
•1205
0.3467
•3201
.2947
'1069
'0946
*2704
•9682
•0068
'0744
•0650
•0565
•0491
0.0057
0.0424
•0047
•0039
-0032
•0027
'0366
•0315
'0270
'0230
0'0022
•ooi8
•0014
•oolz
•0010
0'0196
•0166
'0140
•or 18
•0099
0'0834
'41
•42
'43
'44
'0642
•0560
'0487
•1853
•3825
•1671
•3557
'1502
'3296
'5552
0.45
'46
00008
•0006
0.0083
•oo69
0.0421
•0363
0.1345
•1199
0.3044
•2802
0.5269
•4986
'47
'48
'49
•0005
•0004
•0003
•0057
'0047
•0039
•0312
'0267
'0227
•1066
*0943
'0832
•2570
'4703
-2348
.4423
•2138
0'50
0'0002
0'0032
0'0193
0.0730
0.1938
'31
'32
'33
'34
0.35
•36
'37
•38
'39
0'40
r=x
'9998
'9997
0•10
•18
n = 12
zo
0 '9999
'7052
'19
9
0'9999
'9999
'9999
'9999
'9998
•3225
0.1422
•1234
•1069
8
0'45
•09
•z6
•z7
7
0.9998
.9996
'9991
.9984
'9973
'4186
•1637
6
0'9999
•3677
•1880
'9997
'9990
5
p = 0.44
•o8
•2157
0 '9999
4
0 '9999
•07
0'15
3
0.9938
'9769
'9514
'8405
'7967
.7513
•z3
'14
2
'0733
•2472
0.2253
'2047
'4101
.9882
*9856
'9824
.9787
.9578
'9507
'9915
.9896
.9873
0'9992
'9989
•9986
.9982
'9978
0.8418
.8235
0.9427
.9338
0.9847
-9817
0'9972
•9965
'6111
•8041
'9240
'9782
'9957
'5833
'7836
•7620
'9131
•9012
'9742
'9696
'9947
'9935
0.9997
'9996
.9995
'9993
'9991
0.8883
'8742
•8589
.8425
•8249
0'9644
0.9921
0'9989
'4145
0.7393
.7157
•6911
.6657
'6396
.9585
-9519
'9445
'9362
-9905
•9886
.9863
.9837
.9986
.9983
'9979
'9974
0.3872
0.6128
0.8062
0.9270
0.9807
0.9968
.9641
See page 4 for explanation of the use of this table.
II
'9972
'9964
.9955
0 '9944
.9930
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n = x3
r=o
i
2
3
4
p = 0.01
0.8775
0'9928
'02
•7690
•6730
•5882
'9730
'9436
•9068
0.9997
•9980
'9938
•9865
0.9999
'9995
•9986
0.9999
•06
0'5133
'4474
.07
'3893
.08
.09
•3383
'2935
0.8646
.8186
.7702
•7206
•6707
0.9755
•9608
'9422
•9201
.8946
0.9969
'9940
.9897
'9837
'9758
0.9997
'9993
.9987
'9976
'9959
0•10
0.2542
0.6213
'5730
0.8661
.8349
•8015
•7663
/296
0.9658
'9536
'9391
•9224
'9033
0.9935
.9903
'9861
•9807
'9740
0'9991
.11
o•8820
.8586
•8333
•8061
'7774
0.9658
'9562
'9449
.9319
.9173
0.9925
.9896
.9861
.9817
.9763
0'9987
0 '9998
•9981
'9973
'9962
'9948
'9997
'9996
'9994
'9991
0.7473
.7161
•6839
'6511
•6178
0.9009
•8827
•8629
' 8415
•8184
0.9700
•9625
'9538
'9438
'9325
0.9930
'9907
•9880
.9846
.9805
0.9988
.9983
'9976
.9968
'9957
0.9998
'9998
'9996
'9995
'9993
0.5843
•5507
.5174
'4845
.4522
0 '7940
0'9198
•9056
.8901
•8730
.8545
0'9757
0.9944
.9927
.9907
•9882
'9853
0'9990
0'9999
•7681
'7411
'7130
•6840
'9987
'9982
'9976
.9969
'9998
'9997
'9996
'9995
0.4206
•3899
•3602
'3317
.3043
0.6543
.624o
'5933
•5624
'5314
0.8346
•8133
-7907
•7669
'7419
0.9376
.9267
•9146
0.9960
'9948
'9935
.9918
•9898
0.9993
'9991
'9988
.9985
•9980
0 '9999
•8865
0.9818
'9777
.9729
'9674
•9610
0'5005
'4699
0'7159
0.8705
0 '9538
0 '9874
'2536
'2302
'2083
'4397
'4101
'6889
•6612
•8532
•8346
'9456
•9365
0.9975
'9968
.9960
'9949
'9937
0.9997
'9995
'9994
'9992
'9990
.03
'04
0'05
•12
•2198
'1898
•13
'14
•1636
'1408
0'15
0.1209
•16
•17
•x8
•1037
•0887
.0758
.2920
•19
•0646
•2616
0.6920
.6537
•6152
'5769
.5389
0'20
'21
'22
'23
0'0550
'0467
0'2336
0'5017
•2080
•1846
•1633
'4653
•43ox
.3961
•3636
.0396
•5262
•4814
.4386
0'3983
•3604
•3249
5
6
7
8
9
10
0 '9999
'9999
'9997
'9995
.9985
'9976
'9964
'9947
0'9999
0.9999
'9998
'9997
'9995
'9992
0 '9999
'9999
0 '9999
'9999
•24
'0334
•0282
'1 441
0'25
0.0238
0.1267
•26
•0200
•27
•28
•29
'0167
'0140
•0117
•1 1 11
'0971
'0846
*0735
0•30
'31
•32
0.0097
•oo80
•oo66
o'o637
•0550
.0473
0.2025
•1815
•1621
'33
'34
•0055
•0045
•0406
*0347
.1280
0.35
•36
0'0037
•6327
•8147
'9262
'39
•0025
'0020
•0016
O'1132
*0997
'0875
'0765
•0667
0'2783
'37
•38
0'0296
'0251
'0213
'0179
•0151
•1877
•3812
•6038
'7935
•9149
.9846
•9813
'9775
'9730
0'40
0.0013
o•o126
0.0579
0.1686
0.3530
0.5744
0'9023
0'9679
0'9922
0'9987
'41
'42
'43
'44
•0010
•0008
•0007
•0005
•0105
'0501
•I508
'3258
'5448
•0088
.0431
.2997
'5151
•8886
•8736
.8574
•8400
•9621
'9554
'9480
'9395
.9904
.9883
.9859
.9830
.9983
'9979
'9973
.9967
0.45
•0030
0.3326
•3032
'2755
'2495
.2251
'1443
'9701
.9635
•9560
'9473
'9012
0'9999
'9999
0 '9999
'9999
'9999
'9998
'9997
'0072
'0370
'1344
'1193
'2746
'4854
•oo60
•0316
•1055
' 2507
'4559
0.7712
'7476
•7230
-6975
.6710
0'0049
0'0269
0'0929
0'2279
0'4268
0•6437
0'8212
•0040
•o228
•0815
•2065
•3981
•6158
'0033
•0026
•0021
'0192
•002
•0135
'0712
•0619
'0536
•1863
•1674
'49
0•0004
•0003
•0003
-0002
•0002
0.9302
'9197
•9082
•8955
•8817
0.9797
'9758
'9713
•9662
•9604
0.9959
'9949
'9937
•9923
.9907
0.50
0.0001
0.0017
0.0'12
0.0461
0.8666
0.9539
0.9888
'46
'47
'48
'3701
•5873
'1498
'3427
•3162.
'5585
.5293
•80'2
•7800
'7576
'7341
0.1334
0.2905
0.5000
0.7095
See page 4 for explanation of the use of this table.
12
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n = 13
r = II
12
p=
n = i4
p = o•ox
•02
'02
•03
•04
•03
.04
0.05
•06
.07
•o8
•09
0'05
0•10
r=
0
0.8687
'7536
'6528
'5647
I
2
0.9916
.9690
'9355
'8941
o•9997
'9975
'9923
•9833
3
4
5
6
7
0 '9999
'9994
'9981
0.9998
0-4877
•4205
0.8470
0'9699
0 '9958
0 '9996
-o6
'7963
'07
'362o
'7436
.9522
•9302
•9920
.9864
'9990
•9980
•o8
•09
'3112
'9042
'9786
'9965
'2670
•6900
'6368
. 8745
•9685
'9941
'9998
'9996
'9992
o•ro
0•2288
•1956
•1670
'13
•14.
'1 423
0.5846
'5342
'4859
'4401
'1211
•3969
0.8416
•8061
.7685
'7292
•6889
0.9559
•9406
•9226
'9021
•8790
0.9908
.9863
'9804
'9731
'9641
0.9985
'9976
'9962
'9943
'9918
0'9998
'II
•I2
0.15
•x6
•x7
•x8
.x9
0'15
o•1o28
•0871
.0736
•0621
•0523
0.3567
•3193
.2848
.2531
- 2242
0.6479
•6068
.5659
.5256
0.8535
•8258
.7962
. 7649
0.9533
'9406
'9259
•9093
0.9885
'9843
'9791
'9727
0'9978
.9968
'9954
'9936
'4862
•7321
'8907
'9651
'9913
0.9997
'9995
'9992
•9988
.9983
0'20
'2I
'22
'23
'24
0•20
'21
'22
0'0440
0'1979
0.4481
'0369
•1741
0•6982
'66 34
0.8702
- 8477
'0309
'1527
'4113
'3761
'6281
'8235
-23
•24
•oz58
'0214
'1335
.3426
'5924
'7977
'1163
'3109
•5568
•7703
0.9561
'9457
'9338
'9203
'9051
0.9884
.9848
.9804
'9752
. 9690
0.9976
•99 67
'9955
'9940
'9921
0.25
•26
.27
•28
.29
0'25
0'0178
0•1010
0'2811
0'5213
0'7415
0'8883
0'9617
0'9897
'26
'0148
•0122
'0101
'0083
•o874
'0754
•0556
'2533
' 2273
•2033
•1812
'4864
'45 21
'4187
•3863
•7116
•6807
•6491
•6168
•8699
'8498
•8282
•8051
'9533
'9437
'9327
'9204
•9868
.9833
'9792
'9743
0.30
'31
•32
'33
'34
0'30
0•oo68
•0055
•0045
.0037
o.o475
0.1608
0.3552
'0404
'1423
'3253
'0343
•0290
•1254
•1101
•2968
•2699
•0030
'0244
'0963
'2444
0.5842
'5514
•5187
.4862
'4542
0.7805
'7546
.7276
•6994
.6703
0.9067
'8916
'8750
'8569
.8374
0.9685
.9619
'9542
'9455
'9357
0.35
•36
'37
'38
'39
0.0024
•0019
•0016
•0012
•0010
0.0205
0'9999
0.35
•36
.37
•38
'39
•0172
•0143
•0119
0.0839
•o729
•0630
.0543
0.2205
•1982
•1774
•1582
0.4227
•3920
.3622
'3334
'0098
'0466
'1405
•3057
0.6405
•6101
'5792
'5481
•5169
0.8164
'7941
•7704
'7455
'7195
0.9247
'9124
•8988
.8838
•8675
0.40
'41
'42
'43
'44
0'9999
'9998
'9998
'9997
'9996
0.40
o-0008
•0006
o.0081
•oo66
0.0398
•0339
0.1243
.1095
0'2793
'2541
0'4859
'4550
-42
•43
•44
'0005
•0054
•0044
'0036
'0287
'0242
•0203
•0961
'0839
'0730
'2303
•2078
•1868
'4246
'3948
•3656
0.6925
•6645
-6357
•6063
'5764
0.8499
•8308
•8104
•7887
•7656
0'45
0 '9995
0.0002
•0002
0.0170
0'5461
.5157
•0001
•0001
'0001
'0142
'0117
'0097
'0079
0.3373
•3100
'9999
'9999
'0023
'0019
•0015
'0012
0.0632
'0545
0'1672
'9993
'9991
'9989
'9986
0.45
'46
'47
'48
•49
0.0029
'46
'47
'48
'49
0'50
0'9983
0 '9999
0'50
0•000I
0'0009
0'0065
•I2
•I3
'14
•16
.17
•18
•19
•27
•28
•29
•31
•32
-33
-34
'9999
'9999
'41
0'9999
'0004
•0003
'0648
3
0 '9999
'9997
'9994
'9991
'9985
0'9999
'9999
'9998
•0468
'1322
'2837
'48 52
•0399
'0339
•1167
•io26
•2585
'2346
'4549
'4249
0.7414
•7160
-6895
•6620
.6337
0'0287
0'0898
0'2120
0.3953
0.6047
•1490
See page 4 for explanation of the use of this table.
1
0 '9999
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
t/ = 14
71 = 15
r= 0
I
2
p = o•ox
'02
'03
p = 0.01
o•86oi
r= 8
9
xo
II
12
13
3
.02
'7386
'03
'6333
0.9904
'9647
'9270
'04
.04
.5421
'8809
0.9996
.9970
.9906
'9797
0.05
•06
.07
-438
•09
0.05
-o6
.07
•o8
•09
0.4633
'3953
'3367
•2863
0.8290
'7738
.7168
.6597
0.9638
'9429
•9171
•887o
0.9945
'9896
•9825
.9727
'2430
•6035
•8531
•9601
0•I0
'II
•I2
•I3
crro
•x x
0'2059
0.5490
'4969
'4476
'4013
.3583
0.8159
/762
'7346
•6916
•6480
0.9444
.9258
•9041
•8796
•8524
0.6042
•5608
•5181
0'8227
'14
0.15
•x 6
'17
•x8
'19
.9999
'9999
.9998
'9997
0'20
•2I
•22
0.9996
.23
'24
.9989
'9984
0'25
0'9978
.26
'27
.28
.29
'9971
'9994
'9992
'9962
-9950
'9935
0 '9999
.9999
.9998
'9998
0.9997
.9995
'9993
'9991
.9988
0.9999
'9999
.9999
-9998
•12
'1470
•1238
'1041
0'15
0.0874
•x6
•17
•i 8
•0731
•o611
0.3186
-2821
•2489
'0510
- 0424
'2187
'4766
'7218
•19
'1915
'4365
•6854
0'20
'21
'22
0'0352
•0291
•0241
0'1671
0'3980
0'6482
•1453
.3615
•6105
'1259
'3269
'5726
•23
•24
•0198
•0163
•1087
.0935
'2945
.2642
'5350
'4978
0•25
'26
'27
0'0134
•0109
•0089
•0072
•0059
0'0802
'0685
'0583
'0495
0.2361
'2101
•1863
0'4613
•4258
'0419
•1447
•3914
•3584
.3268
0.0047
•0038
0.0353
•0296
o•,268
•ii07
0.2969
•2686
•28
•29
'9963
'9952
0.9999
'9999
'9999
'31
•32
'33
'34
0 '9757
0 '9940
0 '9989
0'9999
o•35
'9706
'9647
.9580
'9503
'9924
'9905
.9883
'9856
'9986
'9981
.9976
'9969
'9998
'9997
.9997
'9995
•36
0.9417
'9320
'9211
.9090
'8957
0.9825
'9788
'9745
0.9961
'9951
'9939
0'9994
0'9999
'41
'42
'43
'44
*9696
'9924
'9639
'9907
'9992
'9990
'9987
'9983
'9999
'9999
'9999
'9998
0.45
'46
'47
'48
'49
o•88ii
•8652
'480
'8293
'8094
0'9574
0.9886
•9861
•9832
'9798
'9759
0.9978
'9973
•9966
.9958
'9947
0 '9997
•9500
'9417
'9323
'9218
0'50
0.7880
0.9102
0.9713
0.9935
0'9991
0.9917
'9895
0'9983
'9869
'9971
'9837
'9800
0.35
.36
'37
.38
'39
040
'9978
0'30
•0031
'0248
'0962
'2420
-0206
•0833
•2171
'0171
'0719
•1940
0.0016
-00I2
.00 I 0
•0008
•0006
0'0142
'0117
'0096
0'0617
•0528
•0450
0'1727
•1531
•I351
•0078
•oo64
•o382
•0322
•1187
•I039
0'0005
'0004
•0003
0002
0002
0'0052
•0042
•0034
'0027
0021
0•0271
•0227
•0189
*0157
-0130
0'0905
'0785
'0678
' 0583
'0498
0'0001
0'0017
•0013
'0010
'0008
0.0107
0.0424
- 0087
'0071
'0057
*0359
•0303
'0254
'49
•0006
•0046
-0212
0'50
0'0005
0'0037
0.0176
-38
'39
0'40
•41
•42
'43
'44
0.45
'46
'47
•48
0.9999
See page 4 for explanation of the use of this table.
1
4
.1645
•7908
'7571
•0025
.0020
'37
'9997
'9996
'9994
'9993
'9992
'9976
•13
'14
0.9998
'9997
.9995
'9994
'9992
0.30
'31
.32
'33
'34
•1741
0 '9998
•0001
•0001
.000 I
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
Y/
= 15
r= 4
5
6
8
7
9
xo
II
12
13
p = 0. oi
'02
'03
0 '9999
'04
'9998
0.05
•36
0'9994
0 '9999
'9986
.07
'9972
'9999
'9997
'9993
'9987
-o8
'9950
.09
•9918
0'10
'II
'I2
0.9873
'13
•9639
'14
0.15
•16
•17
•18
•19
0'20
'21
'22
.23
'24
0.25
•26
'27
•28
.29
0.30
'31
'32
'33
'34
.9522
0.9997
'9994
'9990
'9985
'9976
0.9383
0.9832
0'9999
'9773
0.9964
'9948
-9926
•9898
.9863
0 '9994
•9222
.9039
'9990
.9986
'9979
'9970
'9999
'9998
'9997
'9995
0'9999
0.9819
*9766
.9702
•9626
.9537
0.9958
'9942
•9922
'9896
'9865
0.9992
'9989
'9984
'9977
'9969
0.9999
'9998
'9997
'9996
'9994
0.9434
.9316
-9183
'9035
•8870
0'9827
.9781
'9726
'9662
'9587
0.9958
'9944
•9927
.9906
.9879
0'9992
'9989
.9985
'9979
'9972
0 '9999
'9998
'9998
'9997
'9995
0 '9999
0.8689
•8491
•8278
•8049
•7806
0.9500
'9401
.9289
.9163
•9023
0.9848
•9810
'9764
.9711
'9649
0.9963
.9952
'9938
.9921
'9901
0 '9993
0'9999
-9991
.9988
'9984
'9978
'9999
'9998
'9997
'9996
0.7548
•7278
•6997
'6705
•6405
0.8868
•8698
'8513
'8313
-8098
0.9578
'9496
'9403
.9298
•9180
0.9876
.9846
.9810
•9768
'9719
0.9972
'9963
'9953
'9941
.9925
0 '9995
0'9999
'9994
'9991
.9989
'9985
'9999
'9999
'9998
'9998
0'6098
0.7869
-7626
'7370
.7102
•6824
0.9050
8905
.8746
•8573
.8385
0.9662
'9596
'9521
'9435
'9339
0.9907
.9884
-9857
.9826
'9789
0.9981
'9975
'9968
-9960
'9949
0 '9997
0.8182
7966
'7735
0 '9745
0 '9937
0'9989
0 '9999
3
4
29
30
0.9231
•9110
.8976
•88
7
69
62
'9695
-9637
-9570
'9494
'9921
.9903
•9881
'9855
.9986
.9982
'9977
'9971
'9998
'9998
'9997
'9996
0.6964
0.8491
0.9408
0.9824
0'9963
0'9995
'9735
•8833
•8606
0.8358
'8090
'7805
'7505
'7190
0.6865
•6531
•6190
.5846
'5500
0.5155
'4813
'4477
'41 48
•3829
.9700
'9613
.9510
0.9389
.9252
•9095
•8921
•8728
0.8516
•8287
•8042
.7780
'7505
0.7216
•6916
'6607
•6291
•5968
•38
'39
•2413
'4989
•4665
'4346
0'40
0'2173
'1948
0'4032
'3726
•1739
•36
'37
'9998
0.9978
'9963
'9943
'9916
'9879
'9813
0.3519
•3222
'2938
•2668
0'35
0 '9999
0'5643
'5316
0 '9999
'9999
'9998
'9996
'43
'44
'1546
'3430
'3144
•1367
•2869
•5786
'5470
'5153
'4836
o•45
0•1204
'46
0.2608
'2359
0.4522
'4211
•2125
'3905
'48
•1055
'0920
.o799
.1905
•3606
0.6535
•6238
'5935
•5626
49
0690
'1699
3316
5314
0'50
0.0592
0.1509
0.3036
0.5000
'41
.42
'47
.77
0'9999
'9999
See page 4 for explanation of the use of this table.
15
'9996
'9995
'9993
'9991
0 '9999
'9999
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n = i6
P=
1' = 0
I
2
3
4
•or
0•8515
0'9891
•02
'7238
•9601
•03
'04
•6143
-9182
'8673
0.9995
.9963
'9887
'9758
0.9998
*9989
'9968
0 '9999
.5204
5
6
0.9999
'9999
*9997
7
8
9
10
'9997
•07
'3131
•o8
•2634
0.81(38
.7511
.6902
'6299
•09
'221 I
'5711
0.9571
.9327
'9031
•8689
•8306
0.10
43.1853
0.7892
'7455
-7001
'6539
•6074
0.9316
'9093
•8838
.8552
'8237
0.9830
'9752
•9652
.9529
'9382
0.9967
'9947
•9918
•9880
-9829
0'9999
'1 550
0.5147
'4614
•4115
.3653
-3227
0 '9995
•xx
•x2
•13
•14
'9991
.9985
.9976
'9962
'9999
.9998
.9996
'9993
0'5614
*5162
.4723
'4302
.3899
0.9209
0 '9765
0 '9944
•9012
'8789
'8542
•8273
'9920
•9888
'9847
'9796
0'9989
'9984
'9976
'9964
'9949
0 '9998
'7540
'7164
'6777
'6381
'9685
'19
0'2839
*2487
.2170
•1885
-1632
0'7899
•0614
.0507
•0418
.0343
0'20
•21
'22
0'0281
'0230
'0188
0'1407
'1209
'1035
0'3518
'3161
0'5981
0'7982
.7673
o'9183
•9008
'7348
•8812
•23
.0153
.0883
.2517
'24
'0124
'0750
'2232
*4797
•4417
'7009
•6659
'8595
*8359
'9979
'9970
'9959
'9944
.9996
*9994
'9992
'9988
0 '9999
•5186
0'9930
'9905
'9873
'9834
'9786
0 '9998
'2827
0. 9733
-9658
•9568
'9464
'9343
0 '9985
•5582
0'25
0'0100
0'0635
0'1971
0'4050
.0535
-0450
'0052
'0042
'0377
'0314
'1733
'1518
.1323
'3697
•3360
•3041
'1149
'2740
0.8103
'7831
'7542
'7239
.6923
0'9204
'9049
'8875
•8683
'8474
0.9729
•9660
'9580
*9486
'9379
0.9925
'9902
'9873
'9837
'9794
0 '9997
•0081
•0065
o•63o2
'5940
'5575
•5212
'4853
0'9984
•26
'9977
'9969
'9959
'9945
•9996
'9994
*9992
.9989
0.0033
•oo26
o.o261
•0216
0.0994
•0856
0 '2459
'0021
•0178
'0734
'33
'34
•0016
•0146
•0626
'3819
'3496
0.8247
•8003
'7743
. 7469
'001 3
'0120
'0533
'1525
'3 187
'5241
'7181
0'9256
•9119
'8965
'8795
'8609
0 '9984
'32
0.6598
*6264
•5926
'5584
0 '9929
•2196
'1953
•1730
0 '4499
.4154
0 '9743
•31
•9683
•9612
'9530
'9436
'9908
.9883
.9852
'9815
'9979
.9972
.9963
'9952
0.35
•36
0.00 10
•0008
0'0451
'37
-0006
0-1339
•1170
. 10 1 8
•0881
0.4900
•4562
.4230
0.6881
•6572
'62 54
'3906
•5930
'3592
•5602
0.8406
•8187
'7952
.7702
'7438
0.9329
•9209
'9074
-8924
'8758
0.9771
.9720
'9659
'9589
'9509
0'9938
-0380
•0319
•0266
0.05
•o6
0.4401
0.15
•16
.17
•18
•27
-28
•29
0.30
.3716
•1293
.1077
•0895
0.0743
0'9930
•9868
.9779
'9658
.9504
0'9991
.9981
-9962
. 9932
'9889
0'9999
.9998
'9995
'9990
*9981
'9588
'9473
'9338
0'9999
'9999
'9997
'9996
'9993
'9990
0 '9999
'9999
'9998
'9999
'9999
.9998
•38
.0005
0.0098
•0079
•0064
-0052
'39
'0004
'0041
'0222
'0759
o'2892
•2613
'2351
•2105
'1877
0.40
0.0003
0.0033
'41
•42
'0002
'0026
'0021
o'o183
•0151
•0101
•oo82
0.0651
.0556
'0473
•0400
•0336
0.1666
. 1471
'1293
•1131
•o985
o'3288
'2997
•2720
. 2457
•2208
0.5272
'4942
•4613
.4289
'3971
0'7161
•6872
.6572
'6264
'5949
0.8577
•8381
•8168
'7940
.7698
0.9417
-9313
'9195
'9064
'8919
0.9809
.9766
.9716
.9658
.9591
o'o281
'0234
•0194
- 0160
•0131
0.0853
'0735
•0630
'0537
•0456
0.1976
'1759
'1559
'1374
•1205
0.3660
'3359
•3068
'2790
'2524
0.5629
'5306
•4981
'4657
'4335
0.7441
•7171
•6889
'6595
'6293
0'8759
•8584
*8393
•8,86
*7964
0 '9514
0'0106
0'0384
0'1051
0'2272
0'4018
0.5982
0.7728
0 '8949
'43
'44
•0002
•000i
•0016
'0001
'0013
0.0001
•0001
0.00,0
-oo08
•0006
'0124
•48
•0005
'49
•0003
0•0066
•0053
•0042
•0034
•0027
0•50
0'0003
0'0021
0'45
'46
'47
See page 4 for explanation of the use of this table.
16
'9921
'9900
'9875
'9845
.9426
'9326
'9214
.9089
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n = 16
7= 11
12
13
14
n = 17
p=
P = o•ox
'02
•03
'04
7= 0
0.0I
'02
0.8429
'7093
.03
'04
'5958
I
2
0 '9877
0.9994
'9956
•9866
'9714
'4996
'9554
•9091
.8535
0.4181
'3493
0.7922
.7283
'07
'2912
•o8
•09
•2423
•6638
'6005
'2012
'5396
0.4818
- 4277
3
4
5
0 '9997
•9986
•9960
0.9999
'9996
•8073
0.9912
.9836
'9727
'9581
'9397
0.9988
'9974
'9949
•9911
'9855
0.9999
'9997
'9993
'9985
'9973
0.9174
.8913
•8617
•8290
0.9779
'9679
'9554
.9402
0.9953
.9925
•9886
0.9497
.9218
•8882
0.05
'o6
'07
•o8
•og
0'05
0•I0
•II
- 12
•I3
0•10
'II
•12
0.1668
.0937
.0770
.3318
'14
'13
'14
0.7618
'7142
•6655
•6164
'2901
'5676
'7935
'9222
0.15
•x6
•x.7
•x8
•19
0.15
•16
•17
•18
•19
0.0631
•0516
•0421
.0343
•0278
0'2525
0'5198
0'7556
0'9013
•2187
•1887
•1621
-1387
'4734
•4289
•3867
'3468
•7159
'6749
'6331
•5909
'8776
.8513
•8225
'7913
0.9681
'9577
'9452
'9305
'9136
0'20
•2I
'22
0•20
'21
'22
0'0225
•0182
0'1182
•1004
0'3096
0'5489
•5073
0'7582
0.8943
'2751
'0146
•0849
'2433
'4667
'7234
•6872
'8727
•8490
'23
'24
•23
•0118
'0094
•0715
•0600
•2141
'4272
•6500
'8230
'24
•1877
•3893
'6121
'7951
0.0075
.0060
0.0501
•0417
0.1637
'1422
0.3530
0'5739
'3186
•0047
'0346
•oz86
'0235
'1229
•Io58
•2863
•2560
'5357
'4977
-4604
0.7653
'7339
.7011
•6671
*0907
'2279
'4240
-6323
0'0193
'0157
'0128
'0104
•0083
0'0774
'0657
•0556
'0468
'0392
0'2019
-1781
•1563
'1366
•1188
0'3887
0'5968
'3547
.5610
•2622
'4895
'4542
0'25
•26
.27
'28
-29
•o6
0'25
0.9999
'9999
'9999
'9998
•26
0'30
0'0023
•0018
0'0007
•0005
0'0067
0'0327
0'1028
0'2348
0'4197
•0054
•0272
•0885
•0004
•0043
•0034
'0225
•0185
-3861
'3535
-1640
•0027
•0151
•0759
-0648
'0550
•2094
.1858
' 1441
•3222
'2923
0'0021
0'0123
0'0464
0•12,60
•0016
•0013
•0100
•oo8o
•0390
•0326
•1096
•0949
'0010
'0065
•0271
- 0817
'2121
'1887
•0008
•0052
•0224
•0699
•1670
0.0041
•0032
•0025
•0020
0.0184
•0151
'0123
•0099
0.0596
•0505
'0425
•0356
0'1471
•iz88
'I 122
•0972
•0015
•oo8o
-0296
-0838
0'0012
0.0064
0'0245
0.0717
0'9999
'9999
'9999
0.35
.36
'37
-38
'39
0'9987
0 '9998
0'35
'9983
'9977
'9970
'9962
'9997
'9996
'9995
'9993
'36
0•40
0.9951
'9938
'9999
'37
•38
'39
0 '9991
0 '9999
0'40
.9988
'41
'42
'43
'44
0'9999
- 0038
•0014
•0011
•0009
'0003
'0002
0'0002
'0001
'9922
'9984
'9902
'9879
'9979
'9973
.9998
.9998
'9997
'9996
0.45
'46
'47
'48
'49
0.9851
'981 7
'9778
'9732
.9678
0.9965
'9956
'9945
'9931
.9914
0.9994
'9993
'9990
'9987
'9984
0.9999
'9999
'9999
'9999
'9998
0.45
0'0006
'46
'47
'48
'49
•0004
•0003
0•50
0'9616
0.9894
0.9979
0 '9997
0•50
O'COOI
'41
'42
'43
'44
*9834
'9766
'0030
'31
•32
'33
'34
'31
.32
'33
'34
'3777
'27
'28
•29
0.9997
'9996
'9995
'9993
'9990
0'30
'1379
'11 3 8
*8497
•0001
•000x
•000x
•0002
•0002
See page 4 for explanation of the use of this table.
17
'3222
•2913
'5251
0.2639
'2372
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
7Z=17
r=6
7
8
0.9999
'9998
'9996
'9993
.9989
0'9999
0.9997
'9995
'9992
.9988
-9982
9
xo
II
12
13
14
15
p = o•ox
'02
'03
'04
0.05
•o6
.07
•o8
0 '9999
'9998
.09
'9996
0'10
0.9992
•II
•12
•13
'9986
.9963
'14
'9944
0'15
0.9917
•16
-9882
•17
•x8
•19
.9837
.9709
0.9983
'9973
'9961
'9943
.9920
0'20
'21
•22
0.9623
0.9891
0 '9974
0 '9995
0 '9999
'9853
.23
•24
'9521
•9402
'9264
•9106
•9680
.9963
'9949
'9930
.9906
'9993
•9989
'9984
'9978
'9999
'9998
'9997
'9996
0 '9999
0•25
0.8929
0.9598
0.9876
0 '9999
•8732
•9501
.9839
•8515
'9389
•28
•8279
•9261
'9794
'9739
•29
•8024
•9116
•9674
0.9969
'9958
'9943
.9925
.9902
0 '9994
•26
.27
'9991
'9987
•9982
'9976
'9998
'9998
'9997
'9995
0 '9999
0'30
0.7752
0 '9993
0'9999
•32
'7162
'8 574
'33
'34
•6847
•6521
•8358
.9508
'9405
•9288
•8123
'9155
0'9873
.9838
'9796
'9746
•9686
0 '9968
'7464
0.8954
.8773
0 '9597
'31
'9957
'9943
.9926
•9905
'9991
.9987
.9983
'9977
'9998
'9998
'9997
'9996
0 '9999
0.35
.36
0.6188
.5848
0.9617
'9536
'9443
'9336
•9216
0.9880
. 9849
•9811
•9766
'9714
0.9970
•9960
'9949
'9934
.9916
0 '9999
•5505
0.9006
'8841
.8659
.8459
•8243
0 '9994
'37
0•7872
•76o5
'7324
•7029
•6722
'9992
.9989
.9985
.9981
'9999
'9998
'9998
'9997
0.6405
•6080
'5750
'5415
•5079
0.801 x
.7762
'7498
.7220
•6928
o•9o81
•8930
. 8764
.8581
•8382
o.9652
•9580
'9497
'9403
'9295
0.9894
.9867
.9835
'9797
'9752
0 '9975
0 '9995
0'9999
.9967
'9958
'9946
'9931
'9994
'9992
.9989
.9986
'9999
'9999
'9998
'9998
0 '4743
'49
•1878
.3448
0.6626
•6313
'5992
.5665
'5333
0.8166
'7934
•7686
'7423
'7145
0.9174
.9038
•8888
•8721
.8538
0.9699
.9637
•9566
'9483
.9389
0.9914
.9892
•9866
.9835
'9798
0.9981
'9976
.9969
.9960
'9950
0 '9997
'48
0.2902
•2623
.2359
•2110
0.50
o•1662
0.3145
0.5000
0.6855
0.8338
0.9283
0.9755
0.9936
0.9988
'9977
-978o
•38
•5161
'39
'4818
0'40
0.4478
'41
'42
'4144
.3818
'3501
•3195
'43
'44
0'45
'46
'47
•9806
'9749
'4410
'4082
•3761
'9999
'9998
0 '9999
'9999
'9998
'9997
See page 4 for explanation of the use of this table.
18
'9996
'9995
'9993
'9991
0 '9999
'9999
'9999
0'9999
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n = 18
r=0
x
2
3
4
5
p = c•in
•02
.03
.04
0•8345
0.9862
'6951
•9505
-578o
'4796
•8997
•8393
0.9993
•9948
'9843
.9667
0.9996
.9982
'9950
0.9998
'9994
0 '9999
0.05
0.3972
0.7735
•o6
•3283
.7055
0.9419
•9102
.8725
.8298
.7832
0.9891
'9799
'9667
'9494
.9277
0-9985
.9966
'9933
'9884
.9814
0.9998
'9995
'9990
'9979
.9962
0 '9999
'9997
'9994
0 '9999
0.9018
.8718
•8382
•8014
.7618
0.9718
'9595
'9442
'9257
.9041
0.9936
.9898
.9846
'9778
•9690
0.9988
'9979
.9966
'9946
.9919
0.9998
'9997
'9994
'9989
•9983
6
7
8
9
xo
•07
•2708
•6378
•o8
'2229
•5719
•09
•1831
.5091
0•10
'II
0.1501
0.4503
•12
•1227
•I002
•395 8
•3460
•13
•0815
•14
•o66z
•3008
•26oz
0.7338
.6827
•6310
'5794
•5287
0.15
0.0536
'17
•i8
•19
.0349
-0225
0.4797
'4327
•3881
•3462
•3073
0.7202
'6771
-6331
•5888
'5446
0.8794
-8518
•8213
•7884
'7533
0.9581
'9449
'9292
•9111
•8903
0.9882
'9833
'9771
.9694
•9600
0.9973
'9959
'9940
'9914
•988o
0 '9999
'0434
0.2241
'1920
'1638
•1391
•1176
0 '9995
•x6
'9992
•9987
•9980
'9971
'9999
'9998
'9996
'9994
0'20
'21
•22
0'0180
•0144
'0114
'0091
'0072
0'0991
'0831
'0694
'0577
'0478
0'2713
•2384
0'5010
'45 86
0'7164
'6780
0'8671
•8414
0'9487
0.9991
0 '9998
-2084
.1813
'4175
•3782
•1570
.3409
•6387
'5988
'5586
•8134
•7832
•7512
0.9837
'9783
'9717
'9637
'9542
0'9957
'9355
•9201
-9026
•8829
'9940
'9917
'9888
.9852
'9986
'9980
'9972
•9961
'9997
'9996
'9994
'9991
0.25
0'0056
•26
•27
•28
•29
0.0044
0.0395
•0324
0.1353
•1161
0.3057
•2728
0.5187
'4792
0.7175
•6824
0.8610
-837o
0.9431
'9301
•0035
'0265
'0991
'2422
'4406
•6462
•8109
'9153
•oo27
•0021
•0216
•0842
•2140
•4032
•6093
'0712
•1881
•3671
•5719
.8986
•8800
0-9946
.9927
•9903
.9873
'0176
'7829
'7531
0.9807
'9751
-9684
'9605
•9512
0.9988
'9982
'9975
'9966
'9954
0.30
'31
•32
0.0016
•0013
•0010
0.0142
•0114
•0092
o.o600
•0502
•0419
0.1646
'1432
0.3327
'2999
0.5344
'4971
0'7217
•6889
0.8593
•8367
0.9404
•9280
0 '9939
'0073
'0348
'4602
•4241
•6550
•6203
.91 39
•0007
-0006
•2691
'2402
•8122
•33
'34
'1241
'1069
•0058
•0287
.0917
•2134
'3889
•5849
•7859
'7579
•8981
•8804
0.9790
'9736
.9671
'9595
-9506
0.35
0'0004
0.3550
0.5491
'0665
•1659
'3224
. 5 1 33
•0561
•1451
•2914
'4776
•0002
0'0236
•0193
'0157
'0127
0.1886
-0003
•0002
0'0046
'0036
'0028
•0022
0.0783
•36
•0472
•1263
'2621
.4424
'39
•0001
•0017
•0103
'0394
•1093
- 2345
'4079
0.7283
.6973
•6651
•6319
'5979
0.8609
.8396
•8165
'7916
•7650
0.9403
•9286
.9153
•9003
.8837
40.9788
'9736
.9675
•9603
'95 20
0.40
0.0001
0.0013
0.0082
•0001
•0001
•0010
'0008
'0066
•0052
•0041
0.0328
•0271
-0223
•0182
0.0942
•0807
•0687
•0582
0.2088
'1849
•1628
•1427
0.3743
•3418
'3105
•2807
0.5634
'5287
'4938
'4592
0.7368
.7072
•6764
'6444
'0032
'0148
'0490
•1243
•2524
•423 0
'6115
0.8653
•8451
•8232
'7996
'7742
0 '9424
'41
0'2258
'2009
0-3915
•1778
-1564
.3272
'2968
0.5778
'5438
'5094
'4751
•1368
'2678
.4409
0.7473
.7188
•6890
-6579
•6258
0.8720
•8530
•8323
•8098
•7856
0.1189
0.2403
0.4073
0.5927
0.7597
•23
-24
'37
•38
'42
•oz81
'43
'44
•0006
•0004
0'45
0'0003
'49
0•0120
•0096
'0077
•0061
'0048
0.1077
•0002
•0002
•0001
•0001
0.0025
'0019
'0015
•0011
'0009
0.0411
'46
'0342
•0283
'0233
•0190
'0928
•0795
'0676
'0572
0.50
0.0001
0.0007
0.0038
0.0154
0.0481
'47
'48
•3588
See page 4 for explanation of the use of this table.
19
0'9999
'9998
'9997
'9836
0 '9999
'9999
•9920
•9896
.9867
•9831
-9314
•9189
.9049
'8893
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n= ig
r
P = o•ox
P= o•oz
'02
'03
'02
•03
•04
•5606
•8900
•4604
0'05
•o6
•07
•08
' 20 5 1
'5440
'8092
•09
'1666
'4798
.7585
•1()
•II
•12
0.1351
0'4203
0'7054
•1092
'3658
'6512
•0881
'0709
•3165
'2723
-5968
.5432
'0569
*2331
•4911
0'0456
-0364
-0290
•0230
•0182
0•1985
-1682
•1419
•1191
-0996
0'4413
0'20
'21
'22
'23
•24
0.0144
o'o829
-0687
0.2369
•2058
0'25
n = 18
r = II
12
13
14
xs
x6
'04
0'05
'o6
.07
•o8
•09
0'10
•I I
•I2
•I3
'I4
•13
'14
0.15
0'15
•16
•x7
•x8
•x6
•x7
•x8
•19
•19
0'20
'21
'22
0 '9999
'23
'24
'9999
'9998
0'25
0 '9998
'9997
•26
'27
•28
.29
'9995
-9993
*9990
0-9999
'9999
'9999
'9998
=
I
2
0.8262
0•9847
'6812
'9454
'8249
0'9991
'9939
-9817
*96 I 6
0 '3774
0 '7547
0 '9335
•3086
-6829
•6i2x
.8979
•8561
0
'2519
'0113
.3941
'3500
-3090
-2713
'0089
•0566
•I 778
'0070
*0465
*1529
'0054
'0381
•I 308
0.0042
•0033
0.0310
•0251
'0203
'0163
•0667
'29
'0025
*00 I 9
•0015
o•iii3
'0943
.0795
•0131
'0557
•26
'27
-28
0.9986
.998o
'9973
'9964
'9953
0 '9997
0'30
0'0011
0'0104
0'0462
.31
'32
'33
'34
'9996
'9995
'9992
'9989
0'9999
•3 I
'0009
'0083
'9999
'9999
'9998
•32
•33
'34
-0007
•0065
•0382
•0314
'0005
•0051
'0257
'0004
•0040
'0209
0'35
0 '9938
•36
•37
.38
'39
.9920
.9898
-9870
'9837
0'9986
•9981
'9974
.9966
'9956
o•9997
'9996
'9995
'9993
'9990
0.40
'41
'42
'43
'44
0 '9797
'9750
'9693
•9628
'9551
0'9942
'9926
'9906
•9882
-9853
0.45
'46
'47
'48
'49
0.9463
.9362
'9247
•9117
'8972
0'50
o•8811
0'30
0.35
0'0003
0'0031
0'0170
0'9999
•36
•0002
'9999
'9999
'9998
•37
•38
-39
•0002
'0024
'0019
' 0137
'0110
'0001
•0014
'0087
•000x
'0011
'0069
0 '9987
0'9998
0'40
o•000i
'9983
-9978
'9971
.9962
'9997
'9996
'9994
'9993
0.0008
•0006
•0005
•0004
•0003
0•0055
'0043
•0033
'9999
'9999
'41
'42
'43
'44
0.9817
'9775
.9725
•9666
'9598
0 '9951
0'9990
'9987
.9983
'9977
'9971
0 '9999
0'45
0'0002
'9937
'9921
'9900
'9875
'9998
'9997
'9996
'9995
•000x
•0001
0.0015
•0012
0 '9999
'46
'47
48
'49
0.9519
0.9846
0'9962
0 '9993
0 '9999
0•50
0 '9999
See page 4 for explanation of the use of this table.
20
'0026
'0020
'0009
'000!
'0007
'0001
•0005
0'0004
TABLE I. THE BINOMIAL DISTRIBUTION FUNCTION
=1
9
r=
3
4
5
6
8
7
10
9
II
12
13
p = 0'01
'02
•03
•04
0 '9995
-9978
*9939
0.9998
'9993
0'05
•o6
0.9868
'9757
0.9980
'9956
•07
'9602
'9915
•98
.09
.9398
'9147
-9853
*9765
o•io
0•885o
•II
•I2
'8510
'8133
0.9648
'9498
.13
-7725
.14
-7292
o•15
•16
•17
0.6841
•638o
.5915
'18
-19
'5451
'4995
0'20
'21
'22
0.4551
.23
-24
.3329
0 '9999
0.9998
'9994
.9986
.9971
'9949
0 '9999
.9998
.9996
*9991
0.9999
'9999
-9096
.8842
0'9914
•9865
-9798
•9710
'9599
0.9983
'9970
'9952
'9924
.9887
0.9997
'9995
'9991
'9984
'9974
0.8556
•8238
-7893
'7524
*7136
0.9463
.9300
.9109
'8890
-8643
0'9837
.9772
-9690
'9589
'9468
0 '9959
0.8369
.8071
0.9324
.9157
0.9767
'9693
'7749
- 7408
.7050
•8966
-9604
'2968
0.6733
.6319
'5900
-5480
.5064
'8752
'8513
'9497
'9371
0.25
0.2631
0.4654
0.6678
0•8251
0.9225
-26
•27
-28
•29
'2320
'2035
'4256
'6295
'7968
'9059
-5907
.7664
.1776
-3871
. 3502
'5516
•1542
'3152
•5125
0.30
0.1332
•32
.1144
•0978
'33
'34
'0831
'0703
0'2822
. 2514
'2227
' 1963
0'4739
•31
0.35
'4123
'3715
.9315
'4359
'3990
.9939
.9911
'9874
.9827
0 '9999
'9998
'9997
'9995
0 '9999
0.9992
.9986
'9979
'9968
'9953
0.9999
-9998
'9996
'9993
'9990
0 '9933
0-9984
'9977
*9966
'9907
'9873
'9831
'9778
'9953
'9934
0 '9999
'9999
'9998
0 '9997
'9995
'9993
'9989
'9984
0'9999
'9997
0 '9999
0'9977
0.9995
0 '9999
'9968
.9956
'9940
'9920
'9993
.9990
'9985
'9980
'9999
.9998
'9997
'9996
0 '9999
'9999
. 9998
'7343
.8871
.8662
0.9713
-9634
.9541
.9432
0.9911
.9881
-9844
'9798
•7005
'8432
'9306
'9742
0.6655
'6295
'5927
'5555
-5182
0.8180
'7909
•7619
'7312
•6990
0.9161
'8997
-8814
'8611
•8388
0 '9674
0-9895
•9863
.9824
'9777
'9720
0.9972
.9962
'9949
'9932
'9911
0 '9994
0 '9999
'9595
•9501
'9392
•9267
.9991
'9988
'9983
'9977
-9998
-9998
'9997
'9995
0.6656
•6310
'5957
'5599
*5238
0.8145
.7884
'7605
'7309
'6998
0.9125
.8965
.8787
'8590
'8374
0.9653
'9574
.9482
'9375
'9253
0.9886
'9854
'9815
'9769
'9713
0.9969
'9959
'9946
'9930
'9909
0 '9993
0.9884
'9854
-9817
'9773
•9720
0.9969
•9960
9948
'993 3
'9914
.1720
'3634
.3293
0.0591
•0495
0- I 500
•1301
0.2968
-2661
'39
•0412
'0341
•0281
•1122
'0962
•0821
'2373
'2105
•1857
0.4812
'4446
'4087
•3739
•3403
0'40
0'0230
0'0696
0'1629
0'3081
-0587
-0492
•0410
•0340
•1421
- 1233
•1063
- 0912
'2774
•2485
•2213
•1961
0.6675
*6340
'5997
.5647
- 5294
0.8139
•7886
•7615
.7328
•7026
0.9648
'0187
- 0151
0.4878
.4520
-4168
•3824
.3491
0'9115
'V
•896o
-8787
.8596
-8387
'9571
'9482
'9379
•9262
0.0777
•0658
•0554
•0463
0.1727
.15 r 2
-1316
•1138
0.3169
•2862
-2570
•2294
•0978
'2036
0.6710
'6383
- 6046
'5701
- 5352
0.8159
'7913
- 7649
-7369
.7072
0'9129
'0385
0.4940
'4587
•4238
•3895
'3561
'8979
- 8813
-8628
.8425
0.9658
'9585
•9500
.9403
. 9291
0.9891
-9863
•9829
.9788
'9739
0'0318
0.0835
0.1796
0.3238
0.5000
0.6762
0.8204
0.9165
0.9682
.36
'37
•38
- 42
'43
'44
'0122
0.45
0.0280
•0229
•0186
•49
0'0077
•006r
'0048
'0037
•0029
0'50
0'0022
0'0096
•46
'47
'48
'0097
-0150
'0121
See page 4 for explanation of the use of this table.
21
'9991
.9987
'9983
'9977
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n = 19
P
r = 14
15
16
n = 20
r
= 0
I
2
0.9990
'9929
'9790
.9561
O• 0 I
•02
'03
'04
p = 0•0i
0.8179
•02
'6676
.03
'5438
0.9831
•9401
•8802
•04
'4420
'8103
0'05
0'05
•o6
-07
•o8
•09
•o6
•07
0.3585
•290I
0.7358
•6605
'2342
•5869
•o8
•1887
•09
•1516
•5169
•4516
&If)
•I I
•12
•X3
0•10
•I I
'12
0'1216
'0972
'0776
0.3917
•13
•0617
•2461
•14
'14
•0490
&IS
0'15
•x6
•x7
•/8
•19
•16
•17
•x8
•19
0'20
'21
'22
'23
•24
0'20
'21
'22
•23
-24
0•25
'26
'27
'28
'29
0'25
•26
•0014
•0123
•0526
'29
•0011
'0097
0'30
0'30
.31
'31
'32
o•0008
.0006
•0004
'33
'34
•0003
'32
'33
'34
'9999
.36
'37
'38
'39
0.9999
.9998
'9998
'9997
'9995
0.40
'V
0-35
4
5
6
0 '9994
'9973
.9926
0 '9997
'9990
0 '9999
0.9245
•885o
•8390
.7879
'7334
0.9841
'9710
'9529
•9294
-9007
0.9974
'9944
'9893
.9817
.9710
0.9997
'9991
•9981
.9962
'9932
0.6769
•6198
'2084
•5080
'4550
0.8670
•8290
.7873
'7427
.6959
0.9568
'9390
.9173
•8917
•8625
0.9887
.9825
'9740
-9630
'9493
0.9976
'9959
'9933
.9897
.9847
0.0388
0.1756
0-4049
0.6477
-0306
•1471
•3580
'5990
•0241
•1227
'3146
•5504
•o189
-0148
•1018
•2748
•5026
'0841
'2386
•4561
0.8298
'7941
'7557
•7151
.6729
0.9327
.9130
.8902
-8644
.8357
0.9781
-9696
'9591
'9463
'9311
0'0115
•0090
•0069
0'0692
'0566
•0461
0'2061
•1770
•1512
04114
•0054
.0374
•1284
-2915
0.6296
•5858
'5420
.4986
0.8042
•7703
'7343
•6965
'0041
'0302
'1085
'2569
'4561
'6573
0.9133
-8929
.8699
'8443
•8162
0'0032
'0024
•0018
0'0243
•0195
•0155
0'0913
'0763
•o635
0'2252
0'4148
0'6172
0•7858
'0433
•1962
•1700
•1466
•1256
•3752
'3375
•3019
•2,685
.5765
'5357
'4952
'4553
'7533
•7190
•6831
•6460
0.0076
•oo6o
•0047
0.0355
•0289
•0235
0.1071
•0908
•0765
0.2375
•2089
•1827
0.4164
•3787
.3426
0.6080
-5695
•5307
'0036
'0028
'0189
•3083
'4921
•0152
'0642
'0535
'1589
'0002
-1374
.2758
'4540
0.35
0'0002
0'0021
0'0121
0.0444
•0001
•0016
-0096
-0366
0.1182
•101 z
0'2454
•36
0.4166
•3803
•27
•28
0 '9999
3
'3376
•2891
•5631
-3690
'3289
•2171
0'9999
'9997
'9994
.9987
'37
•000 I
•0012
•0076
'0300
'0859
•1 9 10
. 3453
•38
•0001
•0009
•oo6o
•0245
•0726
•1671
0 '9999
'39
•0001
'0007
•0047
•0198
'0610
'1453
•3118
•2800
0 '9994
0'9999
0'40
0'0005
0'0036
0.1256
'41
•0004
•0003
-0028
-0021
•0423
'0349
1079
'43
'44
'9999
'9998
'9997
'9996
o•o16o
-0128
0.0510
'9991
'9988
'9984
'9979
0 '9999
0.45
0.9972
0'9995
'46
'47
'48
'49
'9964
'9954
'9940
'9924
'9993
'9990
'9987
'9983
0.50
0.9904
0'9978
0 '9996
. 42
42
'0783
•oo63
.0286
•0233
•0660
0.2500
2220
21925
1959
-1719
•1499
0.0009
0.0049
0.0189
0.0553
0.1299
'0007
'0005
'0152
•0121
•0096
'0076
'0461
•0381
•0313
'0255
•I I I 9
'0958
'o688
0.0059
0.0207
0.0577
•0102
.0080
'44
'0002
0002
0012
0 '9999
0'45
0.000!
'9999
'9999
'9998
'9997
'46
•0001
'47
•000I
43
'49
•0004
•0003
'0038
•0029
•0023
. 0017
0'50
0.0002
0.0013
•48
See page 4 for explanation of the use of this table.
22
.0922
•0814
TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION
n - 20
p=
r = 7
8
9
I0
I1
0.9999
'9999
.9998
'9996
0 '9999
12
13
14
15
16
0.01
'02
•03
'04
0.05
•o6
.07
•o8
'09
0.9999
'9998
0. 10
•II
•I2
0.9996
•13
•14
'9976
.9962
0' 15
0.9941
.16
'17
•18
'19
.9912
'9992
'9986
0.9999
.9999
.9998
*9995
'9992
0 '9999
'9999
'9759
0'9987
.9979
'9967
.9951
'9929
0'9998
.9996
'9993
-9989
'9983
0'20
•2I
0.9679
0.9900
0 '9974
•9862
'22
'23
•24
*9464
'9325
•9165
'9814
'9754
-9680
•9962
*9946
*9925
'9897
0.9994
'9991
'9987
•9981
'9972
0 '9999
'9581
'9998
'9997
.9996
'9994
0.9999
'9999
0'25
'26
0.8982
'8775
.27
•28
-29
'8 545
•8293
'8018
0.9591
'9485
'9360
0.9861
'9817
.9762
'9695
•9615
0.9961
'9945
.9926
.9900
•9868
0.9991
'9986
.9981
.9973
•9962
0.9998
'9997
*9996
'9994
'9991
0'9999
0'30
0.7723
-7409
0.8867
•866o
0'9987
0 '9997
•6732
•6376
-8182
•7913
'9909
-9881
'9846
-9982
'9975
•9966
'9955
'9996
'9994
'9992
'9989
0 '9999
'8432
0.9829
.9780
*9721
'9650
.9566
0 '9949
.9931
'7078
0.9520
.9409
'9281
.9134
•8968
0.35
'36
-37
•38
'39
0•6oio
'5639
.5265
-4892
'4522
0.7624
'7317
'6995
•6659
'6312
o•8782
'8576
'8350
•8103
'7837
0.9468
'9355
'9225
•9077
'8910
0.9804
'9753
'9692
•9619
'9534
0.9940
'9921
'9898
•9868
*9833
0.9985
'9979
'9972
.9963
'9951
0 '9997
'9996
'9994
'9991
'9988
0'9999
0.40
'41
'42
'43
'44
0.4159
'3804
'3461
'31 32
'2817
0.5956
'5594
'5229
'4864
•4501
0 '7553
0'8725
.852o
'8295
•8o5i
•7788
0 '9435
0'9984
0'9997
'9321
'9190
•9042
•8877
0'9790
.9738
.9676
•9603
'9518
0 '9935
'7252
'6936
'6606
•6264
•9916
'9893
•9864
•9828
'9978
'9971
.9962
'9950
'9996
'9994
.9992
'9989
0 '9999
0.45
'46
'47
48
'49
0.2520
'2241
'1 980
'1739
'1518
0.4143
*3793
•3454
'3127
•2814
0.5914
'5557
'5196
'4834
'4475
0.7507
*7209
•6896
•6568
*6229
0.8692
. 8489
-8266
•8o23
•7762
0'9420
- 9306
'9177
•9031
•8867
0'9786
'9735
.9674
'9603
'9520
0 '9936
0 '9985
0 '9997
'9917
'9895
•9867
'9834
'9980
'9973
'9965
'9954
'9996
'9995
'9993
*9990
0 '9999
0'50
0'1316
0.2517
0.4119
o'5881
0.7483
0.8684
0.9423
0 '9793
0 '9941
0'9987
0 '9998
.31
.32
'33
'34
'9873
-9823
'9216
•9052
'9999
'9998
'9999
'9999
'9998
See page 4 for explanation of the use of this table.
23
'9999
'9998
'9998
'9999
.9999
'9998
'9999
'9999
'9999
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
r=
0
I
3
4
5
6
The function tabulated is
0'00
'02
•04
1'0000
0'9802
•06
•o8
0.9418
0•9231
.9992
-9983
•9970
0.10
0.9048
0 '9953
•12
'8869
•14
•x6
•x8
'8694
•8521
.8353
'9934
'9911
•9885
.9856
0'20
'22
0.8187
0.9825
'8025
'9791
•24
•7866
0.9608
2
0'9998
F(r
for r = o, I, 2, ... and ft < 2o. If R is a random variable
with a Poisson distribution with mean it, F(r1/1) is the
r; that is,
probability that R
0 '9999
0.9998
'9997
'9996
'9994
'9992
r} = F(rl,u).
Pr {R
Note that
Pr {R r} =
=I
'26
'7711
'28
.7558
'9754
.9715
'9674
0'30
0.7408
0.9631
•32
'34
.36
-38
'7261
*9585
'7118
.6977
.6839
'9538
'9488
'9437
0.9989
.9985
.9981
-9976
'9970
0'9999
0.9964
.9957
'9949
'9940
'9931
0 '9997
'9999
'9999
'9998
'9998
-
F(r
F(r1/1)- Ffr -
-
r - I}
ilp).
=
r!
Linear interpolation in is is satisfactory over much of
the table, but there are places where quadratic inter-
.9997
'9996
'9995
'9994
polation is necessary for high accuracy. Even quadratic
interpolation may be unsatisfactory when r = o or I and
a direct calculation is to be preferred: F(olp) = e1 and
F(il#) =
0.6703
.6570
'6440
'6313
•6188
0'9384
.9330
'9274
•9217
•9158
0.9921
•9910
•9898
.9885
'9871
0.9992
'9991
'9989
'9987
'9985
0.9999
'9999
'9999
'9999
'9999
0'50
'52
o•6o65
'5945
'54
'5827
-56
.5712
-58
'5599
0.9098
'9037
'8974
•8911
'8846
0.9856
'9841
'9824
*9807
'9788
0.9982
'9980
'9977
'9974
'9970
0.9998
'9998
.9998
'9997
'9997
o•6o
•62
.64
-66
-68
0.5488
0.8781
'8715
'8648
0.9769
'9749
'9727
-8580
.851r
•9705
0-9966
.9962
'9958
'9953
'9948
0.9996
.9995
'9995
'9994
'9993
0•70
•72
0.4966
•9682
Pr {R
equal to
0.40
.42
'44
'46
'48
'5273
•5169
•5o66
-
The probability of exactly r occurrences, Pr {R = r}, is
+p).
R is approximately normally distributed
with mean it and variance p; hence, including 4 for
For # >
.5379
li:
=t
20,
continuity, we have
F(rlis)
Co(s)
where s = (r+4 p)Alp and 0(s) is the normal distribution function (see Table 4). The approximation can
usually be improved by using the formula
-
F(rip)
0:10(s) -
I 2rt
e-"I
(s2 - I) (s5 -7s2 +6s)1
6,/,7
72#
For certain values of r and > 20 use may be made of
the following relation between the Poisson and X2distributions :
F(rlit) = I -F2(r+i) (210
0'9999
where Fv(x) is the x2-distribution function (see Table 7).
Omitted entries to the left and right of tabulated values
'9999
'9999
are o and I respectively, to four decimal places.
'74
'4771
.76
•78
.4677
.4584
0•8442
•8372
•83o2
'8231
.8160
o•8o
- 82
.84
0.4493
'4404
.4317
o•8o88
•8o16
'7943
0.9526
'9497
'9467
-86
-88
.4232
-4148
.7871
.9436
'7798
'9404
0•90
0.4066
'92
'3985
0.7725
'7652
0-9371
'9338
'94
'3906
'7578
'9304
'96
'3829
'98
'3753
'7505
'7431
roo
0.3679
0-7358
'4868
j•
0'9659 0'9942 0'9992 0 '9999
'9634
'9991
'9999
'9937
-9608
.9930
'9990
'9999
'9582
'9924
'9989
'9999
'9998
'9917
'9554
'9987
0.9909
'9901
'9893
'9884
'9875
0.9986
'9984
'9983
'9981
'9979
0.9998
.9998
'9998
'9997
'9997
0'9977
'9974
'9972
'9969
.9966
0'9997
'9269
. 9233
0-9865
'9855
'9845
'9834
.9822
0.9197
0.9810
0.9963
0'9994
'9996
'9996
'9995
'9995
0'9999
'9999
'9999
0 '9999
24
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
ii
r
=
0
I
2
3
4
5
6
0.9197
'9103
'9004
•8901
'8795
0.9810
'9778
'9743
•9704
.9662
0.9963
'9955
'9946
.9935
'9923
0 '9994
0'9999
'9992
'9990
.9988
'9985
'9999
'9999
.9998
*9997
0.9617
.9569
'9518
'9463
'9405
0'9909
.9893
.9876
'9857
'9837
0'9982
'9978
'9973
'9968
'9962
0 '9997
•6092
'59 x 8
'5747
0.8685
•8571
•8454
'8335
'8213
'9996
'9995
'9994
'9992
0.9999
'9999
'9999
'9999
0 '9955
0'9991
.9989
'9987
*9984
'9981
0 '9998
'9948
'9940
'9930
'9920
8
7
9
10
II
I t.
r = 12
3'40
0'9999
'45
'9999
3.50
0-9999
1.00
'05
•xo
•15
0.3679
'3499
-3329
•3166
0.7358
*7174
•6990
•6808
'20
'3012
'6626
1.25
-30
'35
'40
'45
o'2865
0.6446
'2725
.2592
'6268
1.50
'55
•60
•65
•70
0.2231
•2422
•2019
•1920
•1827
0.5578
0.8088
'5412
'7962
0 '9344
'9279
•5249
*7834
.7704
. 7572
'9212
*9141
•9068
0.9814
'9790
'9763
'9735
'9704
1.75
-8o
.85
'90
'95
0.1738
•1653
-1572
'1496
'1423
0 '4779
0'7440
•7306
.7172
'7037
•6902
0.8992
-8913
.883 x
'8747
•866o
0.9671
*9636
'9599
'9559
•9517
0.9909
*9896
'9883
'9868
'9852
0.9978
'9974
'9970
'9966
.9960
0 '9995
0 '9999
•4628
.4481
'4337
•4197
'9994
'9993
'9992
.9991
'9999
'9999
'9998
.9998
2'00
0.1353
•1287
•1225
•1165
•x 108
0.4060
•3926
'3796
.3669
.3546
0.6767
•6631
•6496
•6361
•6227
0.8571
•848o
•8386
*8291
•8194
0 '9473
0'9989
'9987
'9985
'9983
'9980
0'9998
•9427
'9379
*9328
'9275
0'9834
- 9816
'9796
'9774
'9751
0'9955
•05
•10
•15
•20
'9997
'9997
'9996
'9995
0 '9999
2'25
0'1054
0.61293
'5960
•5828
'5697
.5567
0.8094
'7993
•7891
'7787
•7682
0.9220
'9162
•9103
'9041
•8978
0.9726
•9700
'9673
'9643
•9612
0.9916
•9906
*9896
'9884
0 '9994
0 '9999
•1003
'0954
'0907
•0863
0.3425
*3309
'3195
'3084
-2977
0'9977
•30
'35
'40
•45
'9974
'9971
'9967
'9872
'9962
'9994
'9993
'9991
'9990
'9999
.9998
.9998
.9998
2'50
o.o821
'0781
•0743
.0707
•0672
o•z873
'2772
.2674
'2579
'2487
0'5438
'5311
.5184
'5060
'4936
0.7576
'7468
. 7360
•7251
.7141
o'8912
'8844
'8774
'8703
•8629
0.9580
'9546
'9510
'9472
'9433
0.9858
'9844
'9828
•9812
'9794
0.9958
'9952
'9947
.9940
'9934
0.9989
'9987
*9985
.9983
'9981
0'9997
2.75
•8o
•85
.90
'95
0.0639
•0608
•o578
0.2397
-2311
•2227
0.4815
•4695
'4576
0.7030
'2146
'4460
•6696
'0523
'2067
'4345
'6584
0'9392
'9349
'9304
'9258
'9210
0'9776
'9756
'9735
'9713
'9689
0.9927
'9919
'9910
.9901
'9891
0.9978
'9976
'9973
.9969
'9966
0 '9994
'0550
0.8554
'8477
•8398
•8318
'8236
'9993
'9992
'9991
'9990
0'9999
'9998
'9998
'9998
'9997
3'00
0.1991
•1918
•1847
. 1778
'1712
0.4232
•4121
•4012
-3904
'3799
0.6472
•6360
•6248
'6137
'6025
o'8153
•8o68
'7982
. 7895
•7806
0.9161
•9110
'9057
'9002
.8946
0.9665
•9639
'9612
'9584
'9554
0.9881
'9870
'9858
'9845
'9832
0.9962
'9958
'9953
'9948
'9943
0.9989
'9988
'9986
'9984
'9982
0'9997
'9997
'9996
'9996
'9995
0'9999
.15
•20
0.0498
.0474
•0450
'0429
•0408
3.25
'30
'35
'40
'45
0.0388
-0369
'035 1
'0334
'0317
0.1648
•1586
•1526
'1468
'1413
0.3696
'3594
'3495
'3397
'3302
0.5914
'5803
'5693
'5584
'5475
0.7717
•7626
'7534
'7442
'7349
0.8888
'8829
'8768
'8705
'8642
0.9523
'9490
'9457
'9421
'9385
0.9817
•9802
0 '9937
'9786
'9769
'9751
0.9980
'9978
'9976
'9973
'9970
0 '9994
'9994
'9993
'9992
'9991
0 '9999
.9931
'9924
'9917
'9909
3'50
0.0302
0.1359
0.3208
0.5366
0.7254
0.8576
0 '9347
0 '9733
0'9901
0'9967
0.9990
0 '9997
'55
•6o
.65
.70
'05
•10
'2466
'2346
•5089
.4932
.6919
•68o8
'9948
'9941
'9934
'9925
.9998
'9997
'9997
'9996
0 '9999
'9999
See page 24 for explanation of the use of this table.
25
'9999
'9999
'9999
'9997
'9996
'9996
'9995
0.9999
0'9999
'9999
'9999
'9999
'9999
0 '9999
'9999
'9999
'9999
'9999
'9999
'9998
'9998
'9998
'9997
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
IL
r
= 0
/
2
3
4
5
6
7
8
9
10
0.3208
•3117
0.5366
•5259
0.7254
•7160
0'8576
•8509
0 '9347
0 '9733
•9308
.9713
0'9901
.9893
0.9967
•9963
0.9990
-9989
3.50
.55
•6o
.65
•7o
0'0302
0'1359
•0287
•0273
•0260
•0247
•1307
•1257
•3027
•5152
'7064
'8441
'9267
'9692
'9883
'9960
'9987
•1209
•ii62
'2940
•2854
•5046
'4942
.6969
.6872
.8372
•8301
.9225
•9182
.9670
•9648
.9873
•9863
'9956
'9952
•9986
'9984
3'75
•8o
0 ' 0235
0'2771
'2689
'2609
'2531
0'4838
0.6775
•6678
•6581
•6484
.6386
0•8229
•8156
•8o81
•8006
.7929
0.9I37
.9091
'9044
.8995
.8945
0.9624
'9599
'9573
'9546
.9518
0.9852
'9840
•9828
-9815
-9801
0'9947
'9942
'9937
'9931
.9925
0.9983
•9981
'9979
'9977
'9974
0•7851
'7773
•7693
•7613
'7531
0.8893
.8841
.8786
•8731
.8675
0'9489
'9458
'9427
'9394
.9361
0'9786
'9771
'9755
'9738
'9721
0.9919
.9912
.9905
.9897
.9889
0.9972
.9969
.9966
.9963
'9959
0.8617
.8558
0.9702
.9683
•9663
.9642
.9620
0.9880
.9871
•9861
.9851
.9840
0.9956
'9952
'9948
'9943
'9938
•85
•0213
•90
.95
'0202
0•I I17
•1074
•1032
•0992
.0193
.0953
- 2455
4'00
o.o183
- 0174
•o166
•0158
0.0916
•42880
- o845
•0812
0.2381
•2309
•2238
•2169
0'4335
•05
•zo
•15
'20
'0150
'0780
•2102
.3954
o•6288
•6191
•6093
'5996
.5898
4'25
•30
•35
'40
'45
0 '0143
'0136
0 '0749
0.2037
•1974
0.3862
'3772
0.5801
'5704
0'7449
•1912
•1851
'3682
•5608
•7283
'8498
.3594
•0117
'0663
'0636
•1793
'3508
'5512
'5416
•7199
.7114
•8436
'8374
0.9326
•9290
'9253
-9214
'9175
4'50
•55
•6o
•65
.70
0.01 11
•0'06
.0 1 0 1
•0096
•0091
o.o6 i 1
•0586
-0563
'0540
-o51/3
0.1736
-1680
•i626
'1574
.1523
0.3423
'3339
•3257
•3176
•3097
0.5321
•5226
.5132
.5039
'4946
0.7029
.6944
•6858
•6771
•6684
0.831!
•8246
•8180
•8114
•8o46
0.9134
.9092
'9049
-9005
•896o
0'9597
0'9829
.9817
.9805
'9792
'9778
0 '9933
'9574
'9549
'9524
'9497
4'75
•8o
-85
-90
-95
0•0087
•008z
•0078
•0074
0.0497
'0477
'0458
'0439
0.1473
'1425
•1379
'1333
0.3019
'2942
•2867
.2793
0.4854
'4763
•4672
'4582
'0071
'0421
•1289
•2721
'4493
0'6597
•6510
•6423
'6335
.6247
0.7978
•7908
-7838
.7767
.7695
0-8914
•8867
•8818
-8769
.8718
0.9470
'9442
'9413
.9382
'9351
0.9764
'9749
'9733
.9717
•9699
0.9903
.9896
•9888
•9880
.9872
5.00
•05
•zo
•z5
0•oo67
0.0404
•0064
'0388
•oo61
•0058
•0372
-0357
0.1247
•1205
•1165
-1126
0.2650
•2581
.2513
'2446
0.4405
'4318
'4231
.4146
0.616o
•6072
'5984
'5897
0.7622
'7548
'7474
•7399
'20
'0055
'0342
•I088
'2381
'4061
'5809
•7324
0.8666
•8614
•856o
-8505
'8449
0.9319
•9286
.9252
•9217
•9181
0.9682
•9663
. 9644
•9624
•9603
0.9863
. 9854
'9844
'9834
•9823
5'25
0'0052
0'0328
0'1051
0'2317
0.3978
0'5722
0'7248
•30
'35
•40
'45
'0050
•0314
•0302
•0289
-0277
-ior6
•0981
. 0948
•0915
-2254
•2193
-2133
- 2074
'3895
. 3814
•3733
'3654
•5635
'5548
'5461
'5375
'7171
'7094
•7017
•6939
0.8392
-8335
•8276
•8217
-8156
0'9144
•9106
.9067
•9027
•8986
0.9582
.9559
.9536
.9512
.9488
0.9812
•9800
•9788
'9775
.9761
5'50
.55
•6o
•65
0'0041
0'0266
'0255
0'0884
'0853
0'2017
'1961
•70
'0033
•1906
-1853
•z 800
0.6860
•6782
'6703
'6623
'6544
0.8095
•8033
'7970
•7906
'7841
-89o1
'8857
•8812
•8766
0.9462
'9436
'9409
•9381
.9352
0.9747
•0824
'0795
•0768
0.5289
'5204
.5119
'5034
.4950
0 '8944
•0244
•0234
-0224
0. 3575
'3498
'3422
'3346
.3272
5'75
•8o
•85
•90
•95
0.0°32
0.0215
0'0741
0'1749
0.3199
0.4866
•0030
•0029
'0206
'0715
•1700
•3127
•4783
•0197
'0027
'0189
•0026
•0181
•0690
•o666
-0642
•1651
•1604
•1557
•3056
•2987
.2918
'4701
'4619
'4537
0.6464
•6384
•6304
•6224
. 6143
0.7776
•7710
•7644
'7576
-7508
0.8719
•8672
•8623
.8574
•8524
0.9322
•9292
•9260
•9228
'9195
0.9669
•9651
•9633
•9614
'9594
6•oo
0.0025
0.0174
o.o620
0.1512
0.2851
0'4457
0.6063
0.7440
0.8472
0.9161
0 '9574
•0224
•0129
'0123
'0047
•0045
'0043
•0039
•0037
•0035
.0719
-0691
'41735
'4633
'4532
'4433
'4238
.4142
'4047
'7367
See page 24 for explanation of the use of this table.
26
-9928
.9922
•9916
•9910
'9733
.9718
•9702
•9686
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
= II
12
3.50
'55
'6o
•65
.70
0.9997
'9997
'9996
'9996
'9995
0 '9999
3'75
•8o
•85
•90
'95
0 '9995
'9994
'9993
'9993
'9992
4'00
0.9991
'9990
'9989
-9988
'9986
4'25
.30
'35
'40
'45
4•50
'55
-6o
.65
.70
0.9976
'9974
'9971
'9969
4.75
'8o
'85
0 '9963
x3
x4
x5
x6
17
= 0
I
2
6'0
'I
'2
0'0025
0'0174
0'0620
'0022
'0159
'0577
'0020
'0146
'0536
•3
•4
•0018
'0134
•0017
'0123
•0498
'0463
0 '9999
6'5
0'0015
0'0113
0'0430
'9998
'9998
'9998
'9998
'0014
'0103
'0400
•0012
'0095
'0371
'9999
'9999
•6
•7
•8
'9
0 '9997
0'9999
'9997
'9997
'9996
'9996
'9999
'9999
'9999
'9999
0 '9985
0 '9995
0 '9999
'9983
'9995
'9994
'9993
'9993
'9998
'9998
'9998
'9998
0.9992
'9991
'9990
.9989
'9988
'9999
'9999
'9999
'9999
0'9999
'0011
'0087
'0344
•oolo
•oo8o
•0320
7.0
0'0009
0'0073
0'0296
'I
•0008
•0067
•0275
'2
'0007
'0061
'0255
•3
•4
'0007
'0056
'0236
•0006
•0051
•0219
0.0006
0.0047
0.0203
'0005
•0043
'0188
'0005
'0039
•0174
'9999
'9999
7.5
•6
.7
•8
•9
'0004
'0036
•0004
•0033
•0161
•0149
0'9997
0 '9999
8•o
0'0003
0'0030
0'0138
'9997
'9997
'9997
'9996
'9999
'9999
'9999
'9999
•I
'2
'0003
•0028
•0127
'0003
•0025
'0118
•3
.4
'0002
'0023
'0109
'0002
'0021
'0100
0.9987
'9986
.9984
.9983
'9981
0.9996
'9995
'9995
'9994
'9994
0-9999
'9999
'9998
'9998
'9998
8'5
0'0002
0'0019
'0002
•0018
0.0093
•0086
'0002
'0016
'0079
'0002
'0015
'0073
'9999
•6
.7
•8
•9
'000I
'0014
•0068
0.9993
'9992
'9992
'9991
'9990
0.9998
'9997
'9997
'9997
'9997
0 '9999
9.0
o-000i
0'0012
o.0062
'9999
'9999
'9999
'9999
'I
'0001
'0011
'0058
'2
'0001
'0010
'9932
'9927
0.9980
'9978
'9976
'9974
.9972
'3
'4
'000!
'0009
' 00 53
'0049
'000!
'0009
'0045
5.25
•30
'35
'40
'45
0'9922
0'9970
0•9989
0'9996
0'0042
'9988
'9987
'9986
'9984
'9996
'9995
'9995
'9995
'000I
'0007
'0038
'0001
'0007
'0035
•000r
•0001
-0006
•0033
'9999
9'5
•6
'7
•8
•9
0'0008
'9967
'9964
.9962
'9959
0'9999
'9999
'9999
'9998
'9998
o'000z
'9916
'9910
'9904
'9897
'0005
'0030
5•50
'55
•6o
-65
0.9955
'9952
'9949
'9945
'9941
0 '9983
0 '9994
0 '9998
0 '9999
I0•0
0'0005
0'0028
-9982
'9980
'9979
'9977
'9993
'9993
'9992
'9991
'9998
'9998
'9997
'9997
'9999
'9999
'9999
'9999
'0005
'0026
'2
'0004
'0023
'3
'4
•0004
'0022
'70
0.9890
'9883
'9875
.9867
'9859
'0003
'0020
5.75
•8o
.85
0.9850
'9841
'9831
0 '9937
0.9975
'9973
'9971
'9969
'9966
0.9991
'9990
'9989
'9988
'9987
0 '9997
0 '9999
'9996
'9996
'9996
'9995
'9999
'9999
'9999
'9998
0.0003
-0003
•0003
0 '9999
10.5
•6
.7
.8
.9
•0002
0.0018
•0017
•0016
•0014
•0013
0'9964
0 '9986
0 '9995
0'9998
0'9999
I I•0
0'0002
0'0012
.05
•IO
'15
•20
'90
'95
5'00
'05
•10
'15
'20
'9982
.9980
'9978
'9966
'9960
'9957
'9953
'9949
0.9945
'9941
'9937
'90
'9821
'95
'9810
'9932
'9927
'9922
'9917
6•oo
0.9799
0.9912
0 '9999
0 '9999
0'9999
See page 24 for explanation of the use of this table.
27
'0002
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
IL
r= 3
4
5
6
8
7
9
10
II
12
13
0.9964
'9958
'9952
'9945
'9937
0 '4457
0.6063
0 '7440
0•8472
0.9161
0 '9574
0 '9799
'4298
•5902
'7301
'8367
'9090
'9531
'4141
.3988
. 5742
.3
•4
'1342
.1264
.1189
0'2851
'2719
'2592
-2469
'235 1
•3837
'5423
•7160
.7017
•6873
•82.59
.8148
•8033
•9016
•8939
•8858
•9486
•9437
.9386
'9776
'9750
.9723
.9693
0.9912
.9900
•9887
.9873
'9857
6.5
0•1118
0.2237
0.3690
•6
.7
•8
•9
•1052
•0988
•9928
•2127
•2022
'3547
•3406
-0871
•1920
•1823
•3270
•3137
0.5265
•5108
'4953
'4799
'4647
0.6728
•6581
•6433
•6285
.6136
0.7916
'7796
.7673
'7548
.7420
0.8774
•8686
.8596
•8502
•8405
0.9332
'9274
'9214
•9151
•9084
0.9661
.9627
'9591
'9552
•9510
0.9840
•9821
•9801
'9779
'9755
0.9929
-9920
.9909
.9898
-9885
7.0
0•0818
•o767
0'1730
•1641
0 '5987
'0719
.3
.4
.9674
•0632
'1555
.1473
' 1395
'4349
•4204
•4060
'3920
•5838
•5689
'5541
'5393
0.7291
-716o
•7027
.6892
'6757
0.8305
•82o2
•8096
'7988
.7877
0.9015
.8942
•8867
•8788
'8707
0 '9467
•2
0.3007
•2881
.2759
•2640
.2526
0 '4497
•1
0.9730
.9703
.9673
•9642
•9609
0.9872
.9857
'9841
•9824
•9805
7'5
•6
.7
•8
0'1321
0. 2414
•2307
•1181
o.66zo
•6482
.6343
•62o4
•6o65
0.7764
.7649
'7531
'7411
•7290
0•8622
.8535
.8445
•8352
•8257
0.9208
'9148
.9085
•9020
•8952
0'9573
•1249
'9
0'0591
.0554
•0518
•9485
'0453
'9536
'9496
'9454
'9409
0.9784
•9762
'9739
'9714
•9687
8.43
0'5925
0.7166
0'8159
'I
.5786
.5647
•5507
'5369
-7041
'6915
•6788
-6659
•8058
'7955
•7850
'7743
0.8881
•8807
.8731
•8652
.8571
0.9362
•9313
•9261
•9207
.9150
0.9658
•9628
'9595
-9561
'9524
0.5231
•5094
'4958
•4823
.4689
0.6530
•6400
•6269
.6137
•6006
0.7634
•7522
'7409
-7294
•7178
0.8487
-8400
-8311
•82.20
•8126
0.9091
•9029
.8965
•8898
•8829
0.9486
'9445
'9403
'9358
•93"
6•o
0'1512
•I
*1425
'2
•5582
-2203
0.3782
•3646
'3514
•1117
•2103
•3384
•1055
•2006
'3257
0.5246
•5100
'4956
•4812
'4670
0.0424
0'0996
•0396
*0940
O'1912
'1822
01134
•3013
0'4530
'4391
'4254
•4119
'3987
.9420
'9371
'9319
•9265
-2
•0370
•o887
•1736
-2896
-3
'4
.0346
'0323
•9837
•1653
•2781
'0789
•1573
•2670
8.5
•6
-7
•8
•9
0'0301
•9281
•oz62
•9244
•9228
0'0744
•0701
0'1496
•1422
0'2562
'2457
•0660
-1352
.2355
•9621
•9584
•1284
'2256
0.3856
•3728
•3602
•3478
•1210
•2160
'3357
TO
0'0212
0.0550
0'1157
.0517
'2
•0108
•0184
'0486
•1098
•1041
•3
•4
•0172
•0160
'0456
.0429
'0986
•0935
o.2o68
•1978
•1892
•1808
•1727
0.3239
•3123
•3010
•2900
'2792
0 '4557
•I
'4426
'4296
•4168
'4042
0.5874
'5742
•5611
'5479
'5349
0.7060
.6941
-682o
.6699
-6576
0.8030
'7932
-7832
'7730
•7626
0.8758
•8684
•8607
•8529
'8448
0.9261
•9210
•9156
'9100
•9042
9.5
•6
•7
0 . 0149
-0138
'0129
0.0403
0'0885
0'1649
0:22
:
26
2844
5
898 5
7
0'3918
0'5218
0'6453
0'7520
•0378
•0120
.0333
'1574
•1502
•1433
•2388
•3796
•3676
'3558
•5089
8
•0338
.0793
.0750
•9
-0111
- 0312
•07 10
•1
.6329
•6205
.6080
'5955
.7412
-7303
•7193
•7081
0.8364
•8279
•8191
•81o,
•8009
0.8981
•8919
•8853
•8786
-8716
10•0
0.0103
•I
0'0293
•0274
0'0671
•0634
'2
•0096
'0089
•3
•4
•0083
•0077
'0257
•0241
0.5830
.5705
'5580
'5456
'5331
0.6968
'6853
.6738
•6622
-6505
0.7916
-7820
•7722
•7623
.7522
0.8645
•8571
'8494
•8416
•8336
10'5
•6
•7
•8
•9
0.6387
•6269
•6150
0.7420
•7316
-7210
0.8253
•8169
•8083
11*0
'0355
366
0.1301
'3442
'4960
•4832
'4705
0'2202
'2113
0'3328
'3217
0 '4579
'1240
'0599
-xi8o
•2027
•0566
•1123
•1944
•0225
'0534
•1069
•1863
•3108
•3001
.2896
'4332
•4210
'4090
0.0071
-9966
•0062
-0057
•0053
0'0211
0'0504
0.1016
0'1785
0 . 2794
0'3971
•0197
.0475
•0966
•1710
•0185
•0173
'0162
'0448
'0423
•0398
'0918
'0872
•o82.8
•1636
'1566
•2694
'2597
'3854
•3739
0.5207
•5084
'4961
'2502
'3626
•4840
•6031
'7104
'7995
'1498
.2410
•3515
'4719
'5912
.6996
'7905
0.0049
0'0151
0'0375
0.0786
0'1432
0'2320
0.3405
0 '4599
0 '5793
0.6887
0.7813
'4455
See page 24 for explanation of the use of this table.
28
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
6•o
'I
r = 14
15
i6
17
0.9986
'9984
0.9995
'9994
'9993
'9992
'9990
0.9998
'9998
'9997
'9997
'9996
0.9999
'9999
'9999
'9999
'9999
0.9996
-9995
'9994
'9993
'9992
0.9998
-9998
'9998
'9997
'9997
0.9999
.9999
'9999
'9999
'9999
i8
19
20
21
22
23
24
'2
'9981
'3
'4
'9978
'9974
6'5
.6
'7
'8
'9
0.9970
'9966
'9961
'9956
'9950
0 '9988
7.0
'I
0'9943
'9935
'9927
'9918
'9908
0'9976
'9972
'9969
'9964
'9959
0 '9990
0-9996
'9996
'9995
'9994
'9993
0 '9999
'9989
'9987
'9985
'9983
'9998
'9998
'9998
'9997
0 '9999
7'5
•6
'7
.8
'9
0 '9897
0 '9954
'9948
'9941
'9934
'9926
0'9980
'9978
'9974
'9971
'9967
0'9992
'9991
'9989
'9988
'9986
0'9997
'9996
'9996
'9995
'9994
0 '9999
'9886
'9873
'9859
'9844
8•o
'I
0'9918
'9908
'9898
'9887
'9875
0'9963
'9958
'9953
'9947
'9941
0'9984
.9982
'9979
'9977
'9973
0'9993
'9992
'9991
'9990
'9989
0 '9997
0 '9999
'3
'4
0.9827
'9810
'9791
'9771
'9749
'9997
'9997
'9996
'9995
'9999
'9999
'9998
'9998
8'5
0 '9726
0 '9934
.9926
0'9987
'9985
'9983
'9981
'9978
0'9995
'9994
'9993
'9992
'9991
0'9998
'9998
'9997
'9997
'9996
0 '9999
-9701
'9675
'9647
'9617
0.9862
'9848
'9832
'9816
'9798
0 '9970
'6
'7
.8
'9
'9999
'9999
'9999
'9998
0'9999
9.0
-1
-2
'3
'4
0.9585
'9552
'9517
'9480
'9441
0*9780
'9760
'9738
'9715
'9691
0.9889
'9878
'9865
'9852
'9838
0 '9947
- 9941
0'9976
'9973
'9969
'9966
'9962
0'9989
0'9996
'9995
'9994
'9993
'9992
0'9998
0 '9999
'9988
'9986
'9985
'9983
'9998
'9998
'9997
'9997
'9999
'9999
'9999
'9999
9'5
.6
'7
.8
'9
0'9400
.9357
'9312
'9265
'9216
0.9665
'9638
'9609
'9579
'9546
0'9823
'9806
'9789
'9770
'9751
0'9911
'9902
'9892
.9881
'9870
0 '9957
0'9980
'9978
'9975
'9972
'9969
0.9991
'9990
'9989
'9987
'9986
0 '9996
0'9999
'9952
'9947
.9941
'9935
'9996
'9995
'9995
'9994
'9998
'9998
'9998
'9997
0'9999
.9999
'9999
'9999
'9999
IWO
0 '9513
0'9857
'9844
'9830
-9815
'9799
0 '9965
'9921
'9913
'9904
'9895
'9962
'9957
'9953
'9948
0.9984
.9982
'9980
'9978
'9975
0 '9993
'9477
'9440
'9400
'9359
0'9730
'9707
•9684
•9658
'9632
0 '9928
'3
'4
0.9165
'9112
'9057
'9000
'8940
'9992
'9991
'9990
'9989
0'9997
'9997
'9996
'9996
'9995
0'9999
'9999
'9998
'9998
'9998
10.5
•6
'7
•8
'9
0.8879
•8815
.8750
•8682
'8612
0'9317
'9272
'9225
'91 77
'9126
0'9604
'9574
'9543
'9511
'9477
0'9781
'9763
'9744
'9723
'9701
0.9885
'9874
'9863
'9850
'9837
0.9942
'9936
'9930
'9923
'9915
0'9972
'9969
'9966
.9962
'9958
0 '9987
0 '9994
'9994
'9993
'9992
'9991
0'9998
'9997
'9997
'9996
'9996
0 '9999
'9986
'9984
'9982
'9980
i•o
0.8540
0'9074
0 '9441
0'9678
0'9823
0 '9907
0 '9953
0 '9977
0'9990
0'9995
0'9998
'2
'3
'4
'2
I
'2
-9986
'9984
'9982
'9979
'9918
'9909
-9899
'9966
'9962
'9957
'9952
'9934
'9927
- 9919
r = 25
101
'8
'9
0.9999
'9999
'9999
0 '9999
'9999
'9999
'9999
'9999
'9998
'9998
'9998
0'9999
'9999
'9999
0 '9999
'9999
See page 24 for explanation of the use of this table.
29
0 '9999
'9999
'9999
'9999
'9999
'9999
'9998
'9998
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
P,
r =
2
3
4
5
6
7
8
9
xo
II
12
11•0
0'0012
0'0049
0.0151
0'0375
0'0786
0'1432
0'2320
0.3405
0 '4599
0'5793
'2
'0010
'0042
•0132
'0333
-0708
'4
•0009
•0036
•0115
•0295
•0636
•6
•0007
•0261
'0230
-0571
•0512
'3192
•2987
.2791
'2603
'5554
•5316
•5080
•0006
•oioo
•oo87
-2147
•1984
•1830
•1686
'4362
-4131
.3905
•8
-0031
•0027
-1307
•1192
•1085
'0986
'3685
'4847
0.6887
•6666
•6442
•6216
'5988
12'0
0'0005
0'0023
0'0458
0'0895
0'1550
0 '2424
•0004
'0004
'0020
0'0076
•oo66
0'0203
'2
•4
-0017
•0057
•0179
•0158
•0014
•0050
•0139
•8
•0003
'0003
•081x
'0734
•0664
'1424
.1305
•6
'04ro
•0366
•0326
•0012
•0043
'0122
•0291
*0599
•2254
•2092
•1939
•1794
0'3472
'3266
•3067
.2876
'2693
0.4616
'4389
•4167
'3950
'3738
0.5760
'5531
'5303
.5077
'4853
13•0
0.0002
0•00I I
0'0259
0'0540
0.2517
0'3532
0'4631
'0009
•0094
'0230
•0487
0'0998
•0910
0. 1658
•0002
•000z
0'0037
•0032
0'0107
'2
•4
' 1530
. 2349
•0008
•0028
-0083
•0204
•6
•0001
•0007
-0024
'0072
•0,8i
•000i
•0006
•oo2i
•0063
•ox6i
•0828
.0753
•o684
•1410
•1297
'1192
-2189
.2037
•1893
'4413
'4199
.3989
.8
•0438
'0393
.0353
'3332
•3139
14.0
0.0001
0'0005
0'0018
0'0055
0'0142
0'0316
0•062I
0.1094
0'1757
0'2600
0•3585
'2
•0001
•0001
'0004
•0048
.0126
•0283
'0003
'0016
•0013
•0111
•0253
•0003
-0012
•0098
•0226
•1003
-0918
•0839
•1628
•1507
•0001
'1392
•2435
•2277
•2127
'3391
•3203
•3021
•0002
•0010
'0042
•0037
•0032
-0562
•0509
•0460
•0087
•0202
•0415
'0766
•1285
' 1984
•2845
15'0
0'0002
0'0009
0'0028
0'0076
0'0180
0'0374
'2
•0002
•0002
•0007
•0024
•0067
•0160
'0337
0.0699
•40636
•0006
•0021
•0059
'0143
.0005
.0018
-0052
•0127
'0304
•0273
'0579
•0001
•8
•0001
•0005
•0016
•0046
•0113
- 0245
•0478
0.1185
•1091
•1003
•0921
-0845
0.1848
-1718
•1596
•1481
•1372
0.2676
- 2514
'2358
'2209
•2067
16•o
0.000i
0.0004
0'0014
0'0220
.0089
.0197
'4
•6
•0001
•0003
'0012
•0010
.0079
0176
0355
0'0774
.0708
.0647
0.1931
'0003
0'0433
.0392
o.1270
*0001
0'0040
*0035
0•0I 00
*2
'1174
.1084
568:
:116
•000I
'4
•6
•8
'4
-6
•0003
•0009
.8
0002
-0008
170
0'0002
0'0007
'2
'0002
.0006
'4
•6
•0001
'1195
•1093
.0526
'2952
'2773
'3784
•I8o2
'0070
•0158
•0321
•oo6I
•0141
•0290
•0591
.0539
•0920
'1454
0•0021
0'0054
0'0126
0.0261
0•0491
0'0847
0'1350
•0018
'0048
'0112
•0235
•0447
•0778
•1252
'0005
•00'6
-0042
-0I00
-0212
•0406
•0714
•I 160
•0004
•0014
•0037
•0004
•0012
•0033
•0089
-0079
•0191
•0171
•0369
.0335
•0655
•0600
•1074
-8
•000!
•0007
18•o
0'0001
0'0003
0'0010
0'0029
0'0071
0'0154
•2
•0001
•0001
•0033
'0002
'0009
-0008
•0025
'0063
'0138
0'0304
'0275
0.0549
•0502
0.0917
'0846
'0022
•0056
'0124
'0002
'0007
*0020
'01 1 1
'0458
'0418
'0002
-0006
•0017
'0049
- 0044
' 0249
'0225
'0779
•0001
•0099
-0203
•0381
•0659
0.0039
•0034
o'oo89
•0079
0.0183
-0165
0.0347
•0315
0.0606
•0556
'4
•6
r =
.:0000002:4
71
.0999
'0993
'0717
X X•0
0'0002
'2
•0002
'4
•6
•000I
0'0002
0'0005
0'0015
•00ox
•8
•0001
-0005
•0004
•0013
'4
.6
•0001
•0007
•0012
'0030
•0071
•0149
•0287
•0509
•0001
'0003
•0010
•0027
'0063
-0260
•0467
-8
12'0
0.0001
•0001
•0003
•0009
•0024
•0056
•0134
•0120
'0236
'0427
'2
'0001
'4
•0001
0• 0 00 I
0•000 3
0.0008
0•0021
0 ' 00 50
0.0108
0 ' 02 I 4
0•0 3 9 0
.8
19•0
'2
20'0
See page 24 for explanation of the use of this table.
30
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
ti,
Y = 13
14
15
i6
17
i8
19
zo
21
22
23
XII)
0.7813
•7025
0.9074
.8963
•8845
*8719
.8585
0.9441
.9364
•9280
'9190
.9092
0.9678
•9628
'9572
'9511
'9444
0.9823
'9792
'9757
.9718
'9674
0'9907
'9889
'9868
'9845
'9818
0'9977
'9972
'9966
'0958
'9950
0'9990
'9987
'9984
.9980
'9975
0'9995
-7624
'7430
•7230
0.8540
. 8391
•8234
'8069
.7898
0'9953
•2
12•0
'2
0'6815
0.7720
'4
•6
'8
'6387
*7347
'7153
'6954
•8875
'8755
•8629
'8495
0'9370
.9290
'9204
•9111
'9011
0'9626
.9572
'9513
- 9448
'9378
0'9787
*9753
'9715
.9672
'9625
0.9884
'9863
'9840
.9813
'9783
0.9939
'7536
0'8444
•8296
'8140
*7978
*7810
0'8987
•6603
.9927
'9914
.9898
'9880
0•9970
.9963
'9955
'9946
'9936
0.9985
.9982
'9978
'9973
'9967
13'0
0.5730
*5511
0.6751
'6546
'6338
•6128
.5916
0.7636
*7456
.7272
•7083
•6890
0.8355
•8208
.8054
*7895
.7730
0.8905
'8791
.8671
'8545
. 8411
0.9302
'9219
.9130
'9035
'8934
0 '9573
'9516
'9454
*9387
'9314
0'9750
'9713
'9671
'9626
'9576
0.9859
'9836
•9810
'9780
*9748
0.9924
'9910
•9894
'9876
.9856
0.9960
'9952
'9943
'9933
.9921
0.5704
'5492
'5281
'5071
0'6694
'6494
•6293
'6090
0 '7559
•4863
•5886
'7384
'7204
•7020
•6832
0.8272
•8126
'7975
•7818
. 7656
0.8826
•8712
. 8592
•8466
'8333
0.9235
.9150
•9060
•8963
•8861
0.9521
•9461
.9396
'9326
•9251
0.9712
•9671
'9627
'9579
'9526
0.9833
•9807
'9779
'9747
.9711
0.9907
•9891
'9873
'9853
.9831
0.4657
'4453
'4253
0.5681
'5476
.5272
0.6641
'6448
.6253
0.7489
'7317
'7141
0.8195
•8051
.7901
0.9170
.9084
.8992
'8894
•8791
0.9469
.9407
.9340
•9268
*9190
0.9673
'9630
*9583
.9532
'9477
0.9805
'9777
'9746
.9712
'9674
'4
•6
•8
.2
'4
•6
-8
•6169
*5950
'5292
.5074
.4858
14.0
0.4644
'2
'4434
'4
•6
•8
'4227
•4024
•3826
15.0
.2
0.3632
'9943
.9932
'9918
'9902
'9994
*9992
.9991
'9988
•4056
•5069
•6056
•6962
'7747
•3864
'4867
'5858
'6779
'7587
0.8752
•8638
.8517
•8391
•8260
0.2745
0'3675
'3492
0.5660
.5461
•5263
•5067
'4871
0.6593
•6406
•6216
•6025
'5833
0'7423
'7255
•7084
•6908
•6730
0.8122
'7980
. 7833
•7681
•7524
0.8682
•8567
.8447
•8321
•8191
0.9108
•9020
'8927
•8828
.8724
0.9418
'9353
'9284
•9210
.9131
0'9633
'25 85
0 '5640
*5448
•5256
•5065
'4875
0.6550
'6367
•6182
'5996
•5810
0.7363
'7199
•7031
'6859
•6685
0.8055
'7914
•7769
.7619
•7465
0.8615
•85oo
•838o
-8255
-8126
0.9047
•8958
•8864
'8765
•866o
0.9367
•9301
•9230
'9154
'9074
'4
•6
-8
'3444
'3260
-3083
•2911
16 0
'2
•6
•8
•2285
•2144
'2971
0.4667
'4470
.4276
•4085
•3898
17•0
0.2009
0.2808
0.3715
'2
'1880
'2651
'3535
'4
.6
•8
'1758
.1641
•1531
2500
' 2354
. 2215
•3361
'3191
'3026
0.4677
'4486
.4297
'4112
'3929
18.0
0.2081
.1953
•183o
'1714
'1603
0.2867
•2712
.2563
'2419
*2281
0.3751
.3576
'3405
.3239
*3077
0.4686
.4500
'4317
.4136
'3958
0.5622
'5435
*5249
'5063
.4878
0.6509
'6331
•6151
'5970
.5788
0.7307
'7146
•6981
'6814
.6644
0'7991
.7852
•7709
•7561
•7410
0.8551
.8436
•8317
•8193
•8065
0.8989
*8899
'8804
'8704
•8600
0.1497
'1397
0'3784
'3613
•3446
'3283
'3124
0'4695
*4514
'4335
•4158
'3985
0'5606
*5424
'5242
•5061
'4881
0'6472
'6298
•6122
*5946
•5769
0.7255
'7097
'6935
'6772
'660 5
0.7931
'7794
'7653
'7507
'7358
0.8490
'8376
'8257
'8134
•8007
0.2970
0.3814
0.4703
0'5591
0'6437
0.7206
0 '7875
'4
13
2432 '33
•3139
4
0.1426
.1327
'1233
.6
•8
•1145
•xo62
10'0
0.0984
'2
'0911
'4
.6
•8
•0842
.0778
•0717
-1213
0'2148
'2021
•1899
.1782
•1128
-1671
0'2920
'2768
•2621
' 2479
•2342
20.0
0.0661
0.1049
0.1565
0.2211
'2
-1303
See page 24 for explanation of the use of this table.
31
'9588
'9539
•9486
.9429
TABLE 2. THE POISSON DISTRIBUTION FUNCTION
ii.
r = 24
25
mo
0.9998
0 '9999
'2
'9997
'4
'6
'8
'9997
'9996
'9995
'9999
'9999
'9998
'9998
36
37
38
39
fl•
1= 35
IT2
'4
'6
'8
0.9999
'9999
'9999
'9999
x8.0
0.9999
'2
'9999
0 '9999
'4
'6
.8
'9998
'9998
'9997
0.9999
'9999
'9999
'9999
'9999
'9999
'9999
19•0
.2
'4
.6
.8
0.9997
'9996
'9995
'9994
'9993
0.9998
'9998
'9998
'9997
'9996
0.9999
'9999
'9999
'9999
'9998
0 '9999
20'0
0'9992
0'9996
0.9998
0'9999
0'9999
30
31
32
33
34
26
27
0 '9999
'9999
12'0
*2
0.9993
0 '9997
0'9999
'9991
'4
.6
'8
'9989
.9987
'9984
'9996
'9995
'9994
'9992
'9998
'9998
'9997
'9996
13'0
0.9980
'9976
'9971
.9965
'9958
0.9990
'9988
.9985
.9982
'9978
0'9995
'9994
'9993
'9991
'9989
0.9998
'9997
'9997
'9996
'9995
0 '9999
.2
'4
.6
.8
14.0
.2
'4
'6
.8
0.9950
'9941
'9930
.9918
'9904
0 '9974
0 '9987
0 '9994
0 '9997
0 '9999
0 '9999
•9969
'9963
'9956
'9947
'9984
•9981
'9977
'9972
'9992
'9990
'9988
•9986
'9996
'9995
'9994
.9993
'9998
'9998
'9997
'9997
'9999
'9999
'9999
'9998
15.0
0.9888
0.9996
'9995
'9994
'9992
'9991
'9998
'9997
'9996
'9995
'9999
'9999
'9998
'9998
0 '9999
'9851
•9829
•9804
0.9991
'9990
.9987
.9985
'9982
0'9999
'4
.6
-8
0.9983
'9979
'9975
'9971
'9965
0 '9998
'9871
0.9938
.9928
.9915
•9902
'9886
0'9967
'2
16•o
0 '9777
'9747
0'9989
'9986
'9984
'9981
'9977
'9993
'9992
'9990
'9988
0'9997
'9997
'9996
'9995
'9994
0 '9999
'9952
'9944
'9934
'9924
0'9978
'9974
'9969
'9964
'9957
0 '9999
'9713
.9677
'9637
0.9925
•9913
.9900
-9884
'9867
0 '9994
'4
'6
'8
0'9869
.9849
•9828
'9804
'9777
0 '9959
'2
'9998
'9998
'9998
'9997
'9999
'9999
'9999
'9999
17.0
0'9594
'9546
0'9748
'9968
•9962
'9956
'9949
'9983
.9980
'9976
•9972
'9991
'9989
'9987
-9985
0'9996
'9995
'9994
'9993
'9992
0'9998
'9998
'9997
'9997
'9996
0'9999
'9495
.9440
'9381
0.9950
'9942
'9933
-9922
•9910
0 '9993
'4
•6
.8
0.9912
'9898
.9883
•9866
.9848
0 '9986
.9715
'9680
.9641
'9599
0.9848
•9827
'9804
'9778
'9749
0 '9973
'2
x8•o
0.9317
0'9554
'9249
'4
•6
•8
'9177
.9100
•9019
.9505
'9452
.9395
'9334
0.9718
.9683
- 9646
•9606
'9562
0.9827
-9804
'9779
•9751
'9720
0.9897
'9882
•9866
.9847
•9827
0.9941
'9931
'9921
'9909
'9896
0.9967
•9961
'9955
'9948
'9939
0.9982
'9979
'9975
'9971
'9966
0.9990
'9989
'9986
'9984
.9981
0'9995
'2
0.9998
'9997
'9996
'9996
'9995
19.0
0.8933
'8842
'4
•6
•8
'8746
•8646
8541
0.9269
'9199
'9126
'9048
•8965
0.9514
'9463
'9409
'9350
•9288
0.9687
.9651
'9612
'9570
'9524
0.9805
.9780
'9753
.9724
'9692
0.9882
•9865
'9847
.9828
•9806
0.9930
'9920
'9908
'9895
•9881
0'9960
'9954
'9946
'9938
.9929
0.9978
'9974
'9970
'9965
'9959
0'9988
'9986
'9983
•9980
'9977
0 '9994
'2
20'0
0.8432
0.8878
0.9221
0 '9475
0.9657
0'9782
0.9865
0.9919
0'9953
0 '9973
0'9985
.
.996;
'9954
'9945
'9936
0.9999
'9999
'9999
'9999
'9998
z8
'9999
'9999
0 '9999
'9999
'9999
'9999
'9998
'9998
r
=
29
0'9999
'9999
'9999
0 '9999
'9999
See page 24 for explanation of the use of this table.
32
'9999
'9999
'9999
'9994
'9993
'9991
'9990
0 '9999
'9999
'9999
'9999
'9998
'9998
'9992
'9991
.9989
'9987
TABLE 3. BINOMIAL COEFFICIENTS
This table gives values of
n!
n(n — 1). .(n — r + 1)
(n — r)! r! —
r!
(r ) =nC,
when r> in use (1 =
n )• (n) is the number of
r
n r
r
ways of selecting r objects from n, the order of choice being
immaterial. (See also Table 6, which gives values of log10 n!
for n < 300.)
—
2
n
=x
I
3
4
2
I
I
2
3
I
I
3
4
3
6
I
4
5
I
1
10
15
7
8
I
x
21
9
I
5
6
7
8
9
10
6
35
56
84
xo
II
12
I
I
I
12
13
I
13
14
I
15
16
17
18
I
1
I
1
19
I
20
21
22
23
24
4
Jo
II
I
28
36
45
55
66
6
5
I
5
15
35
70
126
20
6
1
21
7
28
84
56
126
120
210
252
210
120
330
495
462
792
462
924
33o
792
1716
1716
3003
3432
14
715
1001
1287
2002
15
105
455
1365
16
17
120
560
1820
18
19
136
153
171
68o
816
969
2380
3060
3876
3003
4368
6188
8568
11628
5005
8008
12376
18564
27132
I
I
I
I
I
20
21
22
190
210
231
I I 4o
1330
1540
23
253
1771
4845
5985
7315
8855
15504
20349
26334
33649
38760
54264
74613
100947
24
276
2024
10626
42504
134596
25
26
27
I
I
I
300
28
29
x
I
25
26
27
28
29
325
351
378
406
2300
2600
2925
3276
3654
12650
14950
17550
20475
23751
53130
65780
80730
98280
118755
230230
296010
376740
475020
30
I
30
435
4060
27405
142506
593775
?I
II
12
= 20
184756
167960
125970
21
22
23
352716
352716
293930
646646
1144066
1961256
705432
1352078
2496144
646646
1352078
2704156
3268760
5311735
4457400
7726160
24
26
27
28
29
8436285
13037895
13123110
21474180
2.0030010
34597290
5200300
9657700
17383860
30421755
51895935
30
30045015
54627300
86493225
25
I
8
36
165
220
286
364
10
9
I
78
91
r
8
7
33
177100
13
77520
203490
497420
6435
1
9
1
45
165
495
xo
55
220
1287
3003
715
2002
31824
50388
6435
12870
24310
43758
75582
24310
48620
92378
77520
x 1628o
170544
245157
346104
125970
203490
319770
490314
735471
167960
293930
497420
817190
13.07504
480700
657800
888030
1184040
1560780
1081575
1562275
2220075
3108105
4292145
2042975
3124550
4686825
6906900
10015005
2035800
5852925
14307150
11440
19448
14
5005
11440
15
2496144
3876o
116280
319770
817190
1961256
15504
54264
170544
490314
1307504
5200300
10400600
20058300
37442160
67863915
4457400
9657700
20058300
40116600
77558760
3268760
7726160
17383860
37442160
77558760
119759850
145422675
155117520
1144066
TABLE 4. THE NORMAL DISTRIBUTION FUNCTION
The function tabulated is 0(x) =
fx
.V2it
dt. 0(x) is
-00
the probability that a random variable, normally distributed
with zero mean and unit variance, will be less than or equal
to x. When x < o use 40(x) = i -0( - x), as the normal
distribution with zero mean and unit variance is symmetric
about zero.
x
0(x)
x
(I)(x)
x
(11(x)
x
I(x)
x
(13( x)
x
0'00
0'5000
'5040
'5080
0'40
'41
'42
0*6554
'6591
0'80
•81
1'20
0'8849
•8869
2'00
0.97725
.9463
*9474
•5120
•5160
•82
•83
x.6o
•61
•62
0'9452
•8907
'8925
'63
'9484
- 44
•6628
•6664
•67oo
0'7881
•7910
'7939
.64
'9495
•0,
•02
•03
•04
'97778
*97831
'97882
*97932
0.8944
•8962
•898o
.8997
1.65
•66
•67
-68
'69
0.9505
•9515
•9525
'9535
'9545
2'05
0'97982
•o6
•07
•o8
•09
•98030
1•70
0.9554
.9564
'9573
.9582
'9591
2•10
•01
•02
•03
•04
'43
'84
.7967
'7995
•21
•22
•23
.24
0'85
0'8023
1'25
•86
•87
•88
•89
•8051
•8078
•8106
•8133
•26
•27
•28
-29
0'05
•06
0.5199
'5239
0.45
.46
.07
.5279
*47
•08
'5319
'09
'5359
•48
.49
0.6736
.6772
•6808
•6844
.6879
0•10
0'5398
0'50
0'6915
0'90
0'8159
1•30
'II
•12
'5438
'5478
•13
'14
.5517
'5557
•51
•52
.53
'54
•6950
•6985
.7019
'7054
'91
•92
•93
.94
•8186
•8212
.8238
.82,64
•3I
-32
.33
'34
0.15
•16
•17
•18
•19
0-5596
•5636
.5675
.5714
'5753
0.55
•56
'57
•58
'59
0.7088
•7123
'7157
•7190
'7224
o.95
•96
.97
•98
.99
0.8289
•8315
.8340
•8365
.8389
0'20
'21
'22
0.5793
0.60
0.7257
100
'5832
'5871
'61
'7291
'01
•62
•02
'03
'04
'5910
'63
'5948
'64
'7324
'7357
'7389
0.25
•26
0 '5987
0'65
0'7422
1.05
•6026
•6064
'6103
•6141
'7454
'7486
.7517
'7549
•06
•27
'28
•66
•67
•68
•o8
•09
.23
'24
•29
'69
'07
•8888
'9015
0'9032
*9049
'98077
•98124
'98169
0'98214
'98257
'98300
•9066
.9082
'9099
.71
'72
.73
'74
1.35
•36
'37
•38
'39
0-9115
•9131
'9147
•9162
'9177
r75
•76
.77
•78
'79
0 '9599
2'15
0'98422
•9608
.9616
•9625
'9633
•16
.17
•18
'19
•98461
•98500
.98537
'98574
0'8413
1'40
0'9192
1.80
0.9641
'41
.9207
'8,
'9649
2'20
'21
0.98610
.8438
'8461
'8485
'8508
'42
'9222
'9236
'925 1
-9656
'9664
.9671
•22
.23
.24
•98679
'43
.44
•82
.83
.84
'98745
0.8531
.8554
'8577
•8599
•8621
1'45
'46
'47
•48
'49
0 '9265
1.85
•86
0'98778
•26
'9292
'87
'27
'98809
'98840
•9306
'9319
•88
'89
0.9678
•9686
.9693
•9699
'9706
2'25
'9279
•28
•29
•98899
1•50
0.9332
'9345
'9357
'9370
-9382
1'90
0.9713
'9719
2'30
0'98928
.31
.98956
'9726
'32
'98983
'9732
'9738
'33
'34
'99010
'99036
0.99061
'99086
•II
•12
.13
•14
0'30
•31
•32
0.6179
0•70
'71
'72
0.7580
1•10
'6217
•6255
'7611
•7642
•I 1
'33
'34
.6293
'6331
'73
'74
'7673
'7704
•13
'14
0.8643
•8665
•8686
'8708
'8729
0.35
•36
'37
•38
'39
0.6368
•6406
'6443
•6480
'65 1 7
0.75
•76
'77
•78
'79
0'7734
r15
•16
•17
•i8
•19
0.8749
•8770
.8790
•8810
•883o
r55
'56
.57
•58
'59
0 '9394
1'95
0 '9744
2'35
•7764
'7794
•7823
'7852
.9406
'9418
•9429
'9441
'96
'97
•98
'99
'9750
'9756
•9761
'9767
-36
.37
•38
'39
0'40
0.6554
0.80
0.7881
1•20
0'8849
1•60
0'9452
2'00
0'9772
2.40
•12
(I)(x)
'51
'52
'53
.54
34
'91
.92
'93
'94
.98341
.98382
•98645
'98713
•98870
'99111
'99134
'991 58
0'99180
TABLE 4. THE NORMAL DISTRIBUTION FUNCTION
x
x
(1.(x)
x
(1)(x)
x
(1)(x)
x
(1)(x)
2'40
0.99180
2'55
0'99461
2'70
'99202
'99224
'99245
'99266
•56
'99477
II
2.85
•86
'57
•58
.72
'73
'74
'99683
•88
•59
'99492
.99506
.99520
0.99653
'99664
'99674
'99693
•89
0.99781
.99788
'99795
.99801
-99807
3.00 0.99865
',II
2'60
0 '99534
2.75
0.99813
3•05
0.99886
-76
0.99702
.9971i
2'90
•61
•62
'63
'64
'99547
•9x
.99819
•o6
.99889
'99560
.77
.99720
'92
'99825
•07
'99893
'49
0'99286
'99305
'99324
'99343
'99361
'93
'94
'9983 1
'99836
•o8
.09
'99900
2'50
0.99379
.52
'99396
'99413
0'99841
- 99846
3.10
. 51
•I2
'53
'54
•13
'99913
'14
'99916
2'55
3'15
0'99918
'42
'43
'44
2'45
'46
'47
'48
'87
'99573
•78
.99728
'99585
'79
'99736
2.65
0'99598
•99609
.99621
2.8o
•8x
2'95
'99430
'99446
•66
.67
•68
.69
-99
.99851
.99856
.99861
0'99461
2.70
3'00
0'99865
.99632
•83
'99643
*84
(3'99744
'99752
.9976o
'99767
'99774
0.99653
2'85
0'99781
•82
.96
.97
.98
•ox
•02
'03
•04
•I
x
(D(x)
.99869
'99874
'99878
'99882
.99896
0.99903
.99906
'99910
0(x)
3'15
•x6
0.99918
•x7
•i8
•19
'99924
.99926
'99929
3'20
'2I
'22
6 23
•24
0 '99931
3'25
0 '99942
'26
'27
•28
-29
'99944
'99946
'99948
'99950
3.30
0.99952
.99921
'99934
'99936
'99938
'99940
The critical table below gives on the left the range of values of x for which 0(x) takes the value on the right,
correct to the last figure given; in critical cases, take the upper of the two values of (1)(x) indicated.
3.075
0.9990
3o5
0.9991
'130 0'9992
3.215
3'174 0.9993
0 '9994
3'263 09994
3-320 09995
3.389 0.9996
3'480 0.9997
3.615 0.9998
0 '9999
When x > 3.3 the formula -0(x) *
xV2IT
0
99990
3 .916 0 '99995
3.976 099996
0
3.826 0.99993
3.867 0.99994
4'055 0.99997
4'173 0.99998
4'417 099999
1•00000
3'73ro .99991
3159
3 .791 99992
0 '99995
I 3 15 1051
is very accurate, with relative error
7+ 78+71x
less than 945/x1°.
TABLE 5. PERCENTAGE POINTS OF THE
NORMAL DISTRIBUTION
This table gives percentage points x(P) defined by the
equation
P
ioo
=
lc° e-It2 dt.
x(P)
If X is a variable, normally distributed with zero mean and
unit variance, P/Ioo is the probability that X x(P). The
lower P per cent points are given by symmetry as - x(P),
x(P) is 2PI loo.
and the probability that IXI
P
x(P)
P
50
0'0000
45
40
35
0'1257
30
25
0'2533
0 '3853
0'5244
0 '6 745
0.8416
x(P)
P
x(P)
P
x(P)
5'0
P6449
2'0
2'0537
I'0
2'3263
0•10
3'0902
1.6646
1'6849
3'0
2'9
1'8808
4.8
4'6
4'4
P8957
2'8
1 .91 10
2.0749
2•0969
2.7
0'07
P7279
2'6
P9268
1.9431
2.3656
2•089
2.4573
4*2
I'6
2'1444
0.9
o•8
0.7
o•6
0.09
0T8
1.7060
1.9
1.8
r7
2.5121
0'06
3'1214
3.1559
3'1947
3.2389
P7507
2.5
P9600
I.5
2•1701
o•5
1'7744
P9774
1 '9954
1.4
2.1 973
0.4
2.5758
2.6521
1 . 7991
2'4
2'3
1'8250
P8522
2*2
2'1
2'0141
r3
1•2
2'0335
I •I
2•2262
2'2571
2'2904
0'3
0•2
0•1
I0
P2816
4'0
3.8
3•6
3.4
5
P6449
3'2
20
15
1.0364
35
2•i20i
P
x(P)
2/478
2•8782
3.0902
P
x(P)
0•0x
0.005
3.2905
3.7190
3.8906
0.001
0.0005
4.4172
0.05
4.2649
TABLE 6. LOGARITHMS OF FACTORIALS
n
log10 n!
n
0
0'0000
i
o•0000
logio n!
n
log10 n!
n
log10 n!
n
loglo n!
100
157'9700
377'2001
250
251
161'9829
267'1177
200
201
202
374.8969
159'9743
15
151
152
26
62
4 '9
7 35. 6
59
9
Ica
379'5054
163.9958
153
269.3024
203
381.8129
252
253
154
271'4899
204
384.1226
254
492.5096
494'9093
497'3107
499'7138
502'1186
227753.'86783043
205
206
255
256
278'0693
280'2679
207
208
282'4693
209
386 '4343
388.7482
39 P 0642
393.3822
395'7024
210
211
212
398'0246
400'3489
402'6752
log10 n!
n
50
64.483 1
5x
102
103
2
0'3010
52
3
4
0.7782
53
66.1906
67.9066
69.6309
I • 3802
54
71'3633
104
166•0128
5
6
7
8
9
2'0792
2'8573
73.1037
74.8519
76.6077
78.3712
80. 1420
105
xo6
107
xo8
109
168.0340
170.0593
3'7024
4'6055
5'5598
55
56
57
58
59
172'0887
156
157
174•1221
176.1595
158
1 59
IO
II
6.5598
6o
81•9202
83'7055
178'2009
180•2462
284'6735
61
II0
III
160
7'6012
161
12
13
14
8.6803
9'7943
10'9404
62
63
64
85'4979
87.2972
89'1034
112
1
xx
13
4
182.2955
118846...43045845
262
163
286.8803
289•o898
15
12'1165
65
90'9163
115
188'4661
16
13.3206
92'7359
94'5619
96'3945
98.2333
xx6
440075..734
360
291'3020
258
259
504'5252
506'9334
509'3433
511'7549
514.1682
260
261
262
263
518.9999
52 r 4182
523.8381
264
526•2597
528.6830
531.1078
533'5344
535.9625
538.3922
257
164 293.5168
214
215
409.6664
297.9544
216
217
265
266
4112:3
4 03
07
03
9
ix 8
165
,66
167
,68
295'7343
190.5306
192.5988
194.6707
2,8 416.6758
267
268
119
196'7462
169
219
419.0162
269
220
221
222
421'3587
423'7031
426'0494
270
271
272
17
14'5511
66
67
x8
15.8063
68
19
17.0851
69
20
18.3861
70
100'0784
120
198.8254
170
306.8608
21
22
23
19'7083
21'0508
22'4125
71
72
101 .9297
200'9082
171
309'0938
202 '9945
24
23'7927
107. 5196
121
122
123
124
172
173
174
311.3293
313'5674
315'8079
223
224
428'3977
430'7480
273
274
25
25.1906
26.6056
28.0370
29%4841
30'9465
75
109'3946
111. 2754
125
209-2748
175
318.0509
225
433'1002
126
211'3751
176
320'2965
226
322'5444
324"7948
327'0477
227
228
229
435'4543
437.8103
4.40.1682
442.5281
275
276
277
8o
81
83
124.5961
34
38- 4702
84
126.5204
329.3030
331.5607
333.8207
336.0832
338'3480
230
231
82
120'7632
122'6770
33
32.4237
33.9150
35'4202
36.9387
85
86
128.4498
130'3843
X85
340'6152
235
186
342•8847
26
27
28
29
30
31
32
35
40'0142
36
37
41'5705
43.1387
38
44 • 7185
39 46.3096
73
74
76
77
78
79
103'7870
105'6503
117
205'0844
207'1 779
113.1619
127
213.4790
115-0540
128
215.5862
177
178
116.9516
129
217.6967
179
118.8547
130
131
132
219'8107
221'9280
224'0485
181
133
134
226 .1 724
228'2995
135
136
137
138
230.4298
232- 5634
234.7001
236 . 8400
87
132-3238
88
89
134.2683
136-2177
180
182
183
184
300'1771
302'4024
304'6303
189
283
284
572.5753
575' 0287
285
:5569:7
02
99
6:
286
577'4835
579.9399
461'4742
463 .8508
288
349'70 71
239
466.2292
289
468'6094
470'9914
473'375 2
475'7608
478.1482
47'91 16
90
138'1719
140
241'1291
190
351'9859
240
49'5244
51' 1477
52'7811
54'4246
91
140.1310
141
243.2783
354.2669
241
45
56.0778
46
47
48
49
57'7406
59'4127
61'0939
62. 7841
50
64.4831
92
142'0948
142
245'4306
93
94
144. 0632
146'0364
143
144
247'5860
249'7443
191
192
193
194
95
96
97
98
148.0141
1 49'9964
151•9831
1 53'9744
145
146
251'9057
254.0700
147
256'2374
368'0003
99
155.9700
148
149
258.4076
260.5808
197
198
1 99
100
57'9700
150
262'7569
200
For large n, logio n!
565.2246
567-6733
237
40
570'1235
584
2;8
39
577
587.3180
290
589/804
291
592'2443
292
293
294
594'7097
597.1766
599'6449
295
296
297
6o2'1147
356'5502
242
358.8358
361.1236
243
244
195
363'4136
245
196
365.7059
246
480'5374
482.9283
370.2970
372'5959
248
2
47
8
249
487
71 5
5'32140
298
609'5330
490' 1116
299
612.0087
374.8969
250
492.5096
300
614.4858
0.39909 + (n+ logio n - 0.4342945 n.
36
553'0044
555'4453
557.8878
560.3318
562 '7774
280
281
282
238
41
42
43
44
540'8236
543.2566
545.6912
548'1273
550'5651
444.8898
447'2 534
449-6189
451.9862
454'3555
3
3 :=
187
139 238.9830
232
233
234
278
279
516'5832
604.586o
607.0588
TABLE 7. THE x'-DISTRIBUTION FUNCTION
F,(x)
The function tabulated is
FAx)
-
jo
tiv-le-igdt
0
(The above shape applies for v > 3 only. When v < 3 the mode is
at the origin.)
for integer v < 25. Fp(x) is the probability that a random
variable X, distributed as X2 with v degrees of freedom, will
be less than or equal to x. Note that F,(x) = 20(xi) - (cf.
Table 4). For certain values of x and v > 25 use may be made
of the following relation between the X2 and Poisson
distributions :
with mean v and variance 2v. A better approximation is
usually obtained by using the formula
Fv(x) * (1)(V-z; - zy -
Fv(x) = 1 - F(iv - x I ix)
where 4(s) is the normal distribution function (see Table 4)•
Omitted entries to the left and right of tabulated values are
r and o respectively (to four decimal places).
where F(rlit) is the Poisson distribution function (see
Table z). If v > 25, X is approximately normally distributed
=
x = 0.0
'I
'2
'3
'4
I
V =
2
V =
x = 4.0
0 '9545
x = 0•0
0.0000
x = 4.0
'I
'2
'9571
•1
I
V =
o•0000
'2482
'3453
.4161
'4729
2
V =
0•8647
•8713
v=
x = 0.0
o•0000
x = 4-o
:2
1
:0
00
22
84
2
'2
-.4
6
0.7385
'7593
:7
79
78
66
5
. 0598
•8
8130
0.5
-6
o.o811
5.0
0.8282
•1036
-2
.99°04963
•7
'9137
'9
'4
•6
•8
'8423
'8553
'9
•1268
•1505
•1746
0'3935
'4231
'4512
5.o
0.9179
•9219
1•0
•I
0'1987
'2229
6•o
•I
'2
'2
'8 977
'3
.2
.3
'2470
'2709
'5034
'4
'4
'2945
'4
'6
'8
•9063
.4780
'9257
'9293
'9328
5'5
'6
0 '9361
1.5
0'3177
6
.7
.3406
.3631
7'0
'2
0'9281
'9392
'8
27
0
'3851
'4066
:4
6
:93
49
508
.8
*9497
0.4276
'4481
8-o
•4681
':4
86
0'9540
'9579
6
199
.. 966647
9-o
0'9707
'2
'4
'9733
'9756
•i
'9596
'2
:0
04
98
58
2
'2
' 8775
.9619
'3
'4
'1393
'8835
'3
-o400
.1813
'3
-4.
.8892
'4
0.9661
•9680
'9698
.9715
'9731
0'5
0'2212
4:5
0 • 8946
•6
'7
•2592
•6
'8997
'2953
'8
'3297
'9
'3624
0'9747
-9761
'9774
'9787
1'0
'I
'9799
'4
•3
'4
'9641
0•5
0'5205
•6
'7
•8
.9
- 5614
'5972
•6289
-6572
4.5
•6
.7
-8
'9
1•0
0.6827
5•0
'I
'7057
'I
'2
'7267
'2
'3
'4
'7458
'7633
'3
'4
I'5
'6
'7
•8
.9
0 '7793
0.9810
•9820
1.5
0.5276
•6
'9830
'9840
.7
•8203
-8319
5'5
'6
'7
'8
'9
'8
-5507
.5726
'5934
'9849
.9
.6133
'9
'9477
2'0
0'8427
6.o
0.9857
2.: 0
3
1
1
O.:68
33
24
6:.!
2
o:.99945520:
•z
' 8527
'I
'65or
'2
•2
-8620
'2
'9865
'9872
'6671
'3
'4875
'4
'6988
'4
•6
.8
'I
'2
'9879
'9886
'9550
'9592
•9631
-9666
2'5
0:55422
457
'794 1
'8o77
.2
'3
'2
3
3
:78
•8
'2
•8672
•8782
o•8884
'9142
'9214
'9342
'3
' 8706
*4
'8787
'3
'4
2•5
0.8862
6.5
o'9892
2•5
0'7135
7.o
0'9698
'6
'8931
'6
'9898
'6
'7275
'2
'9727
'7
'8997
'7
'9904
'9057
'8
'9909
'9
'9114
'9
'9914
'9
'7654
'4
.6
'8
'9753
.9776
'9798
'7
.8
'9
'5598
'8
:78
503:
:774
'5927
:6
8
:97
977
97
3'0
0.9167
-9217
7•0
3-o
•I
0'7769
8:2
o
7
0:9
98314
3• o
.1
o • 6o84
•6235
10.0
•1
'2
'7981
•8o8o
'8173
'4
'9850
'2
•6
'3
'8
•9864
'9877
'4
'6382
'6524
- 666o
'4
'6
•8
0.9814
•9831
'9845
'9859
•9871
o•8262
'8347
'8428
9'0
'2
o•9889
'9899
'4
'9909
'8577
'6
'8
'9918
'9926
o.679z
•6920
'7043
•7161
'7275
I•o
'8504
3.5
-6
'7
-8
'9
o'8647
I0•0
0.9933
4'0
0'7385
'2
'9264
'2
'3
'4
'9307
'9348
'3
'4
0.9918
'9923
'9927
'9931
'9935
3'5
'6
'7
'8
'9
0'9386
'9422
'9456
'9487
7.5
'6
0'9938
'9942
'7
'9945
'8
'9517
'9
'9948
'9951
3.5
'6
'7
'8
'9
4'0
0 '9545
8'o
0.9953
4'0
'I
•3
'4
•7878
37
. 5765
•2
•2
0.9883
•9893
'4
'9903
•6
'8
.9911
.9919
12'0
0'9926
TABLE 7. THE x2-DISTRIBUTION FUNCTION
9
xo
II
.00x 8
•0073
•0190
o'0006
•0029
•oo85
0'0002
0•0001
•0011
•0037
0.0729
0.0383
•1150
'0656
0.0191
.0357
0.0091
•0186
v=
4
5
6
7
8
X = 0'5
1•0
0'0265
•0902
0'0079
0'0022
0'0006
0'000 I
'0374
1.5
.1734
•o869
•0144
•0405
2'0
•2642
•1509
•0803
•0052
•0177
•0402
2'5
3•0
0.3554
0.2235
0.1315
'4422
•3000
•1912
12
13
14
•0004
•0015
0.0001
•0006
0•0002
0.0001
0'0042
0'0018
•0093
•0045
00008
•0021
0.0003
-0009
3'5
4'0
4'5
•5221
'3766
'2560
•1648
•'008
'0589
'0329
'0177
'0091
'0046
'0022
'5940
'4506
'5201
'3233
. 3907
•2202
.2793
•1429
'1906
-o886
•1245
•o527
•0780
•0301
-4471
.ol 66
•0274
•oo88
•0154
'0045
' 0084
5.0
0.7127
0.4562
•5185
. 5768
'6304
.6792
0.3400
•4008
'4603
•5173
'5711
o'2424
•2970
•3528
•4086
*4634
0.1657
•2113
•26o x
•3110
*3629
0.1088
.1446
•1847
•2283
.2746
0.0688
.o954
•1266
•16zo
•2009
0'0248
0'0142
•7603
•8009
0.5841
•6421
.6938
'7394
. 7794
0'0420
5.5
6•o
6.5
7.0
•0608
•o839
•1112
.1424
'0375
.0538
- 0978
'0224
'0335
'0477
'0653
0.8140
.8438
.8693
.8909
*9093
0.7229
.7619
'7963
'8264
'8527
0.6213
'6674
*7094
'7473
•7813
0.5162
'5665
.6138
'6577
•6981
0.4148
'4659
•5154
'5627
•6075
0.3225
'3712
'4199
'4679
.5146
0.2427
•2867
'3321
*3781
'4242
0.1771
•2149
*2551
•2971
•3403
0.1254
•1564
'1904
•2271
•2658
0.0863
•1107
-1383
•1689
-zozz
0'9248
*9378
'9486
.9577
•9652
0'8753
.8949
•9116
.9259
•9380
0.8114
-838o
•8614
•8818
-8994
0'7350
•7683
'7983
•8251
.8488
0.6495
•6885
'7243
•7570
.7867
0'5595
•6022
'6425
•6801
. 7149
0'4696
•5140
'5567
•5976
'6364
0.3840
*4278
. 4711
.5134
'5543
0•3061
'3474
•3892
.4310
'4724
0.2378
.2752
•3140
.3536
'3937
0.9715
*9766
'9809
'9844
'9873
0'9483
0'9147
'9279
'9392
.9488
'9570
o'8697
•8882
.9042
•9182
'9304
0.8134
* 8374
.8587
•8777
.8944
0.7470
'7763
.8030
.8270
*8486
0.6727
.7067
.7381
7 6 7o
'7935
0.5936
.6310
•6662
.6993
'7301
0.5129
. 5522
. 5900
•6262
•6604
0'4338
'9570
'9643
'9704
'9755
0'9896
'9916
'9932
'9944
'9955
0'9797
.9833
•9862
'9887
'9907
0.9640
*9699
'9749
'9791
•9826
0.9409
'9499
'9576
'9642
'9699
0.9091
'9219
'9331
.9429
'9513
0.8679
.8851
•9004
•9138
.9256
0.8175
'8393
•8589
•8764
•8921
0.7586
'7848
•8088
•8306
•8504
o•6926
.7228
•7509
•7768
'8007
o.6218
'6551
•6866
•7162
'7438
0.9924
.9938
'9949
.9958
*9966
0.9856
•9880
'9901
.9918
'9932
0'9747
.9788
'9822
'9851
'9876
0'9586
'9648
'9702
.9748
.9787
0.9360
•9450
'9529
'9597
.9656
0.9061
'9184
*9293
'9389
'9473
0.8683
'8843
*8987
'9115
.9228
0.822.6
'8425
-8606
.8769
•8916
0.7695
.7932
'9992
'9994
0. 9964
'9971
'9976
'9981
.9984
20
21
22
23
0'9995
0'9988
0'8699
'9999
0.9707
'9789
'9849
'9893
'9924
0 '9048
24
0.9821
*9873
'9911
'9938
'9957
0 '9329
'9962
'9975
'9983
'9989
0.9897
*9929
'9951
'9966
'9977
0 '9547
'9992
'9995
'9997
'9998
0.9972
•9982
'9988
'9992
'9995
0 '9944
'9997
'9666
'9756
*9823
'9873
*9496
'9625
'9723
'9797
.9271
'9446
*9583
'9689
*8984
'9214
'9397
'9542
25
26
27
0 '9999
0 '9999
0 '9997
0.9984
'9989
'9993
'9995
'9997
0'9970
'9980
'9986
'9990
'9994
0.9769
.9830
0'9654
'9963
'9974
'9982
'9988
0'9909
'9935
'9954
'9968
'9977
0'9852
'9998
'9999
'9999
'9999
0'9992
'9995
'9997
'9998
'9999
0 '9947
'9999
0 '9999
0'9998
0'9996
0.9991
0'9984
0 '9972
'6575
*8352
-8641
7'5
8•o
8.5
9.0
9'5
o-8883
'9084
10.0
xo.5
x•o
-9251
.9389
'9503
11•5
0.9596
.9672
'9734
.9785
12.0
•9826
12•5
13.0
13.5
0.9860
•9887
'9909
14•0
*9927
14.5
'9941
15.0
15.5
16•o
0 '9953
16.5
17'0
.9976
.9981
17.5
0 '9985
18•o
18'5
19.0
19.5
'9988
28
29
30
'9962
.9970
'9990
'9998
'9999
'9999
38
6
'9893
'9923
'9945
'9961
' 0739
'4735
•5124
•5503
•5868
•8151
.8351
'8533
'9876
'9910
'9935
-9741
-9807
'9858
'9895
0 '9953
0 '9924
TABLE 7. THE f-DISTRIBUTION FUNCTION
v=
X
15
x6
17
18
19
=3
4
0.0004
•0023
0'0002
0'0001
•001I
'0005
0'0002
0'000 I
5
6
7
8
9
0.0079
•0203
•0424
-0762
•I225
0'0042
0'0022
0'001 I
•0 I 19
•oo68
•o267
•0165
'051 I
•0335
•o866
•0597
•0038
•0099
'0214
•0403
10
II
12
0'1803
0.1334
•1905
0.0964
•1434
•2560
•1999
•3977
'4745
•3272
.4013
'2638
•3329
o•o681
•1056
•1528
•2084
'2709
13
'5
16
17
x8
19
0 '5486
0.4754
'5470
.6144
.6761
.7313
0 '4045
0•3380
•4762
.5456
-6112
.6715
20
21
22
23
24
0'8281
0.7708
'821 5
•8568
•8863
'9105
25
26
27
0'9501
14
'2474
'3210
•6179
•6811
'7373
•7863
•8632
•8922
•9159
'9349
20
21
22
23
24
25
0•0006
0'0003
0.0001
•0021
•0058
•0133
•0265
•00II
'0006
•0033
•oo8 1
.0 r 7 1
•0019
•0049
•0'08
0.0001
•0003
•00I0
•oo28
•oo67
0.0001
•0005
-00'6
•0040
0.0001
•0003
•0009
•0024
0.0001
•0005
•0014
0.0471
•0762
•1144
•1614
•2163
0'0318
0'0211
0'0137
0'0087
0'0055
0'0033
-0538
•0839
•1226
•1695
.0372
•0604
•0914
'1304
•0253
•0426
•0668
•0985
•0168
•0295
•0480
.0731
.oi To
•0201
•0339
.0533
'0235
•0383
'4075
'4769
•5443
•6082
0.2774
'3427
•4101
'4776
'5432
0.2236
•2834
•3470
'4126
.4782
0.1770
.2303
•2889
'3510
'4149
0.1378
.1841
•2366
'2940
'3547
0'1054
•1447
•1907
•2425
*2988
0.0792
•x '19
•1513
.1970
.2480
0.0586
'0852
. i '82
'1576
.2029
0.7258
'7737
•8153
•8507
•8806
0.6672
•7206
•7680
•8094
•8450
0.6054
•6632
•7157
•7627
.8038
0.5421
•6029
.6595
-7112
. 7576
0.4787
'5411
'6005
0.4170
'4793
'5401
'6560
* 5983
•7069
.6528
0.3581
'4189
'4797
'5392
. 5962
0.3032
'3613
'4207
.4802
'5384
0.2532
'3074
'3643
•4224
-4806
0.9053
*9255
'9419
'9551
'9655
0.8751
•9002
•9210
'9379
'9516
0.8395
•8698
-8953
.9166
'9340
0'7986
.8342
•8647
•8906
'9122
0.7528
*7936
•8291
•8598
•886o
0.7029
0.6497
'6991
.7440
'7842
'8197
0'5942
'7483
•7888
•8243
•8551
0.5376
'5924
'6441
-6921
'7361
•0071
'0134
28
'9713
'9784
29
'9839
0.9302
'9460
'9585
•9684
'9761
30
31
32
33
0.9881
'9912
.9936
'9953
'9966
0.9820
•9865
.9900
.9926
'9946
0 '9737
0'9626
'9712
.9780
'9833
'9874
0.9482
'9596
'9687
-9760
'98,6
0'9301
'9448
'9567
.9663
'9739
0.9080
'9263
'9414
.9538
'9638
0.8815
•9039
'9226
-9381
'9509
0.8506
•8772
.8999
'9189
'9348
o.8x 52
•8462
•8730
.8959
.9153
0 '7757
•9800
'9850
.9887
'9916
35
36
37
38
39
0'9975
0.9960
'9971
'9979
'9985
'9989
0.9938
'9954
'9966
'9975
.9982
0.9905
'9929
'9948
'9961
'9972
0.9860
.9894
'9921
'9941
-9956
0 '9799
'9846
'9883
.9911
'9933
0'9718
.9781
'9832
.9871
.9902
0.9613
'9696
.9763
'9817
'9859
0'9480
'9587
'9675
'9745
'9802
0.9316
'9451
'9562
'9653
.9727
0'9118
'9284
'9423
'9537
.9632
40
41
42
43
44
0 '9995
0'9992
'9994
'9996
'9997
'9998
0'9987
'9997
'9998
'9998
'9999
'9991
'9993
'9995
'9997
0.9979
•9985
'9989
.9992
'9994
0'9967
'9976
.9982
.9987
'9991
0.9950
'9963
'9972
.9980
'9985
0.9926
.9944
.9958
.9969
'9977
0.9892
'9918
'9937
'9953
'9965
0.9846
'9882
.9909
.9931
'9947
0.9786
.9833
'9871
.9901
'9924
0.9708
.9770
'9820
•986o
'9892
45
46
47
48
49
0 '9999
0 '9999
0'9996
'9997
'9998
'9998
'9999
0 '9973
0 '9960
0.9942
0 '9916
'9995
'9996
'9997
'9998
0'9989
'9992
'9994
'9996
'9997
0'9983
'9999
'9999
0'9998
'9998
'9999
'9999
'9999
0 '9993
'9999
'9987
'9991
'9993
'9995
-9980
'9985
'9989
'9992
'9970
'9978
'9983
.9988
'9956
'9967
'9975
'9981
'9936
'9951
'9963
.9972
0 '9999
0 '9999
0 '9998
0 '9996
0 '9994
0.9991
0'9986
0.9979
34
50
•9620
'9982
.9987
'9991
'9994
39
'6468
'6955
.7400
'7799
-8i io
•842o
•8689
'8921
TABLE 8. PERCENTAGE POINTS OF THE x2-DISTRIBUTION
This table gives percentage points
equation
g(p)
P/100
defined by the
co
-
12 1-1/PN f,p)X1P-1
100 2'
e-i' dx.
If X is a variable distributed as X2 with v degrees of freedom,
Phoo is the probability that X 26(P).
For v > loo, ✓2X is approximately normally distributed
with mean ✓2v-1 and unit variance.
V
=
0 x(P)
(The above shape applies for v 3 3 only. When v < 3 the mode is
at the origin.)
8o
P
99'95
99'9
99'5
99
97'5
95
90
I
0.06 3927
0.00I000
0 . 01528
0'051571
0'002001
0'02430
0'043927
0.01003
0'07172
0'031571
0'02010
0'1148
0'039821
0.05064
0'2158
0'003932
0'1026
0'3518
0'01579
0;6
04
69
418
0;0
26
10
47
0- 5844
0.4463
roo5
0.06392
0 09080
0.2070
0.2971
0.4844
0.7107
2
3
4
5
6
7
8
9
.
0.1581
0.2102
0.4117
0 '5543
0.8312
P145
1.610
2 '343
0.2994
0'3811
0'6757
0.8721
V237
P635
2'204
3'070
0 '4849
0 '5985
0.8571
1.152
0'9893
1.344
P735
1'239
1.646
1.690
2.180
2.088
2/00
2.167
2133
3'325
2.833
3'490
4'168
3.822
4'594
5.380
6.179
6.989
7.807
8.634
9'467
0/104
0.9717
xo
1'265
1 '479
2.156
2'558
II
1 .587
1 . 834
2.214
3-053
3'571
4'107
4'660
3'247
3.816
4'404
5.009
5.629
3'940
4'575
5.226
5.892
6.571
4'865
5'578
6.304
7.042
7'790
7.261
7.962
8.672
9'390
10•12
8.547
9.312
10.09
10•86
11. 65
12.86
13.72
10-85
11.59
12- 34
13.09
13.85
12.44
13.24
14'04
14'85
15.66
12
1 '934
13
14
2 . 305
2. 617
2.697
3.041
2.603
3'074
3.565
4'075
15
16
3-108
3'536
3.980
4'439
4'912
3'483
3'942
4'416
4.905
5'407
4'601
5. 142
5.697
6.265
6.844
5.229
5.812
6.408
7.015
7633
6.262
6.908
7'564
8'231
8.907
5'398
5.896
5.921
6.447
7'434
8.034
8- 643
9-260
9.886
8.260
8.897
9'591
10•28
:0;9689
17
18
19
20
21
22
23
24
6'404
6.983
6- 924
7'453
7-529
8.085
1902
10.86
12.40
70
6o
0 . 1485
0/133
12
94
5
1
2:4
0'2750
P022
3
76
59
2
1.- 8
3-000
3.828
4'671
5'527
6 '393
3.655
4.570
5'493
6.423
7'357
7.267
8.148
99
9:02
3:
8.295
9'237
io.:
Ir3
8
1o•82
12-08
11.72
12.62
13'53
14'44
15.35
13.03
13.98
14'94
15 .89
16.85
14'58
15'44
16. 31
17.19
18.06
16.27
17.18
18.10
17.81
18.77
19.73
19.02
20.69
19'94
21.65
20.87
2P79
22.72
22.62
23'58
10.31
11.15
12'00
25
7'991
8.649
10.52
11.52
13.12
14'61
16'47
18.94
26
8.538
9.222
11'16
12'20
13'84
15'38
17.29
19.82
27
28
29
9.093
18•1i
20.70
21.59
22.48
ir81
1z.88
14'57
12'46
13.12
13'56
15'31
16.05
0.15
16.93
18.94
10.23
10'39
10.99
17.71
19.77
10.80
11.98
13.18
11.59
12.81
14.06
18'49
205,-60
25'51
20'07
22- 27
22
26
53-"93
14
56
-64
38
15'64
16 61
.
.
2:98r:382491
22-88
21.66
23 27
24.88
23.95
15'32
14'95
16.36
17'79
10.23
20 69
16.79
14.40
13.79
15.13
16.50
17 89
19 29
28.73
30'54
27-37
29.24
31 12
32 99
.
27-44
29-38
31.31
33.25
35-19
40
50
16.9z
23-46
30'34
37'47
44'79
17 92
20'71
22 . 16
24'43
26.51
24-67
31'74
39'04
46'52
27-99
35'53
178
4
51
3:2
29-71
32.36
4570-.4185
48.76
3520.
3
64
41'45
34'87
1
81
44
5;3
37'13
66
4
56.-82
63.35
72.92
66.40
76 19
52•28
59.90
54' 16
61.92
59.20
67.33
30
32
34
36
6o
70
8o
90
I00
9'656
9.803
.
.
.
14'26
8
3
47
5:444
53'54
61.75
70.06
65.65
74' 22
40
.
434
37196
51'74
6o 39
27-34
29.05
37-69
5456:3463
23.65
24.58
.
24'54
25.51
26.48
.
64.28
59'90
69.2!
69.13
77'93
73'29
82.36
78.56
82'51
85 . 99
87.95
92-13
95.81
.
TABLE 8. PERCENTAGE POINTS OF THE f-DISTRIBUTION
This table gives percentage points x;,(P) defined by the
equation
rco
100
2142
rq)
A( p)
xiv-1 e- P dx.
0
If Xis a variable distributed as x2 with v degrees of freedom,
Phoo is the probability that X x,2,(P).
For v > ioo, VzX is approximately normally distributed
with meanzi
.s/
and unit variance.
P
50
v=I
40
(The above shape applies for v
at the origin.)
30
20
To
5
3.841
5'991
7'815
9.488
2
3
4
1.386
2.366
3'357
0'7083
1.833
2.946
4'045
1.074
2.408
3'665
4'878
1.642
3'219
4.642
5'989
2.706
4'605
6.251
7'779
5
6
7
8
9
4'351
5.348
6.346
7'344
8.343
5.132
6.2i 1
7.283
8'351
9'414
6.064
7.231
8.383
9'524
7.289
8.558
9.236
10.64
II
9'342
10.34
0 '4549
X,2,'(P)
2'5
I
3 only. When v < 3 the mode is
0'5
0•I
0'05
5.024
7'378
9'348
11.14
6.635
9.210
11'34
13.28
7.879
10•6o
12.84
14.86
10'83
12'12
13.82
16'27
18'47
15.2o
17'73
20.00
11.07
12'59
12.83
14'45
15 '09
20'52
22'46
24'32
26'12
27'88
22'II
24'10
26'02
27'87
29'67
12'02
14'07
16'01
18.48
11.03
13.36
15'51
17'53
10•66
12'24
14'68
16'92
19'02
20'09
21'67
16.75
18.55
20.28
21.95
23.59
10'47
1P53
11.78
12'90
13.44
14.63
15'99
17.28
18.31
20'48
21'92
23.21
24'72
25.19
29'59
31'42
19'68
2616
31'26
14'01
15'12
15•81
16'98
18'55
19'81
21.03
23.34
26.22
22'36
24'74
27'69
16.22
18.15
21.06
23.68
26.12
29.14
28.30
29.82
31.32
32.91
34'53
36.12
33'14
34'82
36.48
38.11
19.31
20'47
z1•61
22.76
23.90
22.31
23'54
24'77
25'99
27.20
25.00
26.3o
2759
28.87
30.14
27'49
28'85
32.80
34'27
35'72
3716
38.58
37'70
39'25
40'79
42.31
43'82
39'72
41.31
30.19
31•53
32.85
30.58
32.00
33'41
34' 81
36.19
28'41
27'10
25.04
26.17
27.30
28'43
29.55
29'62
30'81
32'01
33'20
31.41
32.67
33'92
35'17
36.42
34'17
35'48
36.78
38.08
39'36
37'57
38.93
40'29
41'64
42 .98
40.00
4P40
42.80
44' 18
45'56
45'31
46.80
48'27
49'73
51.18
47'50
49'01
50.51
52.00
53'48
29
28.17
29.25
30.32
31.39
32.46
30.68
31•79
32.91
34'03
35'14
34'38
35'56
36.74
37'92
39'09
3765
38.89
40.11
41'34
42.56
40.65
41'92
43'19
44'46
45'72
44'31
45'64
46 .96
48.28
49'59
46'93
48.29
49'64
50'99
52. 34
52.62
54'0 5
55'48
27'34
28'34
26.14
27.18
28.21
29.25
30.28
56-89
58'30
54'95
56.41
57.86
59'30
60.73
30
32
34
36
38
29. 34
31'34
33'34
35'34
37'34
31'32
33'38
35'44
37'50
39'56
33'53
35'66
37'80
39'92
42.05
36.25
38'47
40.68
42.88
45.08
40'26
42'58
44'90
4721
49'51
43'77
46'19
48.60
51'00
53.38
46'98
49'48
51.97
54'44
56.90
50'89
53'49
56.06
58.6z
61•16
53'67
56'33
58.96
61.58
64.18
5910
62.49
65.25
67.99
70'70
62.16
65•oo
67.8o
70'59
73'35
40
50
6o
39.34
49'33
59.33
69.33
79.33
41.62
51'89
62.13
72.36
82.57
44'16
54'72
65.23
75.69
86.,2
47' 27
58.16
68.97
79'71
90'41
51.81
63.17
74'40
85'53
96.58
55'76
67.5o
79.08
90.53
I01.9
59'34
71.42
83.3o
95.02
106.6
63.69
76.15
88.38
100'4
rI2•3
66.77
79'49
91'95
104.2
116.3
73'40
86.66
99.61
112.3
124.8
76'09
89'56
102'7
89.33
99.33
92.76
102.9
96.52
106.9
124.3
118•1
129.6
124'1
135.8
128.3
140.2
149'4
12
11'34
12'58
13
14
12.34
13.34
13'64
14.69
15
14'34
15'34
15'73
1732
16.78
17.82
18.87
19.91
18.42
19'51
20.60
20'95
21.99
22.77
23.86
24'94
26'02
16
17
18
19
16'34
17'34
18'34
20
19.34
21
20'34
22
23
21 '34
24
23'34
25
26
24'34
25'34
27
28
70
8o
90
zoo
22 '34
26 '34
23.03
24'07
25•11
21'69
9'803
111.7
107.6
118.5
41
16.81
1372
45'97
115.6
128.3
140.8
153'2
TABLE 9. THE t-DISTRIBUTION FUNCTION
The function tabulated is
F„(t)
ray +1-) f
=
,
VV 7T
- co k I m
ds
svoi(v+i) .
F„(t) is the probability that a random variable, distributed as
t with v degrees of freedom, will be less than or equal to t.
When t < o use F„(t) = i F„( t), the t distribution being
symmetric about zero.
The limiting distribution of t as v tends to infinity is the
normal distribution with zero mean and unit variance (see
Table 4). When v is large interpolation in v should be
harmonic.
-
V
=
I
-
V =
V =
Omitted entries to the right of tabulated values are
(to four decimal places).
V =
2
t = o 0 0.5000
•1
'5317
t = 4•0
4.2
0.9220
.9256
'2
'5628
.9289
'9319
'9346
•5700
-2
.5928
•621
4'4
4'6
4.8
'2
.3
'4
•3
'4
-6038
•6361
'3
'4
0.5
0'6476
•6
•6720
.7
'6944
•8
'9
•7148
'7333
5.0
5'5
6.0
6.5
7.0
0.9372
'9428
'9474
'9514
'9548
0.5
-6
'7
-8
'9
0.6667
.6953
•7218
'7462
• 684
I•0
0.7500
I.0
•7651
7.5
8.0
0.9578
I
'2
'7789
8.5
•9627
'2
'3
'7913
•8026
9.0
9.5
.9648
.9666
1.5
•6
•7
•8
'9
0.8128
10. 0
I0•5
-8386
'8458
II-5
I2•0
2•0
0.8524
•,_,
R585
12.5
'2
•3
'4
2'5
0.8789
•6
•7
-8
.9
•8831
-8871
t = 0.0
0.5000
'5353
2
v=
t = 0.0
3
V
=
3
0•5000
t = 4.0
.5367
.1
0.9860
.9869
'2
'5729
'2
'9877
.9760
•3
'4
•6081
•6420
•3
'4
'9891
4'5
'6
'7
-8
'9
0 '9770
0.5
0. 6743
4'5
'9779
-6
.7046
.6
'9788
'9804
.7
•8
•9
-7328
.7589
-7828
'7
.8
'9
5'0
0'9811
I.0
'2
•9818
.9825
•I
'2
0 '8045
- 8242
- 8419
5•0
•I
•2
0.9923
I
'3
'4
0.7887
8070
-8235
-8384
•8518
•3
'4
.9831
'9837
•3
'4
.8578
•872o
'3
'4
'9934
'9938
0.9683
•9698
.9711
'9724
'9735
r5
•6
'7
.8
.9
0.8638
.8746
- 8844
.8932
.9011
5'5
0.9842
•9848
'9853
.9858
•9862
/.5
-6
•7
-8
•9
0.8847
•8960
5.5
.6
'7
.8
'9
0'9941
0'9746
.9756
21)
0'9082
6.0
0.9303
'9367
'8642
13.5
'9765
'2
'9206
•I
'2
'9875
2'0
'I
'2
0 '9954
'9147
0.9867
-9871
6.0
13•0
'9424
.8695
14.0
'9773
14'5
'9781
'9308
'3
'4
'9879
'9882
'3
'4
'9475
.9521
-9960
'8743
'3
'4
•2
•3
'4
. 9961
15
0.9788
2.5
6.5
0'9561
•6
.7
-8
•9
.9598
.9631
- 8943
.6
'7
.8
'9
-9687
6.5
-6
-7
-8
'9
0.9963
•9801
•9813
-9823
.9833
0.9886
-9889
2'5
16
17
i8
19
3•0
•I
'2
0. 8976
•9007
'9036
20
21
22
0'9841
3•0
'I
0.9712
'9734
7.0
•I
-3
'4
•9063
-9089
23
•9862
0.9970
.9971
-9972
'9973
24
3.5
•6
.7
-8
•9
0'9114
25
•9138
-916o
•918z
•9201
30
4'0
0'9220
'4
•8222
-8307
-8908
•9604
t = 4*0
•I
-
•6
'7
•8
•9
'9259
0.9352
-9392
•6
.7
-8
'9429
-9463
'9494
0 '9714
.9727
'9739
'9750
-9796
-9892
.9895
•9062
•9152
.9232
-9661
1
•9884
0.9898
•9903
.9909
. 9914
'9919
-9927
'9931
'9944
'9946
'9949
'9951
-9956
-9958
.9965
-9966
.9967
.9
-9898
7•0
•I
'2
.9753
•2
'9867
'3
'4
0.9901
'9904
.9906
.9909
•9911
3•0
'3
'4
0.9523
'9549
'9573
'9596
•9617
'3
'4
'9771
'9788
'3
'4
3'5
•6
.7
-8
'9
0.9636
- 9654
•9670
•9686
•97ot
7.5
-6
-7
•8
•9
0.9913
•9916
•9918
•9920
.9922
3.5
-6
'7
.8
.9
0.9803
•9816
-9829
'9840
•9850
7'5
.6
'7
-8
'9
0 '9975
45
0.9873
'9894
'9909
•9920
'9929
50
0'9936
4 '0 0.9714
8.0
0.9924
4.0
0.9860
8.0
0.9980
35
40
'9849
'9855
'2
42
2
'9969
'9974
.9976
'9977
-9978
'9979
TABLE 9. THE t-DISTRIBUTION FUNCTION
4
5
6
7
8
9
xo
II
12
13
14
t = 0.0
0.5000
0•50o0
0.5000
0.500o
0.5000
0.5000
0.5000
0.500o
0.500o
0.500o
0.5000
'I
•2
'5374
'5744
'3
'4
•6104
. 6452
'5379
'5753
•6119
'6472
.5382
.5760
•6129
.6485
.5384
'5764
•6136
. 6495
.5386
.5768
•6141
•6502
'5387
'5770
•6145
•6508
.5388
'5773
•6148
•6512
•5389
'5774
•6151
•6516
'5390
'5776
•6153
•6519
'5391
'5777
•6155
•6522
'5391
'5778
.6157
•6524
0.5
•6
'7
.8
0•6783
0.6809
0.6826
•7096
.7387
.7657
.7127
•7148
'9
•7905
'7424
•7700
'7953
'7449
'7729
.7986
0•6838
•7163
'7467
'7750
•8oio
0•6847
'7174
•7481
.7766
-8028
0•6855
-7183
'7492
'7778
•8042
0.6861
.7191
•7501
.7788
•8054
0•6865
'7197
•7508
'7797
•8063
0•6869
•7202
•7514
•7804
•8o71
0•6873
•7206
.7519
•7810
•8078
0•6876
•7210
•7523
•7815
•8083
ro
•x
o•813o
•8335
•8518
•8683
•8829
0•8184
•8393
•8581
.8748
.8898
0.8220
- 8433
•8623
•8793
.8945
0•8247
•8461
•8654
•8826
'8979
0.8267
•8483
•8678
•8851
•9005
0•8283
•85o1
•8696
•8870
•9025
0•8296
'8514
•8711
•8886
•9041
0.8306
.8526
•8723
•8899
-9055
0•8315
•8535
•8734
•8910
.9066
0•8322
.8544
'8742
•8919
.9075
0.8329
.8551
•875o
.8927
-9084
I.5
•6
0.8960
0.9030
•9076
•9148
0'9079
•9196
'7
.8
'9
•9178
•9269
'9349
'925 1
'9341
0.9140
.9259
•9362
0
5 532
:9
94
0.9161
-9280
.9383
'9473
'9551
0.9177
'9297
•9400
'9490
•9567
0.9191
.9310
'9414
•9503
-958o
0.9203
'9322
156
1
5529
994
.9
9421
'9390
9469
0.9114
•9232
'9335
9426
.9 04
0.9212
'9332
'9435
'9525
•9601
0.9221
'9340
'9444
'9533
•9609
2'0
0'9419
'I
'2
'9482
'9537
'3
'4
'9585
•9628
0.9490
'9551
•9605
•9651
'9692
0.9538
'9598
•9649
•9694
'9734
0.9572
-9631
•9681
-9725
•9763
0.9597
•9655
.9705
'9748
•9784
0.9617
•9674
'9723
'9765
•9801
0.9633
'9690
'9738
'9779
•9813
0.9646
.9702
'9750
'9790
•9824
0.9657
.9712
'9759
'9799
•9832
0.9666
'9721
.9768
.9807
•9840
0•9674
.9728
'9774
.9813
-9846
2'5
.6
0.9666
0.9728
'9759
'9730
•9786
•9810
•9831
0•9767
'9797
•9822
'9844
'9863
0'9795
•9700
'9823
•9847
-9867
•9885
0.9815
'9842
-9865
•9884
.9901
0.9831
-9856
•9878
.9896
'9912
0.9843
•9868
•9888
•9906
-9921
0.9852
.9877
•9897
'9914
-9928
0.9860
.9884
•9903
'9920
'9933
0•9867
.9890
-9909
'9925
'9938
0.9873
-9895
'9914
'9929
'9942
0.985o
•9866
-9880
'9893
•9904
0.9880
.9894
'9907
.9918
-9928
0.9900
•9913
.9925
'9934
'9943
0.9915
'9927
'9937
'9946
'9953
0.9925
'9936
'9946
'9954
'9961
0.9933
'9944
'9953
-996o
'9966
0.9940
'9949
'9958
.9965
'9970
0.9945
'9954
.9962
.9968
'9974
0.9949
'9958
.9965
'9971
'9976
0.9952
-9961
'9968
'9974
'9978
0.9914
'9922
'9930
'9937
'9943
0.9936
'9943
'9950
'9955
•9960
0'9950
0.9960
•9965
'9970
'9974
'9977
0.9966
'9971
'9975
'9979
•9982
0.9971
'9976
'9979
•9983
'9985
0'9975
'9956
.9962
.9966
'9971
'9979
•9982
•9985
-9988
0.9978
'9982
'9985
•9987
-9989
0.998o
.9984
.9987
.9989
'9991
0.9982
.9986
'9988
'9990
'9992
0.9948
'9953
'9958
•9961
'9965
0.9964
.9968
'9972
'9975
'9977
0.9974
'9977
.9980
-9982
'9984
0•9980
•9983
'9985
•9987
•9989
0.9984
.9987
-9988
'9990
'9991
0'9987
0'9990
0.9991
0'9992
•9989
'9991
'9992
'9993
'9991
'9993
'9994
'9995
'9993
'9994
'9995
'9996
'9994
'9995
'9996
'9996
0.9993
'9995
'9996
'9996
'9997
0.9979
'9982
'9983
.9985
-9986
0.9986
•9988
'9989
'9990
'9991
0.9990
'9991
'9992
'9993
'9994
0.9993
'9994
'9994
'9995
'9996
0.9994
'9995
'9996
'9996
'9997
0.9995
'9996
'9997
'9997
'9998
0.9996
'9997
'9997
'9998
'9998
0 '9997
0 '9998
'9998
'9998
'9998
'9999
'9998
'9998
'9999
'9999
0.9988
0.9992
0.9995
0 '9996
0'9997
0.9998
0 '9998
0.9999
0 '9999
v=
'2
'3
'4
'7
.8
'9
3'0
'I
'2
'3
'4
3.5
•6
'9756
'9779
0.980o
•9819
'9835
•9850
'9864
0.9876
'9
•9886
•9896
'9904
•9912
4'0
0 '9919
'I
'2
'3
'4
•9926
'9932
'7
.8
'9937
'9942
4.5
.6
0.9946
'7
.8
'9
'9953
'9957
.9960
0.9968
'9971
'9973
'9976
'9978
5'0
0.9963
0 '9979
'9950
.9300
43
TABLE 9. THE t-DISTRIBUTION FUNCTION
v=
15
16
17
i8
19
20
24
30
40
6o
c0
t = 0.0
•1
.2
•3
'4
0.5000
.5392
*5779
•6159
'6526
0'5000
0'5000
0'5000
0'5000
0'5000
0'5000
0'5000
0'5000
0.5000
.5392
'5780
•6160
•6528
*5392
'5781
•6161
•6529
'5393
'5781
•6162
.6531
'5393
'5782
•6163
. 6532
'5393
'5782
•6164
*6533
'5394
'5784
•6166
'6537
'5395
'5786
•6169
'6540
'5396
'5788
-6171
'6544
'5397
'5789
•6174
'6547
0.5000
'5398
'5793
'6179
'6554
0.5
•6
'7
•8
'9
0.6878
•7213
'7527
•7819
•8088
0.6881
•7215
'7530
•7823
•8093
0.6883
•7218
'7533
•7826
•8097
0.6884
•722o
'7536
•7829
-81oo
0.6886
•7222
'7538
. 7832
-8103
0.6887
•7224
'7540
. 7834
•8xo6
0.6892
•7229
'7547
'7842
•8115
0.6896
'7235
'7553
•7850
•8124
0.6901
'7241
'7560
•7858
•8132
0.6905
'7246
. 7567
•7866
'8141
0.6915
'7257
.7580
•7881
'8159
1.0
0.8339
•8562
'8762
'8940
'9097
0.8343
•8567
'8767
'8945
'9103
0'8347
.8571
-8772
*8950
'9107
0 '8351
0 '8354
0 '8364
•8578
'8779
*8958
•9116
•8589
.8791
'8970
•9128
0.8383
•8610
•8814
'8995
•9154
0 '8413
.8575
'8776
*8954
•9112
0.8373
•8600
•8802
'8982
•9141
0 '8393
'3
'4
0'8334
•8557
•8756
'8934
'9091
•8621
•8826
'9007
•9167
'8643
•8849
'9032
-9192
1.5
0'9228
0'9235
0'9240
0'9245
0'0250
0'9254
0'9267
0'9280
.6
'7
.8
'9
'9348
'9451
'9540
'9616
'9354
'9458
'9546
•9622
'9360
'9463
'9552
•9627
'9365
.9468
'9557
•9632
'9370
'9473
'9561
•9636
'9374
'9477
'9565
•9640
'9387
'9490
'9578
•9652
'9400
'9503
'9590
•9665
0.9293
'9413
.9516
•9603
•9677
0.9306
.9426
'9528
•9616
•9689
0.9332
'9452
'9554
-9641
.9713
2'0
0.9680
'9735
0.9691
'9745
'9790
'9828
'9859
0.9696
'9750
'9794
.9832
'9863
0.9700
'9753
'9798
.9835
•9866
0 '9704
0.9715
'9768
-9812
'9848
•9877
0'9727
'9779
•9822
'9857
•9886
0.9738
'9790
•9832
'9866
'9894
0.9750
•9800
•9842
'9875
'9902
0.9772
•9821
•9861
'9893
•9918
•I
•z
'2
'9781
'3
'4
'981 9
'985 1
0.9686
'9740
.9786
'9824
'9855
2.5
'6
'7
.8
'9
0.9877
'9900
'9918
'9933
'9945
0.9882
'9903
'9921
'9936
'9948
0.9885
'9907
'9924
'9938
*9950
0.9888
'9910
'9927
.9941
'9952
0.9891
'9912
'9929
'9943
'9954
0'9894
'9914
'9931
'9945
'9956
0'9902
0'9909
'9921
'9937
'9950
*9961
'9928
'9944
'9956
'9965
0'9917
'9935
'9949
'9961
'9970
o'9924
'9941
'9955
•9966
'9974
0'9938
*9953
'9965
'9974
'9981
3'0
0.9955
.9963
'9970
.9976
'9980
0 '9958
0.9960
.9967
'9974
'9979
'9983
0.9962
-9969
'9975
-9980
'9984
0.9963
.9971
'9976
-9981
'9985
0 '9965
0 '9969
0.9973
0'9977
•9972
'9978
•9982
'9986
.9976
'9981
'9985
*9988
.9979
'9984
.9988
'9990
.9982
'9987
.9990
'9992
0.9980
.9985
*9989
.9992
'9994
0.9987
.9990
'9993
.9995
'9997
3'5
'6
'7
.8
'9
0 '9984
0 '9985
0.9987
'9990
'9992
'9993
'9995
0.9988
'9990
'9992
'9994
'9995
0.9989
'9991
'9993
'9994
'9996
0.9991
'9993
'9994
'9996
'9997
0 '9994
'9988
'9990
.9992
'9994
0.9986
'9989
'9991
'9993
'9994
0 '9993
.9987
'9989
'9991
'9993
'9994
'9996
'9997
'9997
'9996
.9997
'9998
'9998
0.9996
'9997
'9998
'9998
'9999
0.9998
'9998
'9999
'9999
4'0
'I
0 '9994
0 '9995
0 '9995
0.9999
'9998
'9998
'9999
'9999
'9999
'9999
'9999
'9999
'9999
'9999
'9999
'9999
'3
'4
0.9996
'9997
'9998
'9998
'9999
0 '9999
'9996
'9997
'9998
'9998
0.9996
'9997
'9998
'9998
'9998
0 '9998
'9996
'9997
'9997
'9998
0'9996
'9997
*9997
'9998
'9998
0 '9997
'9995
'9996
'9997
'9997
4,5
0 '9998
0'9998
0'9998
0 '9999
0 '9999
0 '9999
0 '9999
•1
•I
'2
•3
'4
'2
.9966
*9972
.9977
'9982
44
'9757
•9801
'9838
•9869
TABLE 10. PERCENTAGE POINTS OF THE t-DISTRIBUTION
This table gives percentage points tv(P) defined by the
equation
P
too
rav +
- vw, r(#y)
tv(p)(1
dt
t 2I p)i(P +1) •
Let Xi and Xz be independent random variables having a
normal distribution with zero mean and unit variance and a
X'-distribution with v degrees of freedom respectively; then
t = XJVX2/P has Student's t-distribution with v degrees of
freedom, and the probability that t t„ (P)is P/Ioo. The
lower percentage points are given by symmetry as - t„ (P),
and the probability that Iti
t,,(P) is 2Phoo.
P (To)
30
40
25
20
V = I
0.3249
0.
7265
I•0000
1 '3764
2
3
4
0'2887
0'6172
0'2767
0'2707
0'5844
0'5686
0.8165
0. 7649
0.7407
-o607
0.9785
09410
5
6
7
8
9
The limiting distribution of t as v tends to infinity is the
normal distribution with zero mean and unit variance. When
v is large interpolation in v should be harmonic.
15
to
5%
2.5
31.82
6.965
4'541
3'747
r 963
I .386
3.078
I'250
1'190
1.638
1.533
6.314
2.920
2.353
2.132
2.015
1'943
1.895
1.860
1.833
2'571
2'262
3.365
3'143
2.998
2.896
2.821
1.812
2. 228
2.764
3.169
2'201
2'179
2'160
2'145
2'718
2'681
2'650
2'624
3'106
1 . 886
0.2672
0'5594
0/267
0. 2648
0- 5534
0'5491
0.2619
o- 26io
0'5459
0'5435
0.9195
0.9057
0.8960
0.8889
0.8834
1.156
1' 134
0'2632
0.7176
0.711I
0.7064
0.7027
1. 108
1•100
1'476
r 440
1'415
1 '397
1.383
0.2602
0.2596
0.5415
0'5399
0'6998
0'6974
0.8791
0'8755
r 093
1. 088
1.372
F363
0'2590
0. 2586
0'2582
0.5386
13
14
0.5375
0.6955
0. 6938
0'6924
0.8726
0'8702
0. 8681
I'083
I' 079
1'076
1'356
1'350
1'345
1'796
1'782
I'771
1'761
15
0.2579
0 '5357
0. 6912
0.69o1
0'6892
0.6884
0. 6876
0.8662
0.8647
0.8633
0.8620
o• 8610
r 074
.071
1069
1.067
r•o66
I • 341
1.337
1'333
1'330
1.328
1'753
1/46
1' 740
1'734
x729
2.131
0'2576
2. 120
2' I I0
2' IOI
2'093
0.6870
0.6864
o.8600
0.8591
0'6858
0. 8583
0'8575
I'064
1'063
1'06'
I .06o
I ' 325
1 . 323
1'321
1'319
I ' 725
I . 721
I'717
1'714
0.8569
r 059
1.318
1.711
1'708
P 706
to
II
12
0'5366
0.5
12.71
4.303
3.182
2.776
'I19
2'447
2.365
2.306
0•X
63.66 318.3
636.6
31.6o
9.925 22'33
12.92
5.841 10.21
8.610
7.173
4'604
4'032
3.707
3'499
3'355
3'250
5'893
5.208
4'785
4'501
4'297
2. 947
2.921
2.898
3'733
3.686
3.646
2'878
3'610
2.861
3.579
4'073
4.0 x 5
3.965
3.922
3.883
2.086
2.528
2.845
2. 080
2'074
2'518
2'508
2.o69
2.064
2.500
2.492
2'831
2'819
2'807
3.552
3.527
3- 505
3.485
3'467
3.850
3.819
3-792
3.768
3'745
2. 060
2. 056
2.052
2'048
2-045
2.485
2.479
3'450
3'435
3.421
3'408
3.396
3.725
3.707
3.690
3'674
3.659
2'042
2'037
2.032
2. 028
2. 024
2.457
2.449
2441
2434
2429
2.750
2. 738
2'728
2'719
3'385
3. 365
3'348
3'333
3.319
3.646
3'551
3'496
3. 460
3'373
3.291
0.2571
0.2569
20
21
22
23
0'2567
0'2566
0'2564
0'2563
0'5329
0'5325
0'5321
0.5317
24
0.2562
0.5314
0.6853
0.6848
25
26
27
28
0. 2561
0. 2560
0' 2559
0.2558
0 ' 2557
0.5312
0.5309
0.5306
0.5304
0.5302
0'6844
0'8562
1. 058
1'316
0.6840
0.6837
0'6834
0.6830
0.8557
0' 8551
0.8546
0.8542
r 058
r 057
i• 056
P315
1 '314
1.313
1.055
1.31
1.699
30
0- 2556
0.5300
0.6828
0.8538
r 055
1.310
32
0'2555
0'5297
0'6822
0'8530
0.2553
0.5294
0.68,8
0.8523
36
38
0'2552
0'2551
0'5291
0'5288
0'6814
0'6810
0'8517
0'8512
1.309
1.307
1'306
1'304
I '697
r 694
34
I'054
I'052
1'052
1'051
1.688
1.686
40
50
6o
0.2550
0- 2547
0.2545
0.5286
0.5278
0.5272
0.6807
0.6794
0.6786
2'403
2'390
2/04
2'678
2•660
0. 6765
1.684
1676
P671
1658
2423
0'5258
1. 303
r 299
1.296
1.2.89
2'009
2'000
0'2539
r 050
r 047
1. 045
1-041
2.021
X20
0.8507
0 ' 8489
0.8477
0. 8446
P980
2.358
2.617
3-307
3.261
3.232
3.160
op
0.2533
0.5244
o•6745
o•8416
i•o36
1•282
r645
1.960
2'326
2'576
3.090
45
1-69.
4.587
4'437
4.318
2. 602
2. 583
2. 567
2'552
2'539
0 ' 2573
29
5'041
4'781
3.055
3.012
2 '977
17
18
19
I.701
6.869
5'959
5'408
4'144
4'025
3'930
3.852
3.787
0- 5350
0 '5344
0.5338
0.5333
1'703
0'05
2'473
2'467
2.462
v 797
2.787
2/79
2. 771
2'763
2.756
2'71[2
4'221
4' 140
3.622
3.6ox
3.582
3.566
TABLE 11(a). 2.5 PER CENT POINTS OF BEHRENS' DISTRIBUTION
0
vz
=
o°
17.36
8'344
7.123
6.771
6.636
6.577
15.56
6.34o
4.960
4'469
4.218
4'074
I I•04
9'065
6'546
3'980
2'365
11.03
9.060
9.055
9.052
6.529
6.511
6-5o1
3'917
3.835
3.786
2.306
9•046
6'485
3'685
2'228
2'179
2'064
9.040
6.473
3.615
1.960
4'563
3'645
3.312
3'145
3'045
2 '979
2 '933
2 '873
2 '835
2•750
2.679
4'414
3.36o
2 '978
2.784
2.667
2.589
2 '534
2460
2 '414
4'303
3.182
2.776
2.571
2.447
3.191
2.816
2.626
2.513
2.437
17.36
11.54
3
4
5
6
7
8
12/ 1
12'71
12'71
I2•29
12'28
12•28
1 I•I x
12.71
12.28
12.28
2
12'71
12/1
12'71
12'71
12/1
12'71
= 2
4'303
4'303
4'303
4'303
4.303
4'303
4.303
I0
4.303
12
4.303
24
4.303
4'303
00
v, = 3
4
5
6
7
8
12'28
12.28
11'03
12'28
12•28
12'28
11.03
I I•03
I I•03
4.190
3'882
4'187
4.186
4.184
3'867
3'857
3'846
3'840
3.828
3.818
4'624
3'903
3'653
3'535
3'468
3'427
3'400
3.366
3'346
3.306
3.276
3.225
3.088
3.026
3.244
3.012
2.897
3.225
2.913
2.756
4'414
4'240
4'205
4'194
4.182
4.180
4.178
4'563
4'100
3'964
3'909
0
go°
12.71
4'303
3.182
2.776
2.571
2 '447
2'305
2•206
2'365
2'306
2'228
2'179
2'064
I•060
3.182
3.191
3'182
3'182
3.182
3.182
3.182
3.182
3.182
3.182
3.182
3'149
3.134
3.127
3.122
3'120
3.117
3.115
3.111
3.1o8
2'992
2'972
2'958
2'831
2'787
2'758
2'663
2'600
V556
2.942
2.933
2.913
2.898
2.719
v696
2.644
2.603
2.498
2.462
2.378
2304
2.312
2.267
2.162
2.067
2/76
2.776
2.776
2.772
2.754
2/46
2 '779
2.717
2.682
2.787
2.675
2.610
2.779
2.625
2532
2.772
2.582
2 '468
2/76
2.571
2 '447
2/76
2/76
2'741
2'738
24
2.776
2.776
2.776
2'365
2'306
2'228
CO
2'776
IO
12
24
00
v1 = 4
5
6
7
8
=4
I•o6
11.04
11•04
17.97
10.14
9.303
9.136
9.090
9.073
15.56
12.41
3
4
5
6
7
8
V2
75
1211
00
=3
6o°
12.71
24
V2
45°
2
IO
=
30°
P1 = I
12
V2
150
IO
12
2'384
2'660
2'567
2•471
2'392
2'734
2.732
2.727
2'646
2.628
2.617
2'594
2.428
2'371
2'335
2.252
2.339
2•268
2•723
2'576
2.537
2.498
2.475
2.421
2'377
If t1and t2 are two independent random variables distributed as t with v1, 1/2 degrees of freedom respectively, the
random variable d = t1sin 0 t2 cos 0 has Behrens' distribution with parameters VI, V2 and 0. The function tabulated
in Table II is dp = dP(vi, v2 0) such that
Pi
2'178
2'223
2'118
2•024
3.182
2.776
2.571
2.447
2'365
V306
2'228
2 '179
2 '064
1.960
2 '179
2 ' 064
1.96o
v2. When v1 < V2 use the result that
dp(vi , 1/2, 0) = dP(P2,
90° - 0)•
-
Behrens' distribution is symmetric about zero, so
Pr (1d1 > dP) = zP/Ioo.
,
Pr (d > dp) = Phoo
Notice that in this table 0 is measured in degrees rather than
radians.
for P = 2.5 and o•5 and a range of values of v1and v2 with
46
TABLE 11(a). 2.5 PER CENT POINTS OF BEHRENS' DISTRIBUTION
0
V2 =
5
=
7
v2 = 8
V2 = 10
V2=12
1/2 = 24
1/2 = CO
75°
90°
2.564
2'562
2'565
2'562
2'564
2'571
2 '554
2.470
2.410
2.367
2•310
2.274
2.449
2. 374
2.320
24
2.571
2.571
2.571
2.549
2 '546
2.541
2 '539
2.533
2'228
2'179
2'064
2'571
2'529
2.191
2'118
2'248
2'203
2'098
00
2.500
2.458
2.428
2.390
2.366
y312
2.266
2.447
2-571
2.527
2.505
2.490
2 '471
2.460
2.436
2.416
2•004
P960
2.435
2'413
2.398
2.379
2.367
2.436
2-394
2.364
2.325
2.301
2.435
2.375
2.331
2.274
2.239
2 '440
2'364
2'310
2'238
2'193
2 '447
2'365
2'306
2'228
2.179
2'342
2'322
2.247
2•201
2.156
2•088
P993
2.064
1.960
2'352
2'322
2'283
2'352
2'309
2'252
2'358
2.304
2'232
2'187
2'082
2'365
V306
2.228
2'179
2'064
P987
1.960
2. 300
2'228
2.183
2.306
2'228
2'179
2.077
1.982
2.064
P960
2'223
2'178
2'072
2179
1.977
1.960
2.571
2'365
2.306
v1 = 6
2 '447
2.440
7
8
2 '447
2 '434
2.447
xo
2 '447
12
24
2.447
2.447
00
2 '447
2.431
2.426
2.423
2.418
2 '413
7
8
2.365
2.358
2'352
2.365
2. 354
2.337
ICI
12
24
CO
2'365
2'365
2'365
2'365
2•350
2'317
2. 347
2.306
2.259
2.216
2'341
2'336
2'280
2'259
2'205
2•158
2•133
2'060
1,1 = 8
I0
12
24
2'306
2'306
2•306
2'306
2.300
2.295
2'292
2'286
2'294
2'274
2'262
2'236
2'292
2'254
2'230
2.175
2'294
2'237
2'201
2'118
00
2.306
2.281
2.215
2.128
2.044
V1 = 10
12
24
CO
2'228
2'228
2'228
V228
2'223
2.220
2'214
V209
2'217
2'205
2•178
2'157
2'215
2'191
2'136
2'089
2'217
2'181
V098
2'024
= 12
2'179
2'175
2'169
2'167
2'179
y179
2.168
2.142
2.112
2.179
2.163
2'120
2'064
2'169
2'085
2.01 1
2'175
24
CO
2.069
1.973
2.064
1.960
V1 = 24
00
2'064
2'064
2'062
2•056
2•058
2'035
2'056
V058
2'062
2'064
y009
1.983
P966
1•96o
vl = 00
1.960
1•96o
1.960
1'960
1.96o
1.96o
i.96o
V1
=
2'064
Pi
Pi
-
j.
V, v2
2'228
-
0 = tan-1 ( s -- - i
s 2 )'
0 being
07
4
measured in degrees. Define r = I - + s2 and d =
2.082
If d > dp the confidence level associated with
it
2 is less
than P per cent, and if d <
dp the confidence level associated with
ft
2 is less than P per cent. (See H. Cramer,
Mathematical Methods of Statistics, Princeton University
Press (1946), Princeton, N.J., pp. 520-523.) Also, the values
of Pi -#2 such that 1(Yi - - (#1-P2)1 < rdp provide a
Ioo 2P per cent Bayesian credibility interval for
#2.
This distribution arises in investigating the difference
between the means #1, P2 of two normal distributions without
assuming, as does the t-statistic, that the variances are equal.
Let o i, p72 be the means and 4., 4 the variances of two independent samples of sizes n1, n2 from normal distributions,
let v1= n1-1, v2 = n2 - 1 and
6o°
45°
2.571
5
6
7
8
12
v2
30°
2 '571
v1 =
xo
v2 = 6
15.
o°
I. -072
.
r
47
pi
-
TABLE 11(b). 0-5 PER CENT POINTS OF BEHRENS' DISTRIBUTION
v2 = I
vl = I
2
3
4
5
6
7
8
io
12
24
c0
V2 = 2
V1 = 2
3
4
5
6
7
8
10
12
24
V2
=3
P2 = 4
o°
x5
3o°
450
6o°
75.
go°
63.66
63.66
63.66
63'66
63.66
63.66
63.66
63.66
63.66
63.66
63.66
63.66
77.96
61.61
61 '49
61'49
61 '49
61'49
61 '49
61 '49
61 '49
61 '49
61'49
61'49
86.96
55.62
55'15
55'14
55'14
55'14
55'13
55'13
55'13
55'13
55'13
55'13
90.02
46.18
45' 08
45'04
45.03
45.03
45.03
45.03
45'03
45.03
45.02
45.02
86.96
34'18
32.04
31.89
31.87
31.86
31.86
31.86
31.86
31.86
31.85
31.85
7796
21'11
17'28
16'70
16'59
16'57
16.56
16.55
16.55
16.54
16'54
16.53
63.66
9.925
5 .841
4'604
4. 032
3.707
3'499
3'355
3.169
3.055
2.797
2 '576
10'01
10'14
8'905
8.717
8.676
8.663
8.657
8.653
8.649
8. 647
8. 642
8.638
10'19
7'937
7'428
7'270
7'210
7'183
7'169
7155
7.148
7'134
7'124
10'14
6.966
6.082
5.716
5'535
5'434
5'373
5.308
5'275
5.223
5'194
10'0!
9'640
9'609
9'604
9'602
9.601
9'600
9.600
9'599
9'598
9'597
6.187
5 .049
4'5 28
4'235
4'049
3'921
3'759
3.660
3'446
3.276
9.925
5'841
4'604
4'032
3.707
3'499
3'355
3.169
3-055
2.797
2.576
5'841
5'841
5'754
5'694
5'640
5'349
5.841
5.681
5.256
5.841
5'841
5.676
5'673
5.218
5'199
5'640
4'720
4'316
4'095
3'958
3.866
3'753
3.686
3'548
3'449
5'754
4'601
4'076
3.782
3'595,
3'467
3'302
3'201
2 '977
2.789
5'841
4.604
4'032
3.707
3'499
3'355
3.169
3.055
2.797
2.576
4'400
3'983
3'755
3.613
3'517
3'395
3'323
3.167
3'045
4'525
4'604
4'032
3.707
3'499
3'355
3.169
3.055
2.797
2.576
9.925
9'925
9'925
9'925
9.925
9.925
9.925
9.925
9'925
9'925
9'925
.
Pi = 3
4
5
6
7
8
xo
5.841
5.671
5.189
5.841
1205
g.
5 'E1
1
5'669
5.668
5.666
5 .664
5'177
5.171
5.159
5.150
5'598
4'986
4'739
4'617
4'548
4'506
4'459
4'434
4'389
4'361
v1 = 4
4'604
4'604
4'604
4'604
4'525
4:505
4'497
4'493
4'490
4.487
4'486
4'482
4'479
4'400
4.283
4.229
4'201
4'184
4.165
4'155
4'135
4'i2i
4'350
4'084
3'945
3.862
3.809
3'745
3.709
3.
64°
3'592
5
6
7
8
I0
12
24
4. 604
4.604
4.604
4'604
4'604
If t1and t2 are two independent random variables distributed as t with v1, v2 degrees of freedom respectively, the
random variable d = t1sin 0- t2 cos 0 has Behrens' distribution with parameters V1, V2 and 0. The function tabulated
in Table it is dp = dP(vi, v2, 0) such that
Pr (d
Pi
3'993
3-694
3'504
3'373
3.206
3'104
2.876
2•685
P2. When v, < V2 use the result that
0).
dp(Vi, v2, 0) = dp(v2, V1, 90° Behrens' distribution is symmetric about zero, so
Pr (1d1
> dp) = P/ioo
> dp) = 213 1 ioo.
Notice that in this table 0 is measured in degrees rather than
radians.
for P = 2.5 and 0.5 and a range of values of vi and v2 with
48
TABLE 11(b). 0.5 PER CENT POINTS OF BEHRENS' DISTRIBUTION
0
V2 = 5
vl
= 5
6
7
8
10
12
24
00
=6
7
8
10
12
v2 = 6
24
00
V2 =7
v, = 7
8
xo
12
24
c0
v2 = 8
v1 = 8
10
12
24
00
o°
xs°
30°
450
6o°
750
900
4.032
4.032
4'032
4.032
4.032
4.032
4.032
4'032
3'968
3'957
3'952
3'949
3'945
3'943
3.938
3'934
3.856
3'794
3'760
3'739
3'715
3 '702
3'677
3'658
3.809
3.663
3'575
3'518
3'447
3'407
3'325
3'266
3.856
3'622
3'476
3'378
3'253
3. I 78
3'016
2.886
3.968
3.666
3'474
3'342
3.173
3.069
2.840
2.646
4.032
3.707
3'499
3'355
3.169
3.055
2'797
2.576
3'707
3'707
3'707
3.707
3.707
3.707
3'707
3'654
3'648
3'644
3'639
3'637
3'631
3'627
3'556
3'519
3'496
3'468
3'453
3.424
3.402
3'514
3'423
3'363
3.289
3'246
3'158
3.093
3'556
3'408
3'308
3.180
3'104
2.938
2'804
3'654
3-461
3'328
3.158
3'053
2.822
2.627
3'707
3'499
3'355
3.169
3'055
2.797
v576
3'499
3'499
3'499
3'499
3'499
3'499
3'454
3'450
3'445
3'442
3'436
3'431
3'369
3'344
3'314
3'298
3.265
3'241
3'331
3'269
3'193
3'149
3.056
2'987
3'369
3.267
3'138
3'060
2.892
2'755
3'454
3.321
3.149
3'045
2.812
2•616
3'499
3'355
3.169
3'055
2.797
2.576
3'355
3.355
3.355
3'355
3.355
3'316
3.310
3.307
3'301
3.295
3'241
3.210
3.192
3.158
3.132
3.206
3.129
3.083
2.988
2.916
3.241
3.110
3-032
2.862
2.723
3.316
4
13
49
3 :0
3'355
3'169
3'138
3.169
3.169
3.169
3.135
3'127
3.121
3.078
3.059
3'021
2.993
3.049
3'002
00
3.055
3.055
3.055
3'029
3.021
3.015
2'978
2.939
2.909
V2 = 24 VI = 24
00
2'797
2.797
2'784
2.777
P2 = CO V1 = 00
2 '576
2.576
V2 = 10
V1 = 10
12
24
00
V2= 12
vl = 12
24
3.078
3.138
3'033
2'904
2'998
2'825
2.828
2.684
2'798
2.600
3.169
3-055
2'797
2.576
2'954
2'853
2.775
2'978
2.803
2.661
3'029
2.794
2.595
3'055
2'797
2.576
2'759
2726
2 '747
2.664
2'759
2'784
2'797
2'613
2'584
2'576
2.576
2.576
2'576
2'576
2'576
If d > dp the confidence level associated with illt-C. /12 is less
than P per cent, and if d <
dp the confidence level associated with 121 11
2 is less than P per cent. (See H. Cramer,
Mathematical Methods of Statistics, Princeton University
Press (2946), Princeton, N.J., pp. 520-523.) Also, the values
of -#2 such that I(x1- x2) - (01-#2)1 5 rdp provide a
200-2P per cent Bayesian credibility interval for it, -#2.
This distribution arises in investigating the difference
between the means ltil #2 of two normal distributions without
assuming, as does the t-statistic, that the variances are equal.
Let oil, x2 be the means and 4, 4 the variances of two independent samples of sizes n1, n2 from normal distributions,
/ S2
•
-
,
let = - V2 = 712 -1 and 0 =
(071 N/V2)
S2 S
2
measured in degrees. Define r = ji- +1and d
vi V2
2.806
2.608
3'169
3.055
2.797
2.576
U being
-X2
r
49
TABLE 12(a). 10 PER CENT POINTS OF THE F-DISTRIBUTION
The function tabulated is F(P) = F(Plv,, v2) defined by the
equation
P
zoo
rco
ITO
“JTI)
Fip.-
r(i-v2) vl v2b2 F(p)(v2 +P,FP( P'+"')c1F'
for P = io, 5, 2.5, I, 0.5 and 0•1. The lower percentage
points, that is the values F'(P) = F'(PIP1, P2) such that the
probability that F < F'(P) is equal to Pim°, may be found
by the formula
F'(P IPi,
VI =
I
P2) = I/F(Plv2,
2
3
(This shape applies only when vl > 3. When v1< 3 the mode is
at the origin.)
4
8
xo
12
24
CO
58.91
9'349
5.266
3'979
59.44
9'367
5.252
3'955
60.19
9.392
5.230
3'920
60.71
9.408
5.216
3.896
62.00
9.450
5.176
3.831
63'33
9.491
5'134
3.761
5
6
7
39.86
8.526
5'538
4'545
49.50
9.000
5'462
4'325
53'59
9.162
5'391
4.191
55'83
9.243
5'343
4.107
57.24
9'293
5'309
4.051
58.20
9'326
5'285
4.010
4'060
3.776
3'589
3.458
3.78o
3.463
3'257
3.113
3.619
3.289
3.074
3.520
3.181
2.961
3'453
3.108
2.883
2'924
2'806
2.726
3'006
2.813
•693
2.611
3'368
3'014
2.785
2.624
2.505
3'339
2'983
2.752
2.589
2.469
3'297
2.937
2.703
2.538
2.416
3.268
2.905
2.668
2.5o2
2'379
3.191
2.818
2.575
2.404
2'277
3.105
2.722
2.471
2'293
3'360
3'405
3.055
2.827
2•668
2.551
3.285
3.225
3.177
3.136
3.102
2.924
2728
2.66o
2•414
2'377
2.304
2.323
2.248
2.284
2'245
2560
2.522
2.195
2'188
2'138
2'209
2'147
2'097
2'178
2'100
2'036
1.983
2 '055
2'451
2'394
v347
2.307
2'461
2'389
2.763
2.726
2.605
2•536
2.480
2.434
2.395
2 '154
2.095
2.0 54
P938
P904
1.846
P797
3'073
3.048
3'026
3.007
2'490
V361
2'273
2'208
2'158
2'119
2'059
2'017
2.462
2'437
2.416
2.397
2.333
2.308
2.286
2.244
2.218
2.178
2.152
2.128
2.102
2'990
2 '695
2.668
2.645
2.624
2.606
2'266
2'196
2'176
2'130
2'109
2.058
2.088
2.061
2.038
2.017
2.028
voox
1.977
P956
P985
P958
P933
I•912
P899
P866
P836
P810
1.787
P755
P718
P686
P657
P631
2.589
2.575
2.561
2'549
2.380
v365
2'249
2'158
2'091
2'040
2.233
2'142
2'075
2'351
2'219
2'128
2'060
23
2.975
2.961
2.949
2.937
2'339
2.207
2.115
2.047
24
2'927
2'538
2'327
V195
2'103
2'035
2.023
2.008
P995
P983
P999
P982
1.967
P953
1.941
P937
P920
P904
1.890
1.877
1.892
P875
P859
1.845
P832
P767
1.748
1.731
1.716
P702
P607
P586
P567
P549
1'533
25
26
27
28
29
2.918
2.528
2'317
2'184
2'092
2'024
2'909
2'901
2'519
2'511
2.307
2.174
vo82.
2.014
2'073
2'005
P952
2.503
2.495
2'299
2'291
2'165
v894
2.887
V I 57
2.283
2'149
2'064
2.057
P996
P988
P943
P935
1.929
P919
P909
p900
x.892
P866
P855
1.845
P836
P827
1•820
p809
P799
P790
1.781
P689
P677
1•666
1.656
P647
1.518
P504
1.491
P478
P467
30
2'881
P980
P967
P819
P773
1.638
P913
P870
1'805
P758
P622
34
36
38
2'859
2.850
2'466
2'252
V049
2'036
2'024
1.884
2.263
2.456
2.243
2'142
2'129
2'118
2'108
P927
v869
2.489
2.477
2'276
32
v014
P6o8
P595
2'234
2'099
2'005
1.858
P847
1.838
P745
P734
2'448
P901
P891
1•881
P793
1.781
2'842
P955
P945
P935
P772
P724
P584
1.456
P437
P419
1404
1.390
40
60
2.835
2/91
2'440
V393
2'226
2'347
P722
1'652
p574
1.511
1.447
P377
P291
1 '193
V303
1.873
P819
P767
1.717
P715
P657
I .601
2/06
P927
P875
1.824
P774
P763
1.707
2.748
P997
P946
1.896
1.847
1.829
p775
120
00
2.091
2.041
1.992
1.945
p670
P599
P546
P383
P000
P2 = I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
2'86o
2'807
2'606
v177
2' I 3o
2.084
2'522
2.331
2.283
2.243
50
2'342
2'283
2.234
2 ' 193
2'079
P971
P961
2 '159
P972
TABLE 12(b). 5 PER CENT POINTS OF THE F-DISTRIBUTION
--1/-?X
If F = X
112
, where X1and X2 are independent random
variables distributed as X2 with v1and v2 degrees of freedom
respectively, then the probabilities that F F(P) and that
F F'(P) are both equal to Pilot). Linear interpolation in
v1and v2 will generally be sufficiently accurate except when
either v1> 1z or v2 > 40, when harmonic interpolation
should be used.
0
F(P)
(This shape applies only when v1>. 3. When v, < 3 the mode is
at the origin.)
I
2
3
4
5
6
7
8
xo
12
24
CO
V2 = I
161 4
199.5
215.7
224.6
230.2
234•0
2
18'51
10'13
19'00
19.16
19.25
19-30
19.33
9.277
6.591
9.117
6.388
9.013
6•256
8'941
24P9
19.40
8.786
5.964
249'1
19.45
8.639
19.50
8.526
6.163
238.9
19'37
8.845
6.041
243'9
19'41
9'552
6.944
236.8
19'35
8.887
6.094
5'774
5.628
4'365
3.669
3.230
2.928
2.707
Ili =
3
4
7'709
8.745
5'912
254'3
5
6.6o8
5.786
5'409
5'192
5.050
4'950
4.876
4.818
4'735
4'678
4'527
6
7
8
9
5.987
5'591
5.318
5'143
4'757
4'534
4'387
4'284
4.207
4'147
4.06o
4.000
3.841
5.117
4'737
4'459
4'25 6
4'347
4.066
3.863
4'120
3.838
3.633
3'972
3.687
3'482
3.866
3.581
3'374
3.787
3.500
3.293
3.726
3'438
3.230
3.637
3'347
3.137
3'575
3.284
3.073
3.410
3.115
2.900
4.965
4.844
4'747
4.667
4.600
4.103
3.982
3.885
3.806
3'739
3.708
3.587
3'490
3.411
3'344
3'478
3'357
3.259
3.179
3.112
3.326
3.204
3.106
3.025
2.958
3.217
3.095
2.996
2.915
2.848
3.135
3.012
2.913
2.832
2.764
3.072
2.948
2.849
2.767
2.699
2.978
2.854
2.753
2.671
2.602
2.913
2.788
2.687
2.604
2.534
2 '737
2.609
2.505
2.42o
2.349
2.538
4'543
4'494
4'451
4'414
4.381
3.682
3.634
3'592
3'555
3.522
3.287
3'239
3.197
3.160
3.127
3.056
3.007
2.965
2.928
2.895
2.901
2.852
2.810
2.773
2.740
2.790
2'741
2.699
2.661
2.628
2.707
2.657
2.614
2.577
2.544
2.641
2.591
2.548
2.510
2'477
2.544
2.475
2.425
2.381
2.288
2.235
2.066
4.351
4'325
4'301
4'279
3'493
3'467
3'443
3.422
2'711
2.685
2.661
2.64o
2.621
2.514
2.488
2.420
3'403
2.866
2.84o
2817
2.796
2/76
2.599
2.573
2.549
4.26o
3.098
3.072
3.049
3.028
3.009
4'242
4.225
2.991
2.975
2.96o
2.947
2.934
2/59
2 '743
2'728
2 '572
2 '459
2.714
4.183
3.385
3.369
3'354
3'340
3.328
2.558
2.545
30
4.171
3.316
2'922
32
34
4'149
3.295
2.901
2.690
2.668
4'130
4.113
4.098
3'276
2'883
2'650
2.534
2.512
2'494
2'399
2.380
3'259
3.245
v866
2.852
2.634
2.619
2463
2.364
2'349
3.232
3.I50
3.072
2.996
2.839
2/58
2.68o
2.605
2.606
2.525
2'449
2'336
2.249
2.18o
2368
2'447
2'290
2. 254
2'175
2'372
2.214
2.099
2'167
2'087
2'010
2'097
2'016
P938
I0
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
36
38
4'210
4.196
40
4•o85
6o
4.001
3.920
3.841
120
00
2/01
2 '494
2.45o
2.412
2-378
2 '404
2.296
2.2o6
2'131
2'010
1.960
2 '342
2'190
2.150
2.308
2.114
2.278
2.250
2.226
2.204
1'843
P917
p878
2'397
2'528
2'464
2 '442
2. 375
2'348
2.321
2'297
2.275
2.508
2.423
2.355
2'255
2'183
2•082
v054
2.028
2.005
P984
2'603
2'490
2'474
2.405
2.388
2.373
2.337
2.321
2.305
2.236
2.587
2•220
2'165
2'148
P964
1•946
1•71 I
1'691
2.204
2.132
1.930
1.672
2'445
2.432
2 '359
2.291
2.346
2.278
2'190
2'177
2'118
2.104
1.915
P901
1.654
1'638
2.421
2.334
2313
2'165
2'142
2'123
2'106
2'091
2'092
2.070
2'050
2.033
2'017
I .887
1.864
P843
P824
1'808
2.077
I '993
2'003
P910
1'831'
P834
P752
P793
x.700
I .608
P509
P389
1.254
1.517
1•000
2 '477
51
2.447
2.266
2. 294
2'244
2'225
2.277
2.209
2'262
2'194
I•917
1 .812
P783
P757
P733
1'622
1.594
P569
1'547
I .527
TABLE 12(c). 2.5 PER CENT POINTS OF THE F-DISTRIBUTION
The function tabulated is F(P) =
the equation
P
r(iPi+ i-v2)
100
F( v1)r(iv2)
V2iv2
v2) defined by
F(p)(P2+
viflttpx+vo dF,
for P = 1o, 5, 2.5, I, 0.5 and o•I. The lower percentage
points, that is the values F'(P) = FTP1v1, v,) such that the
probability that F t F'(P) is equal to P/Ioo, may be found
by the formula
0
F'(Plvi, v2) = IF(PIP2, PO.
v1=
F(P)
(This shape applies only when v1> 3. When v1 < 3 the mode is
at the origin.)
5
6
7
8
ro
12
24
00
921'8
937'1
39'33
14'73
9.197
948.2
39'36
14.62
9'074
956.7
39'37
14'54
8.98o
968.6
39'40
14'42
8.844
976.7
39'41
14'34
8.751
997'2
39'46
14.12
8.511
1018
39'50
13'90
8'257
6.978
5.820
6.757
5.6o0
4'899
4'433
6.619
6'525
5.461
5'366
4'102
4'761
4'295
3'964
4'666
4' 200
3.868
6.278
5.117
4'415
3'947
3'614
6.015
4'849
4'142
3.670
3'333
3.365
3.173
3.019
2.893
2.789
3.080
2.883
2'701
2'625
2'560
2'503
2'452
v395
2.316
2.247
2.187
2-133
2'602
2'570
2'541
2'408
2'368
2'331
2'299
2'269
2'085
2'042
2'003
2.515
2'491
2'469
2'448
2'430
2 '242
2.217
2.195
2'174
2'154
I'906
1.878
2.412
2.38,
2.353
2.329
2.307
2'136
2'103
2'075
2'049
2'027
1.787
1.750
1.717
1.687
1.661
2.288
2'007
1'882
1'760
I'640
1'637
1'482
1'310
1'000
2
3
4
864.2
39.17
15.44
9'979
899.6
39'25
15'10
9'605
39'30
14' 88
9'364
V2 = i
2
647'8
38'51
3
4
12'22
799'5
39.0o
16.04
10•65
5
6
7
8
9
8.813
8.073
7'571
7.209
8'434
7.260
6'542
6.059
5.715
7'764
6.599
5.890
5'416
5.078
7.388
6.227
5.523
5.053
4'718
7.146
5.988
5-285
4'817
4'484
4'652
4'320
6.853
5.695
4'995
4'529
4'197
5'456
5.256
5.096
4'96 5
4'857
4'826
4'630
4'474
4'347
4'242
4%168
4'275
4'121
3'996
3.892
4'236
4'044
3.891
3-767
3.663
4'072
3.881
3.728
3 . 604
3.501
3'950
3'759
3.607
3'483
3.38o
3'855
3.664
3'512
3.388
3.285
3'717
3.526
3'374
3.250
3'147
3'621
13
14
6.937
6.724
6'554
6'414
6.298
Is
6'200
6.115
6'042
5.978
5.922
3.804
3'729
3.665
3.608
3'559
3'576
3.5oz
3'438
3.382
3'333
3'415
36 341
3.277
x8
4'153
4'077
4' 0 II
3'954
3.903
3.293
3.219
3.156
3.100
3.051
3'199
3.125
3.061
3.005
2.956
3.060
2.986
17
4'765
4'687
4'619
4'5 6o
4'508
2'963
z6
2'817
2'769
2'720
4%161
4'420
4'383
4'349
4'319
3'859
3.819
3.783
3'750
3.721
3.515
3'475
3'440
3'408
3'379
3.289
3.128
3.090
3.055
3.023
2.995
3.007
2 '969
2'934
2'913
2 '774
2 '735
2-637
2'902
2'874
2.874
2.839
2.808
2.779
3.129
3.105
3.083
3:063
3'044
2.753
2.729
2.613
2'923
2'903
2'884
2'802
2'782
2/63
2'707
2'687
2.568
2 '547
4'201
3'353
3.329
3'307
3.286
3.267
2'848
2.824
5.588
3'694
3.670
3'647
3.62,6
3.607
2.969
2 '945
5'610
4'291
4.265
4'242
4.221
2.669
2'529
4'182
4'149
4'1zo
4'094
4'071
3.589
3'557
3'529
3.505
3'483
3'250
3.026
2'995
2.968
2'944
2.867
2.836
2.808
2.785
2.763
2/46
2.715
2.651
2.620
2.593
2.569
v548
2'511
2'480
34
36
38
5.568
5.531
5'499
5'471
5.446
2.453
40
6o
5'424
5.286
3'463
3'343
120
co
5'152
5'024
4'05I
3'925
3.805
3.689
2'529
2'412
2'299
2'192
v388
2.27o
2 'I57
2.048
zo
zz
12
zo
20
21
22
23
24
25
26
27
28
29
30
32
17'44
5.871
5'827
5.786
5-750
5'717
5.686
5.659
5'633
3'227
3'116
3'218
3'191
3.167
3'145
3.126
3.008
2. 894
v786
3'250
3'215
3.183
3'155
2'923
2. 904
2.786
v674
2.567
5'119
3'221
3'172
2.688
2.664
v643
2 '744
2.624
2.627
2.515
2.408
2.507
2.395
2.288
52
2'922
2.866
2'700
2.668
2•64o
2'590
2%429
2 407
3'430
3.277
3'153
3.050
2.889
2.825
2'676
2'169
2'055
1.945
2'725
2.595
2.487
r968
1'935
1.853
1'829
1'807
TABLE 12(d). 1 PER CENT POINTS OF THE F-DISTRIBUTION
--2, where X1and X2 are independent random
Xpi11X
F=V2
If
variables distributed as X2 with v1and v2 degrees of freedom
respectively, then the probabilities that F F(P) and that
F < F'(P) are both equal to Nioo. Linear interpolation in
vi or v2 will generally be sufficiently accurate except when
either v1 > 12 or v2 > 4o, when harmonic interpolation
should be used.
0
F(P)
(This shape applies only when v1
at the origin.)
vl =
I
v2 = I
2
3
4
5
6
7
8
9
xo
II
12
2
3
4
5
6
3. When v1< 3 the mode is
7
8
to
12
24
00
5928
99'36
27.67
14.98
5981
99'37
27.49
14'80
6056
99'40
27.23
14'55
6,o6
99'42
27.05
4'37
6235
99'46
26.6o
13'93
6366
99'50
26.13
13'46
9'020
6.88o
4052
98'50
34- 12
21.20
4999
99'00
30.82
18.00
5403
99'17
29.46
16.69
5625
99'25
28.71
15.98
5764
99'30
28.24
15'52
16.26
13.75
13.27
10.92
12.06
9.780
I1•39
9.148
10.97
10.67
10.46
10•29
8.746
8.466
8.26o
8.102
10.05
7'874
9.888
7718
9.466
7313
12.25
11.26
9'547
8'649
8'451
7.591
7'847
7.006
7'460
6.632
7.191
6.371
6.993
6.178
6.840
6.029
6.620
5'814
6.469
5'667
6.074
5'279
10.56
8'022
6'992
6'422
6'057
5'802
5.613
5 .467
5'257
5'111
4.729
4'859
4.31 1
10'04
9.646
6'552
6.217
5'953
5'739
5 .564
5'994
5 .668
5'412
5'205
5'035
5'636
5.386
5.200
5.316
5.064
4'862
4'695
5.069
4.821
4'620
4'456
4.886
4'640
4'441
4'278
5.057
4'744
4'499
4'302
4'140
4'849
4'539
4'296
4'100
3'939
4'706
4'397
4'155
3 .960
3'800
4'327
4'021
3'780
3'587
3.909
3.602
3.361
3 .165
3'427
3'004
3'434
3'666
3'553
3'455
3.371
3'297
3'294
3'181
3'084
2.999
2'925
2.868
2.753
2 '653
2.566
2.489
5859
99'33
27.91
15.21
5.650
13
9'074
7'559
7.206
6'927
6'701
14
8.862
6.515
15
16
17
18
19
8.683
6'359
5'417
4'893
4'556
4'318
4'142
4'004
3805
8.531
8.400
8.285
8.185
6.226
4'773
4'669
4'579
4.500
4'437
4'336
4'248
4'171
4'202
4'102
3'939
4.026
3'927
3-841
3'765
3.890
3'791
3.705
3'631
3'691
3'593
6.013
5.926
5'292
5'185
5'092
5010
20
8.096
8.017
7'945
7.881
7.823
5'849
5'780
5'719
5.664
5.614
4'938
4'874
4'817
4'765
4.718
4'431
4'369
4'313
4'264
4.218
4.103
4'042
3'988
3'939
3.895
3.871
3.812
3'758
3'710
3'667
3'699
3.640
3'587
3'539
3'496
3.564
3.506
3'453
3'406
3'363
3.368
3'310
3'258
3.211
3.231
3'173
3.121
3.074
2.859
2.8o1
2.749
2.702
2.421
2.360
2.305
2.256
3'168
3.032
2.659
2.211
7'770
7'721
7.677
7.636
7'598
5'568
5'526
5.488
5'453
5'420
4'675
4'637
4'601
4'568
4.538
4'177
4'140
4'106
4'074
4'045
3'855
3.818
3'785
3'754
3'725
3'627
3'591
3'558
3'528
3'499
3'457
3'421
3.388
3'358
3'330
3'324
3.288
3.256
3.226
3.198
3'129
3.094
3062
3.032
3.005
2'993
2.958
2.926
2.896
2-868
2.62o
2.585
2.552
2'522
2.495
2.169
2.131
2 '097
2 ' 064
2 '034
5.390
5'336
5'289
5'248
5'211
4'510
4.018
3'699
4'459
4'416
4'377
4'343
3'969
3'927
3'890
3'858
3'652
3'611
3'574
3'542
3'473
3'427
3 .386
3'351
3'319
3'304
3.258
3.218
3.183
3.152
3'173
3.127
3.087
3'052
3.021
2'979
2.934
2.894
2.859
2.828
2'843
2.798
2'758
2.723
2.692
2.469
2.423
2.383
2'347
2.316
2.006
34
36
38
7.562
7'499
7'444
7.396
7'353
1.911
1.872
1'837
40
6o
7'314
7077
5'179
4'977
4'787
4.605
4'313
4'126
3'949
3.782
3'828
3'649
3'480
3.319
3'514
3'339
3'174
3.017
3.291
3'119
2.956
2.802
3.124
2'953
2.792
2.639
2'993
2.823
2.663
2.511
2.801
2.632
2.472
2.321
2.665
2.496
2.336
2.185
2.288
2.115
1•950
1.791
1.8o5
1.601
1.381
1.000
21
22
23
24
25
26
27
28
29
30
32
9.330
120
6'851
00
6.635
4.015
53
3.508
1'956
TABLE 12(e). 0.5 PER CENT POINTS OF THE F-DISTRIBUTION
The function tabulated is F(P) = F(Plv,, v2) defined by the
equation
P
ravi+ -PO v
Ioo r(iv,) ra v2) 1
v iv2
_ co
z
fF(p) (12+ viF)1(P.+92) dF,
for P = 1o, 5, 2.5, I, 0•5 and o•i. The lower percentage
points, that is the values F'(P) = F'(Plvi, v2) such that the
probability that F < F'(P) is equal to P/Ioo, may be found
by the formula
F'(Plvi, v2) = i/F(Plv2, v1)•
Vi. =
I
2
3
4
1/2 = I
20000
199'0
21615
22500
2
16211
198.5
199'2
199'2
3
4
55'55
31•33
49'80
26.28
47'47
24.26
46'19
23.15
5
6
7
8
9
22.78
16.24
14.69
13.61
18.31
14.54
12.40
11'04
IO II
16.53
12'92
10•88
9'596
8.7i7
12.83
12.23
I r 75
1 P 37
ii•o6
9.427
8.912
8.510
8.186
7.922
15
i6
17
18
19
10•80
1o•58
10•38
20
(This shape applies only when vl
at the origin.)
7
8
zo
12
24
(X)
23715
199'4
44'43
2 I .62
24224
43'69
20'97
24426
199'4
43'39
20.70
24940
199'5
42'62
20.03
25464
22'46
23437
199'3
44'84
2 1 •97
23925
199'4
44'13
2r35
15'56
12.03
1o•o5
8.805
7956
14'94
11.46
9'522
8.3oz
7'471
14'51
11.07
9'155
7952
7134
14'20
10.79
8.885
7694
6.885
13.96
/0'57
8.678
7496
6.693
13.62
10.25
8.38o
7.211
6.417
13.38
10'03
8.176
7.015
6.227
12.78
9'474
7-645
6'503
5/29
12 '14
8'879
8.081
7.600
7226
6.926
6.68o
7'343
6.88
6.521
6.233
5'998
6'872
6.422
6.071
5'791
5'562
6'545
6.102
5'757
5'482
5'257
6.3o2
5.865
5'525
5'253
5'031
6.116
5.682
5'345
5'076
4'857
5.847
5.418
5'085
4'820
4'603
5.661
5.236
4'906
4'643
4'428
5.173
4-756
4'431
4'173
3.961
3'904
3'647
3.436
7701
7.514
7354
6 '476
6.303
6'156
5'803
5.638
5'497
10'22
7215
6'028
5'375
10.07
7'093
5'916
5.268
5'372
5.212
5'075
4'956
4'853
5'071
4'913
4'779
4'663
4'561
4'847
4'692
4'559
4'445
4'345
4'674
4'521
4'389
4'276
4'177
4'424
4'272
4'142
4.030
3'933
4'250
4'099
3'971
3.860
3'763
3/86
3.638
3.5
3'402
3'306
3.260
3•112
2 '984
2'873
2.776
6.986
6.891
6.806
6/30
6.661
5.818
5.730
5.652
5'582
5.519
5'174
5.091
5.017
4'950
4.890
4'762
4.681
4.609
4'544
4'486
4'472
4'393
4.322
4'259
4'202
4.257
4'179
4'109
4'047
3.991
4.090
4'013
3'944
3'882
3.826
3.847
3'771
3'703
3'642
3.587
3.678
3.602
3'535
3'475
3.420
3.222
3.147
3'081
3'021
2.967
2'690
23
24
9.944
9.830
9.727
9.635
9'551
2.428
25
9.475
26
27
9.406
5.462
5'409
5.361
5.317
4'835
4/85
4.740
4.698
4-659
4'433
4'384
4.340
4.300
4.262
4'150
4'103
4.059
4.020
3'983
3'939
3'893
3.850
3.811
3'775
3'776
3'730
3.687
3.649
3'613
3'537
3'492
3'450
3'412
3'377
3'370
3'325
3.284
3'246
3.211
2.918
2'873
v832
2.377
2'330
2.287
2 '794
2.247
2.759
2.210
4'228
4'166
3'949
3'889
3'742
3'682
4'112
3'836
3'630
4'065
4'023
3'790
3'749
3'585
3'545
3'580
3.521
3.470
3.425
3'385
3'344
3.286
3.235
3.191
3.152
3'179
3.121
3.071
3.027
2.988
2.727
2.670
2.620
2.576
2.537
2.176
2'114
2.060
2•013
1.970
3'986
3/60
3'548
3'350
3'713
3.492
3'285
3'091
3'509
3.291
3'087
2.897
3'350
3.134
2 '933
2.744
3.117
2 '904
2 '705
2.519
2'953
2'742
2 '544
2.502
1.932
r689
1.431
I•000
io
II
12
13
1
4
2/
22
18.63
z8
9'342
9.284
6'598
6'541
6'489
6•44o
29
9'230
6'396
5'276
30
9.180
6 '355
5'239
32
9'090
6'281
5'171
34
36
38
0.012
8.943
8.882
6.217
6.161
6•
5.113
5.062
5•o16
4'623
4'559
4'504
4'455
4.412
8'828
6.066
5'795
5'539
5.298
4'976
4'729
4'497
4.279
4'374
4'140
3'921
3'715
40
6o
120
00
8'495
8'179
7.879
,5
23056
199'3
45'39
6
3. When v1< 3 the mode is
54
199'4
2'358
2'290
2.089
1.898
199'5
41-83
19.32
7.076
5-951
5.188
4'639
4'226
2.614
v545
2 '484
TABLE 12(f). 0.1 PER CENT POINTS OF THE F-DISTRIBUTION
If
IX ,
F= X
v, V2
-2
where X1and .7C2 are independent random
variables distributed as X' with v1and v2 degrees of freedom
respectively, then the probabilities that F F(P) and that
F < F'(P) are both equal to Piro°. Linear interpolation in
v1or v2 will generally be sufficiently accurate except when
either vl > 12 or v2 > 4o, when harmonic interpolation
should be used.
(This shape applies only when V1
at the origin.)
vl =
I
v2 = I *
2
3
4
5
6
7
8
9
2
3
4053
998'5
167•0
74'14
5000
999'0
148.5
61.25
5404
999'2
141.1
56.18
47'18
35.51
37'12
27.00
33.2o
23.70
4
5
5625
999'2
137.1
53'44
5764
999'3
134.6
51'71
6
5859
999'3
132.8
50'53
3. When v, < 3 the mode is
7
8
I0
12
24
a)
5929
999'4
131•6
49'66
5981
999'4
130.6
49'00
6056
999'4
129.2
48'05
6107
999'4
128.3
47'41
6235
999'5
125.9
45'77
6366
999'5
123'5
44.05
26.92
18.41
26.42
17.99
13.71
11•19
9'570
25.13
16.90
12.73
10.30
8.724
23'79
15'75
11.70
9'334
7-813
8'445
7638
6.847
6 .249
5'781
5'407
6.762
5'998
5'420
4'967
4'604
5'101
31.09
29.75
28.83
28.16
27.65
20•80
16'21
20'03
15'52
19'46
15'02
19'03
29'25
21'69
18'77
21'92
17'20
14'63
14'08
25.41
22.86
18'49
16.39
15'83
13.90
14'39
12.56
13.48
11.71
12.86
11.13
12.40
10.70
12.05
10.37
11.54
9'894
21.04
19.69
18.64
17.82
17.14
14.91
13.81
12.97
12.31
11.78
12.55
1r56
io•8o
10.21
9.729
11•28
10.35
9.633
9'073
8.622
10•48
9.578
8.892
8'354
7.922
9.926
9'047
8.379
7.856
7'436
9.517
8.655
8•ooi
7'489
7.077
9.204
8-355
7.710
7.206
6.8o2
8.754
7922
7292
6-799
6. 404
7626
7.005
6-519
6.130
11'34
10'97
9'335
10•66
10.39
10•16
8.727
8.487
8.280
8.253
7944
7.683
7'459
7.265
7022
18
19
16-59
16.x2
15.72
15.38
15.08
6.808
6.622
6•081
5.812
5'584
5'390
5'222
5.812
5'547
5'324
5.132
4'967
4'631
4'447
4' 288
4'307
4'059
3.85o
3.670
3'514
20
2x
14'82
14'59
9'953
9'772
22
23
24
14'38
9'612
14'20
14'03
9'469
9'339
8•098
7'938
7'796
7669
7'554
7.096
6'947
6.814
6.696
6.589
6'461
6.318
6.191
6•078
5'977
13.88
13'74
13.61
13.5o
13.39
9.223
9'116
9.019
8.931
8'849
7'451
7'357
7.272
7.193
7.121
6'493
6.406
6.326
6.253
6.186
5.885
5.802
5.726
5.656
5'593
13.29
13.12
8.773
8.639
6.125
6•014
5.919
5.836
5.763
5.698
5.307
4'947
4.617
4'757
4'416
4'103
10
II
12
13
14
15
16
17
25
26
27
28
29
9'006
34
36
38
12'97
8'522
I2•83
8.420
8.331
7.054
6.936
6.833
6.744
6.665
40
6o
12.6i
11.97
120
11 . 38
00
1o•83
8.251
7.768
7321
6.908
6.595
6.171
5.781
5.422
3o
32
1211
7'567
7092
6.741
6.471
7.272
6.805
6.460
6.195
6'562
6'223
6.355
6.175
6.021
5'845
5.962
5.763
5'590
6•019
5.881
5'758
5'649
5'550
5.692
5'557
5'438
5'331
5-235
5.308
5.190
5.085
4'991
5.075
4'946
4'832
4'730
4'638
4'823
4'696
4'583
4'483
4'393
4'149
4'027
3.919
3.822
3'735
3'378
3.257
3.151
3.055
2.969
5'462
5.381
5.308
5'241
5.179
5'148
5.070
4'998
4'933
4'873
4'906
4'829
4'759
4'695
4'636
4'555
4480
4'412
4'349
4'292
4'312
4.2 38
4'171
4'109
4'053
3.657
3.586
3.521
3'462
3'407
2'890
2'819
5'534
5'429
5'339
5.26o
5.190
5.122
5.021
4'934
4'857
4'790
4.817
4'719
4'633
4'559
4'494
4.581
4'485
4'401
4'328
4-264
4'239
4'145
4.063
3'992
3'930
4.001
3.908
3.828
3'758
3.697
3'357
3.268
3.191
3.123
3'064
2.589
2498
2 '419
2'349
2.288
5•128
4'731
4'372
4'044
3'743
4'436
4' 086
3-767
3'475
4'207
3.865
3'552
3.266
3'874
3'541
3'237
2.959
3.642
3.315
3'011
2'694
2'402
2'132
v233
5.44o
* Entries in the row v2 = I must be multiplied by ioo.
55
3'016
2'742
4.846
2'754
2.695
2'640
I .890
1.543
I•000
TABLE 13. PERCENTAGE POINTS OF THE CORRELATION
COEFFICIENT r WHEN p = 0
The function tabulated is r(P) = r(P1v) defined by the
equation
2 1)
-
\ITT
1
rp)fr(P)
(1 - r2) 2 dr = P
2
Let r be a partial correlation coefficient, after s variables have
been eliminated, in a sample of size n from a multivariate
normal population with corresponding true partial correlation
coefficient p = o, and let v = n - s. This table gives upper
P per cent points of r; the corresponding lower P per cent
points are given by - r(P), and the tabulated values are also
upper 2P per cent points of For s = o we have v = n and
r is the ordinary correlation coefficient. When v > 130 use
the results that r is approximately normally distributed with
-1
6
7
8
9
xo
II
12
13
14
5
2'5
I
0'5
0.9877
'9000
0.9969
•9500
0 '9995
0 '9999
•9800
.9900
Tables of the distribution of r for various values of p are
given by, for example, F. N. David, Tables of the Ordinates
and Probability Integral of the Distribution of the Correlation
Coefficient in Small Samples, Cambridge University Press
(1954), and R. E. Odeh, ' Critical values of the sample
product-moment correlation coefficient in the bivariate
normal distribution', Commun. Statist. - Simula Computa.
II (I) (1982), pp. x-26. The z-transformation may also be
used (cf. Tables 16 and 17).
0 '9343
0 '9587
0 '9859
•8822
'8329
•7887
7498
•9172
. 8745
. 8343
'7977
'9633
'9350
'9049
-8751
0'5494
•5214
'4973
'4762
'4575
0.6319
•6021
'5760
'5529
'5324
0.7155
•6851
-6581
'6339
•6120
0.7646
'7348
-7079
'6835
•6614
0.8467
0'4409
0'5140
'4259
'4124
'4000
•3887
'4973
'4821
'4683
'4555
20
21
22
23
0.3783
.3687
•3598
'3515
'3438
0.4438
'4329
'422 7
'4132
'4044
'
0.641 I
•6226
•6055
'5897
'5751
0.7301
'7114
'6940
'6777
•6624
0 '5155
0.5614
'5487
•5368
•5256
•5151
0.6481
'6346
•62,19
'6099
.5986
'5034
'4921
'4815
'4716
0 '3365
'3297
•3233
•3172
•3115
0.3961
•3882
•3809
'3739
.3673
0.4622
'4534
'4451
'4372
'4297
30
31
32
33
34
0.3061
•3009
-2960
'2913
•2869
0.3610
'3550
•3494
'3440
•3388
04226
'4158
'4093
'4032
'3972
0.5052
'4958
'4869
'4785
'4705
0-5879
'5776
'5679
'5587
'5499
0.4629
'4556
'4487
'4421
'4357
0 '5415
0.2826
0'3338
0'3916
0 '4296
785
'3862
•3810
'4238
'2709
•3291
•3246
•3202
' 2673
'3160
' 2
'2746
'8199
'7950
-7717
•7501
0.5923
'5742
'5577
'5425
'5285
25
26
27
28
29
35
36
37
38
39
'9980
0.8783
•8114
'7545
.7067
•6664
15
16
17
18
24
0 '999995
o.8054
'7293
•6694
•6215
.5822
'3760
'3712
'4182
'4128
'4076
1
U-shaped.)
zero mean and variance -- (cf. Tables 16 and 17).
v-3
P
r(P)
(This shape applies for v > 5 only. When v = 4 the distribution
is uniform and when v = 3 the probability density function is
zero mean and variance--, or (more accurately) that
v -1
z = tanh-1r is approximately normally distributed with
v --- 3
4
5
0
P
v = 40
5
2'5
0'2638
0'3120
0'3665
42
'2573
'3044
44
46
48
'2512
'2455
'2403
-2973
' 2907
- 2845
'3578
•3496
5c1
52
54
56
58
0'2353
'2306
'2262
'2221
•218I
0'2787
6o
62
64
66
68
0'2144
•2I08
' 2075
'2042
'2012
0'2542
70
72
74
76
78
0.1982
•1954
1927
.1901
•1876
0.2352
'2319
•2287
'2257
80
0.1852
'1829
'1807
•1786
'1765
0'2199
'2172
90
92
94
96
98
0 '1745
•1726
1707
-1689
•1671
100
105
II0
115
120
125
130
0'1478
'1449
82
84
86
88
'5334
'5257
'5184
-5113
0'5045
'4979
'4916
'4856
'4797
56
0%5
O•I
0 '4741
'3420
'3348
0.3281
•3218
'3158
•3102
•3048
0.4026
'3932
'3843
•3761
•3683
0.3610
•3542
'3477
'3415
'3357
0.4267
'4188
'4114
'4043
'3976
0.2997
'2948
0.3301
•3248
0'3912
'2902
'3198
.2858
•2816
.3150
•3104
•3850
'3792
'3736
•3683
0.2776
'2737
'2700
•2664
•2630
0.3060
•3017
'2977
'2938
•2900
0.3632
•3583
'3536
'3490
'3447
0 '2597
0.2864
•2830
'2796
.2764
'2732
0'3405
•2565
'2535
.2505
'2477
0.2072
•2050
•2028
•2006
.1986
0.2449
0'2 702
0' 3215
'2422
'2396
'2371
' 2 347
' 2673
- 2645
'3181
'3148
•2617
'2591
•3116
•3085
0. 1 654
0.1966
'1918
1576
. 1874
. 1541
•1832
'1793
0.2324
-2268
-2216
•2167
0.2565
'2504
•2446
'2393
0- 3054
- 1614
•2I22
'2343
0'2079
-2039
0.2296
1509
'2732
•2681
'2632
•2586
'2500
' 2461
'2423
'2387
'2227
' 2146
'2120
'2096
0.1757
•1723
'2252
'4633
'4533
'4439
'4351
'3364
'3325
•3287
'3251
'2983
'2915
'2853
'2794
0.2738
•2686
TABLE 14. PERCENTAGE POINTS OF SPEARMAN'S S
TABLE 15. PERCENTAGE POINTS OF KENDALL'S K
Spearman's S and Kendall's K are both used to measure the
degree of association between two rankings of n objects. Let
di (1 5 i n) be the difference in the ranks of the ith object;
-51-02(n+ 1)2(n— I) for S and ,72,1-n(n — t)(2n + 5) for K, and
when n > 4o both statistics are approximately normally distributed; more accurately, the distribution function of X =
[S n)]I[]n(n +1)'
”
.17---1]
-- is approximately equal to
Spearman's S is defined as E 4. To define Kendall's K, re-
Y
(130 (x)—
order the pairs of ranks so that the first set is in natural
order from left to right, and let mi (1 5 i
n) be the number
of ranks greater than i in the second ranking which are to the
e
241/27T
SPEARMAN'S S
2'5
5
I
KENDALL'S K
0'5
0•I
5
6
7
8
9
2
0
6
16
3o
48
4
0
2
12
22
6
4
O
14
36
26
IO
20
4
Io
58
84
118
160
42
64
92
128
170
34
54
78
108
146
20
165
zo
34
52
76
104
220
II
286
364
455
222
194
140
616
284
354
436
53o
248
312
388
474
184
236
298
370
72
102
O
2'5
5
n=4
I0
—
P
i(n3—
0
II
—o.c:14(ign2+ 5n-36)
*(n3—n)
and 110(x) is the normal distribution function (see Table 4).
A test of the null hypothesis of independent rankings is
provided by rejecting at the P per cent level if S x(P), or
K x(P), when the alternative is contrary rankings. The
other points are similarly used when the alternative is
similar rankings. To cover both alternatives reject at the 2P
per cent level if S, or K, lies in either tail. Spearman's rank
correlation coefficient rsis defined as 1 — 6S/(n3 — n), and has
upper and lower P per cent points I — 6x(P)/(n3— n) and
— [1— 6x(P)/(n3— n)] respectively. Kendall's rank correlation
coefficient ric is defined as 4K I[n(n —
1, and has upper
and lower P per cent points 4x(P)/[n(n — 1)] r and
I} respectively.
— {4x(P)I[n(n—
right of rank i. Kendall's K is defined as E mi.
i =1
For Table 14 the tabulated value x(P) is the lower percentage point, i.e. the largest value x such that, in independent
rankings, Pr(S < x) P/ loo; in Table 15, K replaces S and
the upper percentage point is given. A dash indicates that
there is no value with the required property. The distributions are symmetric about means (n3— n) for S and in(n-1)
for K, with maxima equal to twice the means; hence the
upper percentage points of S are -i(n3 — n) — x(P) and the lower
percentage points of K are in(n-1)— x(P). The variances are
P
(x3 — 3x), where y =
I0
n=4
20
5
6
7
8
9
I
0'5
9
I0
10
14
14
15
22
19
24
3o
20
25
27
18
23
28
12
13
14
33
39
46
53
62
34
41
48
56
64
56o
68o
816
969
1140
15
16
17
18
19
7o
79
89
99
II0
73
83
93
103
114
1330
121
133
146
126
138
151
159
164
172
178
144
157
171
185
216
232
248
266
120
5
31
26
33
7'5
I0.5
14
18
36
43
51
59
67
37
44
52
61
69
40
47
55
64
73
22.5
27.5
33
39
45.5
77
86
97
1(38
79
89
I00
83
94
105
117
129
6o
68
76-5
142
156
170
184
200
95
zos
xxs.s
126.5
138
238
254
272
216
232
249
267
285
150
162.5
175'5
189
203
303
323
342
363
384
217.5
232.5
248
264
280.5
297.5
315
333
351'5
370.s
390
2I
I2
142
13
14
188
244
xs
x6
17
18
19
388
478
58o
694
20
21
22
23
24
824
970
1132
1310
1508
736
868
to18
1182
1364
636
756
890
1040
1206
572
684
8o8
452
544
65o
1771
948
1102
768
2024
900
2300
20
21
22
23
24
25
26
27
28
29
1724
1566
1272
1460
1664
1888
2132
1584
1796
260o
2925
3276
3654
4060
25
26
27
28
29
186
201
216
232
2794
1784
2022
2282
2564
1388
1588
1806
2044
2304
It:48
1958
248
193
208
223
239
256
3o
31
32
33
34
3118
3466
3840
4240
4666
2866
3194
3544
3920
4322
2584
2884
3210
3558
3930
2396
2682
2988
3318
3672
2028
2280
2552
2844
316o
4495
496o
5456
5984
6545
3o
31
32
33
34
265
282
300
318
337
273
291
309
328
347
283
301
320
340
359
290
308
328
347
368
35
36
37
38
39
512o
5604
6118
666z
7238
4750
5206
5692
6206
675o
4330
4754
5206
5686
6196
4050
4454
4884
5342
5826
3498
3858
4244
4656
35
36
37
38
39
356
376
397
418
440
367
388
409
430
452
380
401
410
5092
7140
7770
8436
9139
988o
444
467
432
454
477
405
428
450
473
497
40
7846
7326
6736
6342
5556
10660
40
462
475
490
501
522
310
2214
2492
210
268
338
418
512
1210
1388
1540
57
n(n — I)
3
13
35
56
84
0•I
6
119
131
200
422
III
123
135
148
161
176
190
205
221
388
52.5
85.5
TABLE 16. THE z-TRANSFORMATION OF THE
CORRELATION COEFFICIENT
The function tabulated is
coefficient p, and let v = n-s. Then z is approximately
normally distributed with mean tanh-1p+plz(v I) (or, less
accurately, tanh-1p) and variance - 3). If s = o we have
v = n and r is the ordinary correlation coefficient. For p = o
the exact percentage points are given in Table 13.
z = tanh-1 r = loge (I
. If
r < o use the negative of the value of z for -r. Let r be a
partial correlation coefficient, after s variables have been
eliminated, in a sample of size n from a multivariate normal
population with the corresponding true partial correlation
-
I
r
z
r
z
r
z
r
z
r
z
0'00
•OI
•02
'03
0'0000
'0 I 00
'0200
0.500
.5o5
•po
'515
•52o
0.5493
.556o
•5627
•5695
•5763
0'750
0.9730
0 '9845
0.9962
z•oo82
1.0203
0•9I0
•912
'9,4
•916
.918
1.5275
'5393
'5513
•5636
.5762
0.9700
•9705
•9710
'9715
•9720
2.0923
•xoo8
•1o95
•1183
.1273
09950
'995x
'9952
'9953
'9954
'755
•760
.765
.770
r
z
2 '9945
3.0046
3'0149
3'0255
'04
•0300
•0400
0.05
o6
•07
•o8
•09
0.0500
•o6oi
•0701
-o8o2
•0902
0'525
0'5832
0'775
.5901
'5971
•6042
-611 2
•78o
.785
'790
'795
1.0327
.0454
.0583
•0714
. o849
0'920
'922
'924
'926
'928
1.5890
-6022
.6157
-6296
'6438
0.9725
.9730
'9735
'9740
'9745
2.1364
'1457
•1552
•1649
'1747
0'9955
•530
'535
'540
'545
0'I0
•II
•I2
0'1003
•II04
•1206
0'550
'93,
'932
'933
'934
1.6584
-6658
.6734
•681x
•6888
0.9750
'9755
'9760
.9765
'9770
2'1847
•I950
- 2054
•2I6o
•1409
-56o
.565
.57o
1.0986
•1127
•127o
1417
•1568
0'9960
1 307
o.800
.8o5
•810
.815
•820
0'930
.13
'14
0.6184
•6256
•6328
•64ox
.6475
0.15
x6
'17
18
•19
0'1511
'1614
•1717
•1820
'1923
0.575
.58o
.585
'590
'595
0. 6550
0'825
'830
'835
1'1723
0'935
.936
'937
.938
'939
1.6967
'7047
-7129
-7211
.7295
0'9775
2.2380
'2494
-2610
.2729
0.9965
•9966
'9967
•285I
•9969
'2027
•218I
'2340
0'20
•2I
'22
'23
'24
0'2027
•2I32
•2237
o.600
•6os
.6 zo
.615
•62o
0.6931
•70I0
'7089
•7169
•7250
o. 85o
.852
'854
.856
-858
1•2562
0.940
'94x
'942
'943
'944
1.738o
'7467
'7555
.7645
'7736
0.9800
•9805
•98,o
•9815
•9820
2'2976
0.9970
'9971
'9972
'9973
'9974
3.2504
.2674
. 2849
•3031
-3220
0'25
'26
'27
0 '2554
0•625
'630
'635
-640
0.7332
'7414
'7498
-7582
•7667
o 86o
•862
'864
•866
•868
1-2933
•3011
•3089
•3169
'3249
0'945
r7828
'7923
•8019
•8117
-8216
0.9825
-983o
.9835
•9840
- 9845
2.3650
'3796
'3946
'4101
'4261
0'9975
'946
'947
'948
'949
'9976
'9977
'9978
'9979
3'3417
•3621
'3834
'4057
-4290
0•870
'872
0.950
'95x
.952
'953
'954
1.8318
•8421
•8527
-8635
- 8745
0.9850
.9855
•9860
•9865
•9870
2'4427
•876
.878
r333r
'3414
'3498
•3583
•3670
'4597
'4774
'4957
'5147
0.9980
•9981
•9982
'9983
'9984
3'4534
'4790
•5061
'5347
•5650
2'5345
'5550
'5764
.5987
•6221
0.9985
-9986
•9987
'9988
.9989
3'5973
-6319
-6689
•7090
.7525
0.9990
'9991
'9992
'9993
'9994
3.8002
3'8529
3.9118
3.9786
4'0557
4'1469
.2585
'4024
•6o51
'9517
•28
.29
'2342
.2448
•2661
•2769
•2877
-2986
'555
'645
•6625
-6700
•6777
.6854
840
'845
•1881
'2044
•2212
•2384
•2634
•2707
•2782
.2857
•9780
.9785
'9790
'9795
-2269
•31o3
-3235
.3369
•3507
0.30
'31
.32
'33
'34
0 '3095
o.65o
-655
-66o
•665
•67o
0'7753
•3205
-3316
'3428
'3541
0.35
•36
'37
•38
'39
0. 3654
•3769
•3884
'4001
'4118
0.675
•68o
.685
•690
.695
0.8199
•8291
-8385
-8480
-8576
o.88o
•882
.884
•886
•888
1•3758
•3847
'3938
'4030
'4124
0'955
.956
.957
.958
'959
1.8857
-8972
•9090
.9210
'9333
0.9875
•9880
040
0.4236
'4356
'4477
'4599
0'700
0.890
.892
1.4219
'4316
1'9459
•720
.9588
•9721
'9857
'9996
0.9900
'9905
•9910
'9915
•9920
2'6467
- 6724
'6996
'7283
'4722
0.8673
-8772
•8872
'8973
•9076
0.4847
'4973
•510I
•523o
-5361
0'725
0.9181
•46
'47
'48
'49
'730
'735
'740
'745
0.50
0'5493
0'750
'41
'42
'43
'44
0'45
'
705
.7 zo
'7,5
'7840
.7928
-8017
-81o7
'874
'9885
-9890
•9895
'9956
'9957
'9958
'9959
'9961
•9962
'9963
'9964
'9968
'894
'441 5
'896
.898
'4516
•4618
0.960
'961
•962
'963
'964
0'900
•902
'9395
.9505
•9616
'904
'906
'908
1.4722
'4828
'4937
'5047
.5 x 6o
0.965
•966
'967
•968
'969
2.0139
-0287
.0439
.0595
.0756
0.9925
'9930
'9935
•9940
'9945
v7911
-8257
-8629
•9031
'9467
0'9995
'9287
0.9730
0'910
1.5275
0.970
2'0923
0.9950
2'9945
I'0000
58
•7587
'9996
'9997
.9998
'9999
3'0363
3.0473
.0585
•0701
-0819
'0939
3.1063
-I190
•132o
•1454
.1591
3'1732
•1877
00
TABLE 17. THE INVERSE OF THE z-TRANSFORMATION
The function tabulated is r = tanh z =
e22 -
+
. If z < o, use the negative of the value of r for -z.
z
r
z
r
z
r
z
r
z
r
z
r
0'00
'01
•02
0'0000
'0100
0'50
•52
'53
'54
0.7616
•7658
-7699
'7739
'7779
0.905!
•9069
•9087
•9104
-9121
2'00
•02
'04
0.9640
.9654
•9667
-9680
.9693
3'00
•02
'0200
•0300
'0400
1'00
•OI
'02
•03
'04
x.50
'51
'03
•04
0.4621
'4699
'4777
'4854
'4930
0.9951
'9952
'9954
'9956
'9958
0'05
0'0500
.0599
1.05
-06
-07
-08
.09
0.7818
-7857
•7895
'7932
-7969
x-55
•56
•57
0.9138
'9154
2•I0
•I2
'9170
-14
- 58
•9186
'59
- 9201
•x6
-18
0.9705
-9716
-9727
'9737
'9748
3'10
•I2
.0798
•o898
0.55
-56
'57
•58
'59
0.5005
•o6
•07
•08
•09
0•10
0.0997
*14
x.60
•6x
-62
-63
- 64
0.9217
-9232
'9246
-9261
.9275
3'20
'22
'1293
•I39I
0.8005
•8041
-8076
•8110
- 8144
2'20
- 22
'I3
0.5370
'5441
•5511
•5581
. 5649
0.9757
•1096
'1194
o•6o
•61
•62
-63
- 64
PIO
•II
•I2
- 24
'26
'28
'9776
-9785
'9793
.28
0'15
0'1489
o•65
•66
-67
•68
-69
0.5717
'5784
•585o
•5915
•5980
I•I5
I•65
0.9289
'9302
2'30
- 32
'67
.9316
'9329
'9341
-36
•38
0.9801
-9809
-9816
-9823
•983o
3.3o
•32
- 34
•36
•38
0 '9973
-66
•17
-x8
•x9
0.8178
-8210
8243
-8275
•8306
0.70
'71
-72
. 73
'74
0.6044
•6107
-6169
•6231
•6291
I'20
•2I
'22
•23
'24
0.8337
•8367
-8397
- 8426
- 8455
1.70
'71
'72
'73
'74
0 '9354
2.40
- 42
'44
'46
•48
0.9837
.9843
.9849
.9855
•9861
3'40
- 42
.44
- 46
•48
0.9978
'9979
'9979
-9980
'9981
0.75
•76
'77
•78
'79
0.6351
- 6411
- 6469
.6527
- 6584
1'25
0.8483
•8511
-8538
.8565
-8591
1'75
0 '9414
- 9425
2'50
'9436
'9447
'9458
•52
'54
-56
•58
0.9866
•9871
•9876
-9881
•9886
3-50
'55
-60
-65
•70
0.9982
'9984
-9985
-9986
-9988
0.6640
-6696
•6751
•68o5
-6858
1'30
x.8o
•32
'33
'34
0.8617
- 8643
•8668
•8692
-8717
-82
-83
-84
0.9468
'9478
'9488
'9498
-9508
2.60
-62
- 64
•66
-68
0.9890
-9895
-9899
.9903
•9906
3.75
•8o
.85
•90
'95
0.9989
- 9990
'9991
'9992
'9993
4.00
•05
•xo
.x5
0 '9993
- 20
'9994
'9995
'9995
'9996
•r7
-x8
•I9
0'20
•2I
'22
•o699
•1586
•1684
'1781
'1877
0'1974
'2070
.2165
•23
•2260
'24
'2355
0.25
0 '2449
-26
•27
•28
'2543
'2636
'29
•282I
0'30
0.2913
•3004
'2729
-5080
'5154
.52,27
.5299
•II
'12
- 13
-26
•27
-28
•29
'52
'53
'54
-
-68
-69
•76
'77
•78
'79
-9366
'9379
'9391
'9402
-06
•08
- 34
•9767
'04
-06
•08
'14
•x6
•x8
'24
'26
0 '9959
.9961
•9963
'9964
-9965
0.9967
-9968
•9969
'9971
'9972
'9974
'9975
'9976
'9977
'32
•3095
'33
'34
•3185
'3275
o•80
•81
-82
.83
-84
0.35
•36
- 37
•38
'39
0.3364
•3452
'3540
•3627
'3714
0.85
-86
.87
•88
-89
0.6911
-6963
•7014
'7064
•7114
1-35
0'8741
•36
- 37
•38
'39
-8764
-8787
.8810
-8832
1-85
-86
-87
-88
.89
0.9517
'9527
'9536
'9545
'9554
2'70
'72
'74
76
•78
0.9910
'9914
.9917
-9920
'9923
0'40
0.3799
.3885
•3969
'4053
•4136
0.90
'9I
.92
.93
'94
0.7163
-7211
'7259
•7306
'7352
1 40
0.8854
•8875
-8896
•8917
.8937
x•90
'9,
.92
'93
'94
0.9562
'9571
'9579
-9587
'9595
2.80
•82
'84
-86
88
0.9926
-9929
'9932
'9935
'9937
4'25
.30
'35
. 40
'45
0.9996
'9996
'9997
'9997
'9997
0.45
'46
'47
'48
'49
0'4219
0.95
•96
'97
•98
'99
0.7398
'7443
'7487
'7531
'7574
1'45
1'95
•96
'97
-98
'99
0.9603
-9611
•9618
-9626
-9633
0 '9940
•46
'47
•48
- 49
0.8957
.8977
-8996
•9015
.9033
2'90
'4301
.4382
'4462
'4542
•92
'94
-96
.98
'9942
'9944
'9946
'9949
4'50
- 55
•60
.65
-7o
0.9998
'9998
'9998
'9998
'9998
0'50
0'4621
1'00
0.7616
I•50
0•905I
2'00
0'9640
3'00
0.9951
4'75
0'9999
'31
'41
'42
'43
'44
.31
'42
'43
'44
59
TABLE 18. PERCENTAGE POINTS OF THE DISTRIBUTION OF
THE NUMBER OF RUNS
with the required property. When n, and n2 are large, R is
Suppose that th A's and n2 B's (n1 n
2) are arranged at
random in a row, and let R be the number of runs (that is,
sets of one or more consecutive letters all of the same kind
immediately preceded and succeeded by the other letter or
the beginning or end of the row). The upper P per cent point
x(P) of R is the smallest x such that Pr {R x} 5 P/ioo, and
the lower P per cent point x'(P) of R is the largest x such that
Pr IR P/ioo. A dash indicates that there is no value
approximately normally distributed with mean
zni n2
+ and
ni+ n2
2n022(2n1n2— — n2)
. Formulae for the calculation
(ni +n2)2 (ni +n2 — I)
of this distribution are given by M. G. Kendall and A. Stuart,
The Advanced Theory of Statistics, Vol. 2 (3rd edition, 1973),
Griffin, London, Exercise 30.8.
variance
UPPER PERCENTAGE POINTS
=3
4
112
P
5
=4
4
5
6
7
8
9
9
9
7
5
5
6
7
8
9
5
6
6
7
7
7
8
8
8
xo
6
7
8
9
xo
I
P
o•x
5
n1 = 8 n2 = 17
18
19
20
9
9
9
9
II
I0
II
12
13
II
14
9
I0
I0
I0
II
II
II
II
12
12
9
12
12
13
13
13
II
12
13
12
13
13
13
14
13
7
12
13
14
8
9
13
14
13
15
15
9
xo
15
16
I7
18
19
20
TO
II
12
13
10
II
13
14
14
15
15
15
10
14
15
16
17
i8
10
19
14
15
15
II
20
II
15
15
15
II
18
19
8
9
13
14
14
15
16
16
xo
II
12
13
14
15
15
15
17
17
14
16
15
16
16
17
17
15
,6
16
16
17
17
12
14
13
14
14
15
16
17
II
12
12
16
16
16
17
14
I
17
17
17
18
18
17
18
18
19
19
19
18
16
16
17
17
19
17
18
18
19
17
19
19
20
20
20
19
18
19
zo
16
16
16
17
18
18
18
18
19
19
19
13
17
14
14
18
19
20
14
15
16
17
I2
13
19
19
20
20
21
14
15
16
17
18
18
19
19
19
20
20
20
2I
21
2I
21
22
22
22
23
19
20
12
13
14
20
20
18
18
19
22
22
19
20
21
23
23
21
22
22
15
19
21
23
I
o•x
20
20
21
21
21
22
22
22
23
23
23
19
20
20
21
21
21
21
22
22
23
23
23
21
22
22
20
21
23
24
24
22
23
25
25
26
21
22
22
23
23
23
25
24
24
24
24
24
24
25
24
24
25
24
24
25
26
26
27
21
22
22
23
23
23
24
24
25
25
25
26
26
27
27
19
24
23
23
24
24
26
24
25
26
26
28
z6
27
28
28
16
20
25
26
29
17
17
18
24
24
26
26
19
25
27
28
29
20
25
27
29
18
19
20
25
25
26
27
27
28
29
30
30
18
19
20
15
15
16
17
18
19
20
20
5
24
18
19
20
20
20
2I
21
21
13
14
15
x6
13
19
19
17
17
18
6o
n2 =
n1 =
16
17
17
18
18
15
15
P
0•I
15
20
16
16
17
x8
x8
19
20
28
19
26
28
20
27
29
30
31
20
27
29
31
TABLE 18. PERCENTAGE POINTS OF THE DISTRIBUTION OF
THE NUMBER OF RUNS
LOWER PERCENTAGE POINTS
P
=
2
2
n2
=
8
9
3
3
n1 =
8
n2 = 19
5
x
8
8
6
6
6
6
6
4
5
5
5
5
3
3
3
7
7
7
8
8
5
6
6
6
6
4
4
4
4
5
8
8
8
9
6
7
7
7
7
5
5
5
5
5
4
7
7
8
8
8
5
6
6
6
7
4
4
4
5
5
8
9
9
9
7
7
7
8
8
5
5
6
6
6
4
5
5
5
5
12
13
12
2
14
5
2
15
x6
17
18
2
19
5
5
5
5
5
4
4
4
4
4
5
3
4
4
4
4
2
3
—
3
3
3
2
I0
3
4
4
4
4
2
10
2
12
12
13
14
5
5
5
5
5
3
3
3
'3
15
16
z7
18
19
6
6
6
6
6
4
4
5
5
5
3
3
3
3
3
xo
20
5
3
3
4
4
4
II
2
14
8
10
6
4
4
5
5
3
15
9
6
6
6
7
7
II
12
13
Ls
15
5
6
6
6
6
4
4
5
5
5
3
3
3
3
3
II
16
17
18
x9
9
9
io
xo
xo
7
8
8
8
8
6
6
6
6
7
16
17
18
19
6
7
7
7
7
5
5
5
6
6
4
4
4
4
4
12
8
9
9
9
io
7
7
7
8
8
5
5
5
6
6
5
5
6
6
6
4
4
4
5
5
2
12
3
3
3
3
x8
19
lo
io
zo
6
7
7
7
5
5
5
5
6
6
6
13
2
14
2
2
=5
5
2
2
2
20
2
2
5
6
2
2
7
8
9
2
2
2
10
3
3
2
12
17
x8
19
4
4
2
4
5
6
7
8
9
3
3
3
3
zo
5
6
6
2
3
3
3
3
3
2
6
2
2
2
2
2
6
7
2
2
2
2
7
2
2
2
2
2
14
3
3
3
2
2
15
16
4
4
3
3
2
12
13
20
6
7
8
9
zo
II
3
3
4
4
4
II
n2= I°
II
18
19
20
5
2
P
2
2
3
3
3
3
4
5
o•x
II
16
4
I
I0
15
4
5
3
3
3
3
3
13
14
3
P
0' I
4
4
4
4
II
3
I
2
2
15
x6
17
2
5
2
7
2
7
8
9
20
17
4
3
2
8
9
xo
18
19
4
4
3
3
2
II
2
12
20
4
3
3
3
3
3
2
5
6
7
8
9
4
2
8
2
x6
2
17
6
7
7
7
7
x8
8
8
13
2
14
2
15
3
2
8
61
20
2
2
9
9
2
I0
2
II
2
9
2
3
3
3
12
13
14
15
x6
9
17
x8
19
20
2
I0
II
15
16
17
x8
19
20
II
12
2
2
20
12
13
14
xs
16
9
7
8
8
20
II
13
13
9
8
8
9
9
7
4
4
4
4
4
13
14
15
x6
17
x8
9
io
io
Jo
ix
8
8
8
9
9
6
6
6
7
7
4
13
19
II
9
7
/7
TABLE 18. PERCENTAGE POINTS OF THE DISTRIBUTION OF
THE NUMBER OF RUNS
LOWER PERCENTAGE POINTS
P
nI= 13
722 = 20
14
15
16
17
14
18
15
19
20
15
5
I
0•I
II
10
I0
II
II
10
8
8
9
9
8
6
7
7
7
II
I2
12
9
10
10
9
9
7
8
8
7
7
II
16
II
P
=
15
5
I
o•x
II
I0
I0
8
8
8
8
8
th = /7
18
19
20
I2
12
I2
IO
16
16
II
10
16
17
18
I2
12
13
13
12
o
To
19
20
17
17
II
nI = 17
II
II
I0
5
19
n2 =
18
18
8
8
9
9
8
P
19
20
I
0•I
13
II
13
II
9
9
9
9
9
20
13
II
18
19
13
II
14
I2
20
19
20
20
14
14
14
15
12
12
12
13
I0
IO
10
II
TABLE 19. UPPER PERCENTAGE POINTS OF THE TWO-SAMPLE
KOLMOGOROV-SMIRNOV DISTRIBUTION
.. d(P). When
rejecting at the P per cent level if nin2 D(ni, n,)?.
This table gives percentage points of
D(ni, n2) = sup I Fi(x) F2(x)I,
where F1(x) and F2(x) are the empirical distribution functions
of two independent random samples of sizes n, and n2
respectively, nI < n2 < 20 and n, = n2 loo, from the same
population with a continuous distribution function; the
function tabulated d(P) is the smallest d such that Pr
{nin2 D(ni, n2) d} Phoo. A dash indicates that there is
no value with the required property. A test of the hypothesis
that two random samples of sizes n, and n2 respectively have
the same continuous distribution function is provided by
— D(ni, n2) are
n1and n2are large, percentage points of 4/n1n2
ni+ n2
—
5
=
2
"2 =
5
6
7
8
9
2
I0
II
12
13
14
2
16
17
x8
19
2
20
3
3
4
5
6
3
3
7
8
9
xo
14
16
18
P
o•x
ni.
16
18
24
26
24
26
24
26
28
26
28
3o
z8
3o
32
34
36
3o
32
34
36
38
38
38
40
40
32
32
34
9
12
15
15
18
2I
21
12
13
27
30
14
33
33
36
=3
I0
5
2'5
I
0•I
36
39
42
42
16
42
45
45
48
16
45
48
51
51
—
48
51
54
57
—
—
16
18
21
20
20
24
20
24
28
—
—
24
27
28
28
28
32
32
36
14
28
29
36
35
38
3o
33
36
39
42
36
36
40
44
44
15
16
17
18
19
40
44
44
46
49
44
48
48
5o
53
45
52
20
52
6o
64
5
6
7
8
20
25
25
24
25
27
24
28
3o
3o
35
35
36
40
35
40
39
43
45
n2 = 17
18
19
20
18
20
22
24
II
16
I
10
12
24
27
15
2'5
approximately given by those in Table 23 with n = co.
Formulae for the calculation of this table are given by
P. J. Kim and R. I. Jennrich, ' Tables of the exact sampling
distribution of the two-sample Kolmogorov—Smirnov
criterion D,„„, m < n', Selected Tables in Mathematical
Statistics, Vol. 1 (1973), American Mathematical Society,
Providence, R.I.
20
22
4
4
4
5
6
7
8
9
4
xo
II
12
13
15
18
18
21
21
24
27
30
2I
24
27
30
30
27
3o
33
3o
33
36
36
39
33
36
39
39
42
36
39
42
42
45
4
4
5
5
9
to
II
12
13
6z
24
28
36
40
44
48
48
—
52
56
52
56
6o
6o
64
6o
64
68
72
76
76
3o
3o
32
68
25
3o
35
35
36
40
44
45
47
40
45
45
5o
52
45
5o
55
6o
65
52
54
57
TABLE 19. UPPER PERCENTAGE POINTS OF THE TWO-SAMPLE
KOLMOGOROV-SMIRNOV DISTRIBUTION
P
nI
=5
5
5
n2 = 1 4
15
x6
17
x8
42
46
51
56
70
50
52
55
54
55
6o
55
59
6o
65
6o
64
68
70
70
75
8o
85
19
56
6o
61
65
66
75
30
28
30
36
30
30
34
35
36
71
8o
36
36
40
85
90
—
—
48
39
40
43
48
13
46
52
44
48
54
54
45
48
54
6o
6o
54
6o
66
66
72
10
12
33
36
38
48
14
48
51
54
56
66
54
57
6o
6z
72
58
63
64
67
78
64
69
72
73
84
78
84
84
85
96
10
x5
x6
17
x8
II
20
6
6
6
7
8
9
I0
Ix
6
6
7
7
9
9
n1 =
9
xs
17
9
76
83
96
34
36
40
42
41
45
88
42
48
49
Ioo
49
56
63
xo
40
46
49
53
63
II
12
44
46
50
56
48
53
56
63
52
56
58
70
59
6o
65
77
7o
72
78
84
56
59
61
62
68
68
73
77
75
77
84
90
96
98
65
69
72
76
So
84
87
91
107
93
56
55
6o
64
112
64
64
7o
77
x3
13
64
72
79
8
9
40
48
40
44
48
46
48
53
86
48
48
54
58
5z
54
58
6o
6o
62
64
67
64
65
70
74
68
72
76
81
8o
72
8o
8o
88
104
x7
18
19
68
77
8o
88
72
8o
86
94
III
112
74
8z
90
98
117
20
8o
54
88
54
96
63
104
63
124
72
To
50
II
52
57
53
59
6o
63
63
70
8o
81
63
69
75
87
9
12
72•
76
81
85
90
t•x
78
84
90
94
99
91
98
105
no
117
90
99
io8
126
98
Ioo
70
68
107
III
8o
77
126
133
90
89
6o
64
68
75
66
70
74
8o
72
77
82
90
8o
84
90
Ioo
96
Ioo
io6
115
76
84
go
Ioo
x18
18
19
79
8z
85
89
92
94
96
xoo
103
io6
108
113
126
132
133
20
I00
II0
120
130
150
ix
66
77
77
88
99
xo
12
17
12
64
72
13
14
15
16
67
75
82
84
76
84
87
94
86
91
96
99
1 08
115
102
120
89
96
io6
127
132
140
73
76
8o
92
102
III
122
12
20
12
96
72
107
84
116
96
127
96
146
154
120
12
13
71
95
117
78
81
86
84
14
94
104
120
15
16
17
84
93
99
io8
129
88
go
96
100
104
io8
116
119
136
141
x8
x9
96
99
io8
1o8
20
104
116
13
14
91
78
91
89
15
x6
87
91
101
17
iz
88
90
97
85
93
102
II0
88
97
107
118
96
Ito
126
15o
120
124
104
130
140
117
156
Ioo
104
164
130
129
104
III
115
121
137
143
114
127
152
131
138
156
164
18
19
96
99
105
110
104
114
120
126
13
20
108
14
x4
15
x6
17
98
120
112
130
II2
143
126
154
140
14
18
19
20
63
65
70
75
78
82
x
17
x8
x9
Ica
20
12
Ix
59
63
69
69
74
2'5
89
93
70
6o
15
16
78
42
5
81
x3
14
xx
lo
8o
84
6o
57
II
70
x5
16
17
18
x9
x8
x9
20
xo
72
42
7
8
9
n2 = 13
14
16
64
13
14
x5
x6
8
42
O•I
66
35
I0
II
8
50
I
19
14
7
8
48
2'5
20
13
7
P
Jo
169
92
98
no
123
96
Ioo
106
III
116
126
152
122
134
159
104
116
126
140
166
II0
114
121
126
133
138
148
152
176
180
TABLE 19. UPPER PERCENTAGE POINTS OF THE TWO-SAMPLE
KOLMOGOROV-SMIRNOV DISTRIBUTION
P
ni = 15 n2 = 15
16
17
18
19
110
5
2'5
I
0'1
105
POI
120
114
135
133
165
162
105
III
116
123
135
119
129
135
142
165
147
114
127
141
152
P
xo
5
2.5
I
128
140
136
133
156
153
148
168
200
170
164
204
187
141
166
174
n j. = 16 n2 = 20
17
17
18
19
136
118
126
180
20
130
146
151
160
175
200
209
18
18
144
162
162
18o
216
19
133
20
136
152
142
152
171
159
166
190
176
182
190
212
214
228
144
16o
169
187
225
840
854
868
882
896
900
1020
1080
1320
915
992
ioo8
1024
1037
1054
1071
io88
1098
1178
1197
1216
1342
1364
1386
1408
910
1040
1495
1056
1072
1105
1122
1235
990
1005
4206
112273
54
1088
1104
1224
1242
1292
1380
1564
1587
1190
1207
1224
1260
1278
1296
1400
1420
1440
1610
1704
1728
15
20
125
135
150
160
195
x6
16
17
18
19
zi2
109
116
128
124
128
144
136
140
16o
143
154
176
174
186
19
120
133
145
160
190
20
ni = n2 = 20
160
168
198
180
189
198
230
200
210
220
230
264
220
231
242
253
288
260
21
22
23
24
273
286
299
336
ni = n2 = 6o
6z
62
63
64
275
z86
300
350
65
312
364
324
364
377
405
420
435
66
67
68
69
1020
1035
450
496
70
71
1050
1065
1080
207
216
240
25
26
27
28
29
225
234
2 90
308
319
297
336
348
30
300
310
330
341
360
372
390
403
31
243
280
250
260
270
o•x
19
;54
518
1
32
320
352
384
416
512
72
33
34
330
374
396
408
396
442
462
476
528
544
73
74
1095
1241
1314
1460
1752
II I0
1258
1332
1480
1776
1125
1216
1232
1248
1800
35
385
420
455
490
595
75
36
396
432
468
504
612
76
37
38
39
407
418
429
444
456
468
481
494
546
518
570
585
629
646
702
77
78
79
560
574
588
6oz
616
600
615
630
688
704
720
738
756
774
836
8o
8i
1280
1440
1296
1458
82
1312
1476
83
84
1328
1344
1264
1275
1425
1500
1292
1444
1596
1824
1309
1326
1422
1463
1482
1501
1617
1638
1925
1659
1975
1520
1680
2000
1539
1701
2025
1558
1722
2050
1494
1512
1660
1680
1743
1848
2075
2184
1700
1720
1870
1892
1740
1760
1780
1914
1936
1958
2210
2236
2262
1729
1748
1767
1786
1800
1820
1932
1953
1974
1980
2002
2416
2139
2162
2484
2511
2538
1995
2016
2037
2058
2185
2208
2231
2254
2565
2592
2716
2744
40
440
520
41
42
43
44
492
504
516
528
533
546
559
572
45
46
47
48
49
540
585
675
720
855
85
1360
1530
552
564
576
637
644
690
705
736
752
874
893
720
768
912
735
833
980
86
87
88
89
1462
1479
1496
1513
1548
1566
1672
1691
90
91
92
1530
:5
546.4
7
93
94
1581
1598
658
672
686
50
650
700
750
51
663
676
689
714
728
742
765
832
848
850
867
884
901
1020
1040
1060
702
810
864
918
1134
55
715
56
57
728
798
825
84o
855
88o
896
912
990
1008
1026
1155
1176
1197
95
96
1615
1632
52
53
54
1000
1710
1950
2288
2 314
2430
2457
58
812
870
928
1044
1218
1764
59
8z6
885
1003
1062
1298
9
99
1805
1824
1843
1960
1782
1980
2079
2277
2772
6o
84o
900
1020
1080
1320
I00
I 800
2000
2100
2300
2800
64
TABLE 20. PERCENTAGE POINTS OF WILCOXON'S
SIGNED-RANK DISTRIBUTION
cent level if W+
x(P); a similar test against u > o is provided by rejecting at the P per cent level if W- 5x(P), and,
against # o, one rejects at the 2/:' per cent level if W, the
smaller of W+ and W-, is less than or equal to x(P). When
n > 85, W-F is approximately normally distributed.
Formulae for the calculation of this table are given by
F. Wilcoxon, S. K. Katti and R. A. Wilcox, ' Critical values
and probability levels for the Wilcoxon rank sum test and
the Wilcoxon signed rank test', Selected Tables in Mathematical Statistics, Vol. I (1973), American Mathematical
Society, Providence, R.I.
This table gives lower percentage points of W+, the sum of the
ranks of the positive observations in a ranking in order of increasing absolute magnitude of a random sample of size n from
a continuous distribution, symmetric about zero. The function tabulated x(P) is the largest x such that Pr {W+ < x}
P/Ioo. A dash indicates that there is no value with the required property. W-, the sum of the ranks of the negative
observations, has the same distribution as W+, with mean
ln(n+ 1) and variance -Nn(n + 1) (n + ). A test of the hypothesis that a random sample of size n has arisen from a continuous distribution symmetric about p = o against the
alternative that j < o is provided by rejecting at the P per
P
n= 5
6
7
8
9
10
II
12
13
14
zs
5
0
2
3
5
8
10
13
17
2I
25
2'5
0
2
3
5
8
10
17
21
0'5
I
I
o
3
I
5
7
9
3
5
7
9
I
n = 45
371
46
47
48
49
389
407
426
446
343
361
378
396
415
312
328
345
362
379
322
339
355
249
263
277
292
307
so
51
52
53
54
466
486
507
529
550
434
453
473
494
514
397
416
434
454
473
373
390
408
427
445
323
339
355
372
389
12
15
12
573
595
618
642
666
536
557
579
602
625
493
514
535
556
578
465
484
504
525
546
407
425
443
462
482
I
2
4
6
19
23
17
18
19
20
21
22
23
24
6o
67
75
83
91
25
loo
26
27
110
119
107
92
83
64
28
130
116
IoI
91
29
140
126
I TO
100
71
79
120
130
140
151
162
109
118
128
138
148
159
171
182
0'5
291
307
0.1
32
37
15
19
23
27
32
8
I
14
18
21
55
56
57
58
59
52
58
65
73
8,
43
49
55
62
69
37
42
48
54
61
26
30
35
40
45
6o
61
62
63
64
690
715
741
767
793
648
672
697
721
747
600
623
646
669
693
567
589
61i
634
657
501
521
542
563
584
89
98
76
84
68
75
51
58
65
66
67
68
69
8zo
847
875
903
931
772
879
718
742
768
793
819
681
705
729
754
779
6o6
628
651
674
697
86
94
103
70
960
990
1020
907
936
846
873
805
831
964
112
121
73
74
1050
994
901
928
io81
1023
957
884
912
721
745
770
795
821
131
75
1112
141
76
1144
151
162
173
77
78
79
1209
1053
1084
1115
1147
"79
986
1015
1044
1075
1105
940
968
997
'026
1056
847
873
900
927
955
185
1276
1310
1345
1380
1415
1211
1244
1277
1311
1136
Io86
1116
"47
1345
1168
1200
1232
1265
1178
1210
983
MI
1040
1070
1099
1451
1380
1298
1242
1130
27
151
137
163
147
175
187
200
159
35
36
37
38
39
213
227
195
208
221
235
249
173
40
264
279
294
238
252
43
44
286
302
319
336
353
310
327
45
371
343
41
42
2'5
0
25
29
34
40
46
241
256
271
5
O
30
35
41
47
53
3o
31
32
33
34
P
O'I
170
182
185
198
211
224
194
207
71
72
266
281
296
220
233
247
261
276
209
222
235
8o
81
8z
83
84
312
291
249
85
197
65
1176
1242
798
825
852
858
TABLE 21. PERCENTAGE POINTS OF THE
MANN-WHITNEY DISTRIBUTION
x(P), and a similar test against
level P per cent if UB
/LA < ,I.tB is provided by rejecting at the P per cent level if
UA x(P). For a test against both alternatives one rejects
at the 2P per cent level if U, the smaller of UA and UB, is less
than or equal to x(P). If n1and n2 are large UA is approximately normally distributed. Note also that UA +UB = n1n2.
Consider two independent random samples of sizes n1and n2
respectively (n1 < n2) from two continuous populations, A
and B. Let all n, + n2 observations be ranked in increasing
order and let RA and RB denote the sums of the ranks of the
observations in samples A and B respectively. This table gives
lower percentage points of UA = RA -ini(ni+ 1); the
function tabulated x(P) is the largest x such that, on the
assumption that populations A and B are identical,
Pr {UA
x} < P/too. A dash indicates that there is no
value with the required property. On the same assumption,
UB = RB — in2(n2+ t) has the same distribution as UA, with
mean ini n2 and variance 12-n1n2(n1+ n2 + 1). A test of the
hypothesis that the two populations are identical, and in
particular that their respective means /tit, itB are equal,
against the alternative itA > AB is provided by rejecting at
P
=
2
2
5
2'5
I
0'5
Formulae for the calculation of this distribution (which is
also referred to as the Wilcoxon rank—sum or Wilcoxon/
Mann—Whitney distribution) are given by F. Wilcoxon,
S. K. Katti and R. A. Wilcox, ' Critical values and probability levels for the Wilcoxon rank sum test and the
Wilcoxon signed rank test', Selected Tables in Mathematical
Statistics, Vol. I (1973), American Mathematical Society,
Providence, R.I.
0'1
12
13
14
I0
15
16
I7
x8
19
O
I
O
I
15
16
17
18
19
3
I
0
3
I
0
3
2
0
2
2
0
I
0
2
I
0
20
4
3
4
5
6
0
7
8
9
To
II
12
13
14
15
16
17
18
I9
20
4
5
6
7
8
9
4
5
4
4
2
4
5
6
7
8
9
0
3
4
7
8
9
to
II
II
=4
3
3
n2 = ro
O
O
2
3
2.5
6
7
8
9
12
3
5
n2 = 5
II
2
P
5
0
I
o
2
I
—
12
I0
14
II
I
0.5
3
2
O
4
2
O
5
5
6
3
3
4
O
7
7
8
9
9
5
5
6
6
7
15
II
16
12
17
13
20
5
6
7
8
18
4
14
2
to
8
I
O
5
6
8
3
5
6
2
3
4
I
2
9
ro
II
9
II
7
8
9
5
6
7
8
9
3
4
5
6
7
12
12
13
15
12
16
18
13
I0
15
14
II
II
2
I
O
2
O
2
I
O
19
15
12
3
3
I
O
20
17
13
7
8
9
to
I
O
22
18
14
II
5
6
7
7
8
4
4
5
5
6
2
I
2
I
2
I
3
3
2
15
i6
3
4
6
9
9
4
4
4
5
2
O
2
O
3
3
0
II
6
7
7
8
I
O
IO
18
5
6
2
6
I
0
2
I
O
3
4
4
I
O
2
I
3
I
6
7
8
9
ro
II
12
13
0
6
2
3
4
5
6
19
20
66
14
15
16
17
18
23
19
25
7
8
to
20
12
14
IO
i6
17
19
13
14
16
5
6
8
II
21
17
23
19
25
21
26
22
28
24
I
I
I
2
2
3
3
3
I
3
4
4
5
5
0•x
I2
13
0
I
I
2
2
3
3
4
5
5
6
7
7
2
3
4
O
7
8
9
II
5
6
7
9
12
I0
2
3
4
4
5
15
16
8
19
12
II
13
15
16
I
6
7
8
9
to
TABLE 21. PERCENTAGE POINTS OF THE
MANN-WHITNEY DISTRIBUTION
P
n1 = 6
7
7
2'5
I
0'5
n2 = 19
30
25
20
17
II
20
32
II
27
22
18
I2
8
10
14
15
50
12
4
6
7
2
15
6
7
9
I
13
3
x6
17
19
14
16
17
13
21
24
14
z6
18
20
22
15
28
16
30
17
7
8
9
10
II
12
7
x8
19
7
8
20
8
9
I0
II
8
P
5
II
9
12
14
16
10
12
O•I
n1 = II
5
2'5
I
0'5
38
42
46
54
33
37
40
44
47
28
31
34
37
41
24
27
30
33
36
57
61
65
69
42
51
55
58
62
37
44
47
50
53
31
39
42
45
48
27
29
32
34
37
47
51
55
6o
64
41
45
49
53
57
35
38
42
46
49
31
34
37
41
44
23
25
28
31
34
18
19
68
61
53
47
37
72
65
56
51
40
20
13
77
51
14
56
69
45
50
6o
39
43
54
34
38
42
z6
29
i5
61
65
70
75
8o
54
59
63
67
72
47
51
55
59
63
42
45
49
53
57
32
35
38
42
45
84
61
66
71
77
76
55
59
64
69
67
47
51
56
6o
6o
42
46
50
54
48
32
36
39
43
20
82
87
92
74
78
83
65
69
73
15
16
72
64
56
77
70
61
58
63
67
51
55
46
50
54
40
43
18
19
83
88
94
75
8o
85
90
75
66
70
75
8o
66
6o
64
69
73
6o
47
51
55
59
48
81
86
92
98
87
71
76
8z
87
77
65
70
74
79
70
52
56
6o
65
57
93
99
105
99
io6
82
88
93
88
94
75
81
86
81
87
61
66
70
66
71
112
113
119
127
I00
101
92
76
107
93
99
114
105
77
82
88
n2 = I2
13
II
17
13
15
5
6
7
8
9
12
12
24
19
16
zo
12
13
26
21
18
II
14
33
35
37
28
30
3z
23
19
13
15
24
21
22
14
x6
15
17
39
15
18
34
13
15
17
z8
9
II
13
24
16
19
15
7
9
II
13
4
5
6
8
22
17
20
22
20
23
z6
18
19
20
12
13
15
9
II
16
12
17
24
26
17
18
20
22
14
15
19
34
36
38
41
28
3o
32
34
24
z6
28
30
17
18
20
21
15
21
17
14
II
7
17
24
27
20
23
13
16
8
10
26
28
14
36
31
26
18
20
22
12
13
30
33
16
18
21
15
x6
17
x8
19
39
42
45
48
51
34
37
39
42
45
28
31
33
36
38
24
27
29
31
33
23
25
9
20
54
10
I0
27
40
19
36
16
26
10
II
22
24
18
21
12
29
13
31
34
37
48
23
26
33
27
24
17
17
14
41
30
33
36
38
41
26
29
31
34
37
17
44
48
51
55
36
39
42
45
48
19
15
23
25
27
44
47
39
42
29
32
25
21
15
26
z8
31
33
36
24
26
29
31
20
39
41
44
47
9
xo
12
13
14
15
16
8
17
18
19
9
9
II
12
9
12
xo
16
17
18
10
XX
19
58
52
20
6z
34
55
30
II
23
13
18
13
20
14
14
x6
14
x8
19
15
14
15
15
17
19
21
17
20
I00
i6
16
83
x6
17
z8
19
89
95
101
20
I07
17
96
14
21
18
18
102
19
109
20
115
x8
109
116
19
18
20
19
19
20
67
20
123
123
130
20
138
0•I
17
20
22
24
27
20
TABLE 22A. EXPECTED VALUES OF NORMAL
ORDER STATISTICS (NORMAL SCORES)
The values E(n, r) are often referred to as normal scores;
they have a number of applications in statistics. In carrying
out calculations for some of these applications the sums of
squares of normal scores are often required: they are provided in Table 22 B.
Suppose that n independent observations, normally distributed with zero mean and unit variance, are arranged in
decreasing order, and let the rth value in this ordering be
denoted by Z(r). This table gives expected values E(n, r) of
Z(r) for r 5 En+ r); when r > En+ I) use
E(n, r) = - E(n, n+ 1- r).
n=
r= I
I
2
3
0'0000
0.5642
2
6
4
5
0.8463
1. 0294
1.163o
1•2672
•3522
•0000
0'2970
0'4950
0.6418
02015
0 '7574
0.0000
3
4
7
0.3527
0.0000
8
9
xo
1 4136
0'8522
0%4728
0'1525
1 '4850
0'9323
0*5720
0•2745
1.5388
P0014
0'6561
0*3758
0'0000
0'1227
5
n=
II
12
13
14
15
16
17
18
19
20
1'8445
1.8675
P4076
ply's,
0.9210
Y = I
1'5864
1'6292
1.6680
1.7034
P7359
P7660
2
•0619
1•1157
P1641
P2079
P2479
3
4
0.7288
0.7928
0.5368
0.8498
0.6029
0.9011
0•6618
0 '9477
0.4620
0.7149
1.2847
0.9903
0.7632
P7939
1.3,88
1.0295
0.8074
1.8200
1•3504
1.0657
0.8481
P3799
P0995
0.8859
5
6
0'2249
-0000
0'3122
•1026
0'3883
•1905
•0000
0•4556
.2673
•0882
0'5157
0'5700
0'6195
•3962
'4513
'2338
•0773
'2952
0.6648
•5016
•3508
•2077
•o688
0.7066
'5477
.4016
•2637
•1307
0 '7454
'3353
•1653
•0000
0.0000
0.0620
7
8
•1460
•0000
9
I0
n=
•5903
•4483
•3149
•1870
21
22
23
24
25
26
27
28
29
30
1.8892
1.9097
1.4582
P9292
1'9477
•5034
P9653
.5243
P9822
'5442
P9983
.5633
2'0285
.4814
2'0428
1'6156
1.1582
r=x
2
1'4336
3
4
i•16o5
1.1882
•21414
'2392
•2628
'2851
'3064
2'0137
1.5815
1 '3267
0.9538
0.9846
•0136
•0409
•0668
.0914
•1147
1.1370
5
6
0.7815
0.8153
0'8470
7
8
'491 5
•3620
•2384
0.9570
•82o2
•6973
•5841
.4780
P0261
•7012
.5690
.4461
0.9317
'7929
•6679
.5527
'4444
1•0041
•6667
.5316
0.9050
•7641
•6369
'5193
•4086
0'9812
.6298
•8462
.7251
-6138
•5098
0.8708
0.7515
0.6420
0.5398
0.894.4
0.7767
0.6689
0.5683
04110
•3160
•2239
-1336
•0444
0.4430
•3501
•26o2
.1724
•0859
0 '4733
0'0000
0'0415
9
10
II
12
13
'4056
•2858
•3297
0.8768
'7335
•6040
'4839
•3705
0•1184
0.1700
0.2175
0.2616
0.3027
0.3410
0.3771
•0000
'0564
•1081
'1558
•0518
'2001
•0995
•2413
•2798
•1439
•0478
•1852
•0922
'0000
•0000
•0000
14
15
68
1.5989
1'3462
1•3648
1.1786
•3824
•2945
•2088
•1247
TABLE 22A. EXPECTED VALUES OF NORMAL
ORDER STATISTICS (NORMAL SCORES)
=
31
32
33
34
35
36
37
38
39
40
T = I
2.0565
2.0697
2'0824
2'0947
2'1066
2'1181
2. 1293
2'401
2'1506
2
1'6317
1• 647 I
1.6620
P6764
1.6902
1.7036
1.7166
P7291
P7413
z• 1608
1.7531
3
4
1•3827
1'3998
1 '4323
1 '4476
1'1980
1.2167
1.4164
1.2347
1•2520
1•2686
1.4624
1.2847
1.4768
1.3002
1.4906
1.3151
1'5040
1.3296
1 '3437
5
6
7
8
9
1.0471
0.9169
0.8007
x.o865
0.9590
1.1051
0.9789
p1230
1'1402
0 '9384
0 '9979
1.0162
1.1568
r0339
0.8455
0.7420
0.6460
0.8666
0.7643
0.6695
0.8868
0.7857
0.6921
0.9063
0'9250
1.1883
1.0674
0.9604
1.2033
1.0833
0'9772
0.8063
0.7138
0r8261
0.7346
0.8451
0.8634
0'5955
0.8236
0.7187
0.6213
1.1728
x.0509
0.9430
0'7547
0'7740
0.881
0.7926
10
0.5021
0.5294
0 '5555
0'6271
0.6490
0'6701
0.6904
0'7099
'4129
'3269
*44 I 8
'4694
13
14
'2432
.1613
'3575
'2757
•1957
•3867
•3065
.2283
0.5804
'4957
'4144
'3358
•2592
0.6043
II
12
•5208
•4409
•3637
•2886
'5449
•4662
•3903
•3166
.5679
'4904
'4158
'3434
•5900
•5136
.4401
.3689
•6113
'5359
4635
'3934
-6318
'5574
'4859
'4169
15
0.0804
•0000
0.1169
•0389
0.1515
.0755
•0000
0.1841
•xi0r
-0366
0.2151
•1428
-0712
-0000
0'2446
0'2727
0 '2995
•1739
•2034
•2316
0.3252
•2585
0.3498
•2842
•1040
•0346
•1 351
'0674
'0000
'1647
'0985
•0328
'1929
'1282
•0640
•2199
0'0000
0'0312
x6
17
18
0.6944
1.0672
19
20
n=
1•5170
•1564
•0936
41
42
43
44
45
46
47
48
49
50
r=I
2.1707
2'1803
2'1897
2.2077
2.2164
2'2249
2'2331
I .7646
1•5296
1'7757
1.7865
1.8073
p8173
•827,
1.8366
2'2412
1'8458
2'2491
2
2.1988
P7971
1.5419
1•3705
1.5538
1•3833
1.5653
1. 5875
1. 6187
1•4196
1.5982
1.4311
1.6086
P3957
1'5766
I.4078
1'4422
1'4531
1.6286
1. 4637
1.2456
1•1281
1•0245
0.9308
0.8447
1.2588
P1421
p0392
0.9463
1.2717
1.1558
1.0536
0.9614
1.2842
1.1690
1.0675
0.9760
1.2964
p1819
1.0810
0.9902
1.3083
1.1944
1'0942
I.0040
0.8610
0.8767
0.8920
0.9068
0'9213
1.3198
1.2066
1.107o
1.0174
0 '9353
1'3311
1'2185
P1195
I'0304
0.9489
0.7645
•6889
•6171
'5483
•4820
0.7815
•7067
-6356
-5676
-5022
0'7979
0.8139
'7405
-6709
-6044
'5405
0.8294
.7566
.6877
•6219
'5586
0.8444
-7723
'7040
•6388
'5763
0.8590
-7875
.7198
•6552
'5933
0.8732
-8023
.7351
-6712
'6099
0.4389
'3772
•3170
0'4591
0.4787
'4187
-3602
•3029
•2465
0.4976
'4383
•3806
'3241
•2686
0 '5159
0 '5336
'4573
'4003
'3446
•2899
'4757
'4194
'3644
•3105
0.5508
'4935
'4379
•3836
'3304
0. 1910
•1360
0'2140
•1599
0'2361
0'2575
0'2781
•0814
•1064
'1830
'1303
•2051
'1534
•2265
'1756
'0271
'0531
*0781
•1020
'1251
•0000
•0260
-0509
'0749
0.0000
0'0250
3
4
1 '3573
7
1.2178
1•0987
0 '9935
8
9
0.81 06
1.2319
1.1136
1.0092
0.9148
0.8279
10
II
12
0.7287
0 '7469
•6515
-5780
'6705
13
14
•5075
'4394
•5283
•4611
15
16
1.7
18
19
0'3734
•3089
.2457
•1835
•1219
0.3960
•3326
'2704
-2093
'1490
0.4178
'3553
' 2942
'2341
'2579
•1749
•1997
20
21
22
23
24
0'0608
0.0892
0•1163
O'1422
'0000
'0297
'0580
'085,
'0000
'0283
5
6
0.8982
'5979
.7238
.6535
•5863
•5217
•3983
'3390
•2808
•2236
0.1671
•IIII
'0555
'0000
25
69
1 '8549
TABLE 22B. SUMS OF SQUARES OF NORMAL SCORES
This table gives values of S(n) = E [E(n, r)]2.
r=1
S(n)
n
S(n)
3
4
o•0000
0•6366
1.432
2'296
xo
xx
12
13
14
7.914
8.879
9.848
1o•82o
11.795
5
6
7
8
9
3'195
4.117
5'053
5'999
6.954
15
16
17
10
7.9x 4
n
2
n
S(n)
n
S(n)
n
S(n)
43
44
37479
38'473
39-466
40.460
41'454
20
17'678
30
27'558
40
2x
18.663
3x
32
33
34
28'549
29'540
30'531
3P523
41
32'515
33'507
34'500
35'493
36'486
45
46
47
48
49
42.448
43'443
44'437
45'432
46.427
37'479
50
47'422
22
19.649
23
24
20'635
21.623
12.771
I3.750
14'730
25
26
18
15-711
28
22.610
23'599
24'588
25'577
19
16 '694
29
26'567
35
36
37
38
39
20
17.678
30
27'558
40
27
42
TABLE 23. UPPER PERCENTAGE POINTS OF THE ONE-SAMPLE
KOLMOGOROV-SMIRNOV DISTRIBUTION
n'D(n) d(P). The distribution of n1D(n) tends to a limit
as n tends to infinity and the percentage points of this
distribution are given under n = co. This table was calculated
using formulae given by J. Durbin, Distribution Theory for
Tests Based on the Sample Distribution Function (1973), Society
for Industrial and Applied Mathematics, Philadelphia, Pa.,
Section 2.4.
If F„(x) is the empirical distribution function of a random
sample of size n from a population with continuous distribution function F(x), the table gives percentage points of
D(n) = sup IFn(X)-F(X)1;the function tabulated is d(P)
such that the probability that niD(n) exceeds d(P) is P/Ioo.
A test of the hypothesis that the sample has arisen from
F(x) is provided by rejecting at the P per cent level if
10
5
2'5
I
0•I
P
10
5
2.5
x
o•x
0.950
r098
0.975
1'191
0.9875
P256
0.995
P314
n = 20
P184
3
4
1'102
1' 130
1•226
1'248
1•330
1'348
1436
1468
0.9995
P383
1. 595
21
22
23
•185
•186
•187
24
•188
P315
•316
'317
•318
'319
1.434
'435
'436
'438
'439
1.576
•578
'579
•58o
-582
1.882
.884
.887
•889
•890
5
P139
P260
P370
•382
•391
•399
'404
1'495
'510
•523
.532
'540
P747
'775
'797
.813
•825
25
26
27
1•188
•189
•190
P320
28
•190
29
•1 9 1
'322
•323
•323
P440
'440
'441
'442
'443
P583
•584
'585
•586
'587
1.892
*894
*895
•897
.898
1 '546
1 '835
'551
•556
'559
.563
'844
*851
-856
.862
30
P192
.196
P324
•329
•332
.335
•337
1•444
'449
'453
'456
'458
P588
'594
.598
-601
•604
P899
.908
'914
•918
•921
r565
•568
.570
.572
'574
1866
•87o
'874
.877
.88o
P338
'339
•340
'346
'358
1'459
'461
'462
'467
'480
x•605
•6o7
•6o8
'614
'628
1.923
•925
•927
'935
'949
P
n=I
2
6
•146
-272
7
•154
'279
•285
•290
8
59
• 1
9
•162
xo
1. x 66
I/
12
•169
•171
13
14
•174
•176
•303
•3 05
P409
'41 3
'417
'420
'423
15
P177
•179
•180
•182
•183
P308
•309
•311
•313
.314
P425
'427
'429
'431
'432
x6
17
18
19
P294
'298
•301
1'701
40
50
•1 99
6o
•20I
70
•203
8o
P205
90
100
200
'206
'207
00
20
1'184
p315
p434
p576
P882
70
'212
•224
•321
TABLE 24. UPPER PERCENTAGE POINTS OF FRIEDMAN'S
DISTRIBUTION
Consider nk observations, one for each combination of n
blocks and k treatments, and set out the observations in an
n x k table, the columns relating to treatments and the rows to
blocks. Let the observations in each row be ranked from I to
k, and let Ri (j = I, 2,
k) denote the sum of the ranks in
the jth column. This table gives percentage points of
Friedman's statistic
M
12
k
3n(k + 1)
nk( + 1) =
on the assumption of no difference between the treatments;
the function tabulated x(P) is the smallest value x such that,
on this assumption, Pr {M x} < Pima. A dash indicates
that there is no value with the required property. A test of
the hypothesis of no difference between the treatments is
provided by rejecting at the P per cent level if M x(P).
The limiting distribution of M as n tends to infinity is the
X'-distribution with k - I degrees of freedom (see Table 8)
and the percentage points are given under n =
k=3
P
lo
5
n=3
6•000
6•000
6•000
6.50o
8.000
8•000
5.200
5'333
5'429
5.25o
5'556
6.400
7.000
7'143
6.25o
6.222
7.600
8.333
7'714
7'750
8.000
8.400
9.000
8.857
9•000
9'556
10'00
12'00
12'29
12'25
12'67
9
5-000
5.091
5.167
4'769
5'143
6.2oo
7.800
7.818
8.000
7'538
7'429
9.600
12.6o
13.27
12.67
I0
II
12
12'46
13'29
13
14
4
5
6
7
8
9
I0
II
12
13
14
6'545
6.500
6.615
6.143
2.5
6.400
6.5oo
6.118
6.333
6.421
7.600
4'900
4'95 2
4'727
4'957
5.083
6'300
6•095
6'091
7'500
7'524
7'364
7.913
25
z6
27
z8
29
4'88o
6.080
6.077
6•000
6.5oo
6.276
7'440
7'462
7'407
7'71 4
7.517
3o
31
32
33
34
4'867
4'839
4'750
4'788
416 5
6•zoo
6.000
6•063
6•061
6•059
7400
7548
7.563
7.515
4'605
5.991
4'933
17
20
21
22
23
24
4'875
5.059
4'778
5.053
4'846
4'741
4'571
5. 034
6'348
6'250
7.625
7'41 2
7'444
7.684
7'750
I
9'455
9'500
9.385
9'143
o.z
5
n =3
4
6-600
6.3oo
740o
7.800
8.2oo
8.400
9•o00
9.60o
5
6
6.36o
6.400
6.429
6.3oo
6•zoo
7.800
7.600
7.800
7.650
8.76o
8.800
9.000
cr000
7.667
8.867
9'960
10'20
10'54
10'50
10'73
1 3'46
13'80
14'07
6.36o
6.273
6.300
6.138
7.680
7.691
7.700
7.800
10•68
14'52
7'714
10•75
10•80
10'85
10'89
1 4.80
6. 343
9.000
9.000
9.100
9.092
9.086
7.720
7.800
7.800
7'733
7.863
9.160
9.150
9.212
9.200
9-253
10'92
10'95
11'05
10•93
11'02
7.800
7'815
9.240
9'348
I 1'10
15'36
11'34
16-27
7
8
8.933
9'375
9. 294
9.000
9'579
12.93
13.50
13.06
13.00
1 3'37
18
19
6.280
6.300
6.318
6-333
6'347
9.300
13.3o
20
6.240
9.238
13. 24
CO
6'251
9.091
9'391
9.250
13.13
13.08
8.960
9.308
1. 10
12•60
12'80
1 4'56
14'91
15.09
15.08
15.15
15.28
15.27
1 5'44
1 3'45
5
n=3
4
13'52
13'23
9'407
13'41
9'172
13'50
13.52
9. 214
k= 4
2.5
5
6
7
7467
7.600
8.533
8.800
7680
8.96o
9.067
9'143
9'200
k=5
2.5
9.600
9.800
I
43•x
i0•13
11.20
13'20
'0.40
10'51
10'60
11'68
11'87
12'11
12'30
14'40
15.2o
15.66
16•oo
10'24
I I .47
13'40
8
13'42
13.69
9
7733
9'244
10.67
12'44
16.36
1 3'52
13'41
00
7'779
9.488
11.14
I3.28
18.47
7'471
9.267
9.290
9.25o
9.152
9.176
7'733
7'771
7.700
7'378
9.210
13.82
k=6
c1:4
71
P
zo
5
n= 3
4
8.714
9.000
9.857
10.29
io.81
11.43
11.76
i2-71
13.29
15.29
5
6
9•00o
9.048
10.49
1(3.57
11'74
16'43
00
9'236
11'07
13'23
13'62
15'09
2.5
12'00
12'83
I
O•I
1705
20'52
TABLE 25. UPPER PERCENTAGE POINTS OF THE KRUSKAL-WALLIS
DISTRIBUTION
Consider k random samples of sizes n,, n2, ..., nk respectively,
n1 n
2 ... nk, and let N = n1+ n2 + ... nk. Let all the
N observations be ranked in increasing order of size, and let
R, (j = I, 2, ..., k) denote the sum of the ranks of the observations belonging to the jth sample. This table gives percentage points of the Kruskal-Wallis statistic
k
12
H
N(N+ r)j
tinuous population; the function tabulated, x(P), is the smallest
x such that, on this assumption, Pr {H P/ioo. A dash
indicates that there is no value with the required property.
The limiting distribution of H as N tends to infinity and each
ratio n;IN tends to a ppsitive number is the x2-distribution
with k I degrees of freedom (see Table 8), and the percentk). A test of
age points are given under n;= oo (j = 1, 2,
the hypothesis that all k samples are from the same continuous population is provided by rejecting at the P per cent
level if H x(P).
-
R2
E
3(N+ I)
on the assumption that all k samples are from the same con-
k
n1, n2, ns
2,
=
re.
=3
5
I
cvx
ni, n2,
123
6, 5,
6, 6,
5
540
4:007
5:7
92
459
4
6,
6, 6
6, 4
3
2
4'438
4'558
4'548
5.410
5.625
5'724
6, 5
6, 6
I,
44
4...2L47
32
7
5:8o65i
2,
I
4'200
2, 2
4'526
4'571
4'556
5'143
5'361
5.556
3,
4,
4,
4,
4,
3, 3
4'622
4'500
4'458
4'056
4'511
5.600
5.956
5'333
5.208
5'444
5.500
5'833
6.000
6.444
6,
6,
7,
7,
7,
4,
4,
4,
4,
4,
3,
4,
4,
4,
4,
3
4'709
4'167
4'555
4'545
4'654
5'791
4.967
5'455
5'598
5.692
6 '155
6.167
6.327
6 '394
6.6i5
6 '745
6. 667
7.036
7'144
7.654
7,
7,
7,
7,
7,
5, 2,
I
4.200
4'373
4'018
6.000
6.044
6-004
6. 533
4'533
5.000
5.160
4.960
5.251
5.648
4'985
5.273
5.656
5.657
5.127
5'858
6.068
6.410
6 '673
6•000
6 '955
6.346
6.549
6•760
6.740
5.600
5'745
5'945
6.136
6.436
5-856
5,
3, I
3, 2
I
2
3
4
2, 2
5, 3, I
3, 2
5, 3, 3
5,
I
4.714
4'65 1
4,
4,
4,
4,
3
4
5, 5,
I
3'987
4'541
4'549
4'668
4'109
5,
5,
5,
5,
6,
5,
5,
5,
5,
2
3
4
5
4'623
4'545
4'523
4'560
2,
I
4'200
5'338
5'705
5.666
5.780
4.822
6,
6,
6,
6,
6,
2,
3,
3,
3,
4,
2
I
2
4'545
3'909
4'682
4'500
4'038
5'345
4'855
5.348
5'615
4'947
6,
6,
6,
6,
6,
4,
4,
4,
5,
5,
4'494
4'604
4'595
4'128
4'596
5.340
5.610
5.681
4'990
5.338
4'535
5.602
4'522
5•66 1
5,
5,
5,
5,
2
3
I
2
3
4
z
2
6, 5, 3
6, 5, 4
I
-
7.200
8.909
9-269
=3
5
3, I
3, 2
7
P=
Ic•
4'571
4
5 6
42080
2, I
2, 2
k
.o.
2'5
2
I
2
3,
3,
3,
3,
2,
2,
2,
P
4.706
5'143
2
3, 3
4, I
4, 2
6.848
6.889
5'727
5.818
5-758
6.201
6.449
I
0.1
8.028
7.121
7.467
7.725
8•000
10.29
9.692
9.752
10.15
10'34
8.124
10'52
8'222
10-89
7.000
7.030
-
6.184
6.839
7.228
6.986
7.321
8. 654
9.262
9.198
5 .'6520
3
6:7
5078
7
7
7:5
14
0
4:0
415
5 :0
39
634
1
:9
2:3
06510
7:4
3, I
3,
2'5
6.788
5-923
6.2to
6.725
6.812
7
4
5'3952
3
4.'5
1 2
4'603
4'121
4'549
4'986
5191
5'376
4:55 6
2 27
5.620
4, 3
4, 4
5, I
8-727
7,
7,
7,
7,
7,
5, 3
4'535
5'607
6.627
7.697
9. 67o
9'841
9.178
9.640
9.874
7.205
7'445
7'760
7.309
8.591
8'795
9.168
-
7,
7,
7,
7,
7,
5,
5,
6,
6,
6,
4
5
I
4'542
4'571
4'033
6'738
6.835
6.067
6'223
6.694
7.931
8.108
7'254
7- 490
7.756
io•16
10'45
9'747
10.06
to.26
7.338
7'578
7.823
8•000
-
8.938
9'284
9.606
9.920
-
7,
7,
7,
7,
7,
6,
6,
6,
7,
7,
2
6.970
7.410
7.106
8.692
-
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
6.667
5.951
6.196
7.340
7.500
7'795
7.182
7.376
8.827
9.170
9.681
9.189
6.667
6.750
7'590
7.936
9.669
9.961
6'315
6.186
6%538
6-909
7.079
-
6'655
6 '873
5, 2
2
4'500
3
4'550
5'733
5-708
5.067
5'357
5.680
4
5
6
4'562
4'560
4'530
3'986
4'491
5.706
5170
5'730
4.986
5-398
6.787
6 '857
6.897
6.057
6.328
8.039
8'157
8.257
7.157
7.491
10.46
10'75
11.00
9.871
10 '24
3
4
5
6
7
4'613
4'563
4'546
5.688
5.766
5.746
5'793
5.818
6.708
6.788
6.886
6.927
6.954
7.810
8.142
8.257
8-345
8.378
10 '45
10.69
10.92
11.13
11.32
8, x, I
4'418
4.011
4.010
4'451
4.909
5.356
4.881
5.316
5.420
5:0
86
14
7
6
6.195
4...050
31
5(D364
9134
7
6.588
g.•189
835
8,
2,
I
I
8, 2, 2
8, 3, x
8, 3, 2
4'568
4'594
4.587
8, 3, 3
8, 4, 1
8, 4, 2
72
6
6..8
60
64
3
7.022
8.791
6
7...3
3
97
50
3
89%940
921
36
TABLE 25. UPPER PERCENTAGE POINTS OF THE KRUSKAL-WALLIS
DISTRIBUTION
k=3
P = to
nb n2, n5
8,
8,
8,
8,
8,
k =4
-
5
2.5
I
0• I
ni, n2, n3, n4
9'742
icror
9'579
9.781
4, 4, I,
4, 4, 2, I
P=
3
4
I
2
3
4'529
4'561
3.967
4'466
4'514
5.623
5'779
4'869
5'415
5.614
6.562
6750
5.864
6.260
6. 614
7'585
7853
7.110
7'440
7.706
8, 5, 4
5
8, 6,
8, 6, 2
8, 6, 3
4'549
4'555
4'015
4'463
4'575
5.718
5.769
5.015
5'404
5.678
6.782
6. 843
5'933
6. 294
6658
7'992
8.146
7.256
7'522
7'796
10. 29
8,
8,
8,
8,
8,
6, 4
6, 5
6, 6
7,
7, 2
4.563
4'550
4. 599
4'045
4'451
5'743
5.750
5'770
5'041
5'403
6'795
6.867
6.932
6'047
6 '339
8-045
8.226
8.313
7.308
7. 571
10.63
10.89
11.10
10.03
10.36
8,
8,
8,
8,
8,
7,
7,
7,
7,
7,
3
4
5
6
7
4'556
4'548
4'551
4'553
4'585
5.698
5'759
5.782
5.781
5.802
6.671
6.837
6. 884
6.917
6.98o
7.827
8.1,8
8. 242
8.333
8.363
10•54
10.84
11.03
44.28
11.42
2,
2,
2,
2,
I,
2,
I,
I,
I
I
2,
2,
2,
2,
2,
2,
2,
2,
I
2
3,
2,
X,
X,
X
8,
8,
8,
8,
8,
8,
8, 2
8, 3
8, 4
8, 5
4'044
4'509
4'555
4'579
4'573
5-039
5.408
5.734
5'743
5761
6'oo5
io•16
10.46
10.69
10.97
1.18
3,
3,
3,
2,
2,
2,
I,
I
2,
I
2,
2,
2,
2,
6.920
7'314
7.654
7889
8.468
8.297
8, 8, 6
8, 8, 7
8, 8, 8
4'572
4'571
4'595
4.582
5'779
5'791
5.805
5.845
6.953
6.98o
6995
7'041
8'3 67
8.41 9
8.465
8.564
11.37
44.55
4,
4,
5,
5,
5,
8, 5,
9, 9, 9
4' 60 5
00, 00, OD
5'991
6.351
6.682
6.886
7'378
9. 210
10'04
10.64
9'840
io.n
I0 '37
n4
2,
3,
2,
2,
2,
2,
2,
I,
I
2
I
3,
3,
2, 2,
2, 2,
I
2,
3, 3,
3,
3,
3,
3,
2
I,
3, 2, I
3, 2, 2
3, 3,
3, 3, 2
3, 3, 3, 3
4, 2, I,
4,
2,
2,
4, 2, 2, 2
4, 3, I, I
4,
4,
4,
4,
4,
3,
3,
3,
3,
3,
2, I
2,
2
3, I
3, 2
3, 3
P=
5'357
5.667
5.143
5.556
5'644
5
25
I
5'679
6.467
6.667
6.667
5.833
6.333
6.250
6.978
6 '333
6. 244
6 '527
6 .600
6.727
6'333
6.689
7'055
7.036
7.515
7.200
7.636
7.400
8.015
6.026
7'ooO
7.667
8.538
5.250
5'533
5'755
5.067
5.833
6.133
6'545
6.178
6.533
7'064
6.741
7.000
7'391
7.067
5'591
5'750
5.689
5.872
6•016
6-309
6'955
6.621
6.545
6.795
6. 984
7.326
7.326
7'455
7.871
7'758
8.333
8.659
7'564
7'775
2.5
I
0•I
5'945
6.386
6.731
6.635
6'874
6'955
7.159
7.538
7.500
7'747
7.909
7.909
8.346
8.23,
8.621
8.909
9'462
9.327
9'945
8876
8.588
8.874
9'075
9.287
10'47
9.758
10.43
10'93
1I-36
4: 4,3 2:
3,
4, 4, 3,
2
I
2
5.182
5.568
5.808
5.692
5'901
4,
4,
4,
4,
4,
3
I
2
3
4
6.019
5.654
5'914
6.042
6•088
7.038
6.725
6 '957
7.142
7.235
7.929
7.648
7'914
8'079
8.228
oo, 00, oo, oo
6.251
7'815
9'348
4,
4,
4,
4,
4,
3,
4,
4,
4,
4,
n4, n2, n3, n4, n5
3, 3,
41.95
5.786
6.250
6.600
11.34
6'982
6'139
6.511
6.709
6955
6.311
6.6o0
2
I,
I
5
2'5
6.750
7.333
7.964
7.533
8.291
6'583
--
--
6.800
7'309
7.200
7'745
7'600
7.682
8.182
7.1n
7.200
7.467
7.618
8.538
8-06 z
8'449
8.8,3
8.703
9.o38
9'233
2,
I
2
6.788
7.026
6.788
2,
I
6'910
2,
2
7'121
7'591
7'910
7'576
7'769
8.044
3, 3, 3, 3, I
3, 3, 3, 3, 2
3, 3, 3, 3, 3
7.077
7.210
7'333
8.0o0
8.2oo
8.333
CO, 00, 2), CO, 00
7'779
9'488
3, 2,
3, 2,
3, 3,
3, 3,
3, 3,
I
6.750
7.133
7.418
2,
3,
3,
3,
3,
3,
11'70
2,
P = zo
I
I,
3, 3, I,
16.27
0'I
-
8.12,
-
--
8'127
8.682
8.073
9'364
-
8.576
9.115
8'424
9.051
9.505
9.303
10403
9'455
9'974
9'451
9.876
10'59
10-64
13'82
0.1
k=
P=
2-5
-
I
7.600
8•348
8'455
7.800
8.345
8.864
8'455
2,
2
8.154
8.846
9.385
I,
I
7'467
7'945
8.348
8'731
9'033
7.667
8.236
8.727
9.248
9.648
7.909
8.303
8'564
I,
2,
I,
I,
2,
2,
2,
2,
2,
I,
2,
2,
2,
2,
2,
I,
2,
2,
2,
2,
2,
2,
I
I
I
Io
• 17
10'20
11'67
13.28
18'47
6
5
I,
2,
11.14
6.833
7.267
7.527
7.909
/42,
7'133
5'333
5.689
5'745
5.655
5.879
5
k=5
k=4
nb nz, n3,
70
0. I
8.648
9'227
9'846
9'773
I 0'54
8.509
9.136
9.692
9.682
I0'38
9'o30
3, 2,
3, 2,
3, 2,
3, 2,
3, 2,
9.513
-
2,
I,
2,
I
I
7.133
7418
7.727
7.987
2,
2,
2
8'198
I,
I,
2,
2,
I, I, I
2,
2,
2,
3, I, I, I, I moo
3, 2, I, I, I 7.697
3, 2, 2, I, I 7'872
3, 2, 2, 2, I 8.077
3, 3,2, 2, 2, 2 8.305
3,
3,
3,
3,
8.909
9.482
9'455
8.667
10'22
11'11
8'564
9'045
8'615
9.128
8'923
9'549
10-15
n -oi
9.190
9.914
10-61
11'68
15.09
20*52
9.628
10.31
10'02
CO, 00, 00 , 00, CO, 00
73
9.236
11.07
12.83
•
TABLE 26. HYPERGEOMETRIC PROBABILITIES
Suppose that of N objects, R are of type A and N— R of
type B, with R N—R. Suppose that n of the objects,
n < N—n, are selected at random without replacement and
X are found to be of type A. Then X follows a hypergeometric distribution with the probability that X = r given by
Here the rows correspond to types A and B and the columns
to ' selected' and not selected' respectively, and the marginal
totals are given.
Fisher's exact test of no association between rows and
columns, or of homogeneity of types A and B, is provided by
rejecting the null hypothesis at the P per cent level if the sum
of the probabilities for all tables with at least as extreme
values of X as that observed is less than or equal to Pj roo.
More extreme' means having smaller probability than the
observed value r of X, given the same marginal totals. This
test may be either one- or two-sided, as shown below.
p(rjN, R, n) =
(':)(Nn — T(Nn)
This table gives these probabilities for N < 57 and n < R
(if not, use the result that p(rIN, R, n) = p(rIN, n, R)). For
N > 57 these probabilities may be calculated by using
binomial coefficients (Table 3) or logarithms of factorials
(Table 6). When N is large and RI N < c•r, X is approximately binomially distributed with index R and parameter
p = n/N (see Table r); similarly, if N is large and n/N < o•1,
X is approximately binomially distributed with index n and
parameter p = R/N. If N is large and neither R/N nor n/N
Example.
I 5
4 4
5 9
(X+.', — nR I N)I[R(N — R) n(N — n)I Nz(N — 1)11
is approximately normally distributed with zero mean and
unit variance; a continuity correction of 2 , as with the binomial distribution, has been used.
A representation of the data in the form of a 2 x z contingency table is useful:
✓
R— r
R
n—r N—R—n+r N — R
n
N—n
N
N
2 I I
= 0 0'5000
I
'5000
5
=
I
2
Rn
2 2
I
0'7500
•6000
.r000
I
.1429
I
•2500
0.8333
•1667
O
I
O 06667
4
O
I
2
0'5000
•5000
4
2 2
O
0•1667
x
•6667
2
'1667
N R n
2 I
= 0
6 2 x
I
8
0-8571
4 I I
0'7500
'2500
I I
= 0
6 x
O
7
N R n
0'3000
3
O 06667
I
'3333
N R n
7
2 I
8
2 2
O
I
01143
2857
O
I
0.5357
2
-0357
7
2 2
•4286
2 2
I
0.0714
'4286
2
'4286
0
3
8 3
2
I
*4762
O
06250
2
•0476
I
•3750
3
4
O
0'4000
I
'5333
2
•0667
7 3
8 3
03571
'5357
2
'1071
001 43
•2286
'5143
•2286
•0143
028 57
6 3
I
. 571 4
0
O
I
2
•1429
I
2
5 x
O
O cr8000
I
.2000
•
-6000
2
'2000
0'2000
5 2 I
6 3 3
O o.6000
I
•4000
O
I
0 0500
2
'4500
*4500
3
•0500
8 3 3
o•1786
2
7 3 3
6 3
9 I z
O 0.8889
I
'III'
2
O
o•5000
-5000
2
O 0.1543
I
'5143
2
'3429
3
•o2.86
8
O 0.8750
•125o
I
3
'5357
2679
•0179
8 4
O o.s000
I
-5000
8 4 2
O 0. 2143
I
'5714
2
74
'2 1 43
2
0
0.4167
I
•5000
2
'0833
9
2 I
O 0.7778
I
'2222
O
0'2381
I
2
'5357
' 2143
3
'0119
9 4 I
O
I
2 2
O
I
0.5833
'3889
2
'oz78
0.5556
'4444
2
O
I
0'2778
*5556
2
'1667
9 4 3
I
2
3
9 3 I
0
O 06667
I
'3333
I
0
O o 8000
'2000
I
I0 2 2
O
I
0'6222
2
'0222
'3556
zo 3 I
I
'3000
xo 3
2
O 04667
'4667
•
2
'0667
0'1190
. 4762
•3571
'0476
9 4 4
I
I0 I
= 0 0'9000
I
' 100
O 0.7000
9 4
0
9
N R n
I0 2 I
.0714
8 4 4
I
O
I
9 3
r=
9 3 3
O 04762
7 3 I
O 0. 5714
I
•4286
N R n
8 4 3
r=
'3333
6
14
From the tables p(1114, 6, 5) = •2098. A more extreme onesided value is r = o, giving a total probability of •2378, not
significant evidence of association or of inhomogeneity. If a
two-sided test is required, r = 4 and r = 5 have probabilities
•0599 and •003o respectively, less than •2098; the total is
now .3007.
When N > 17 and nR/N is not too small, a (two-sided)
test of the hypothesis of no association, or of homogeneity, is
provided by rejecting at the P per cent level approximately
if x2 = N[rN — nR] 21[R(N — R) n(N — n)] exceeds A(P) (see
Table 8). (Cf. H. Cramer, Mathematical Methods of Statistics
(1946), Princeton University Press, Princeton, N.J., Sections
30.5 and 30.6.)
is less than o•1,
N R n
6
8
013397
'3175
2
'4762
•i587
3
•0079
4
xo 3 3
O
I
0'2917
•5250
2
-175o
•0083
3
I0
0
I
4
o.6000
'4000
TABLE 26. HYPERGEOMETRIC PROBABILITIES
N R n
10 4
r=
2
N R n
II 3
2
N R n
12 I I
0
0.3333
1' = 0
0.5091
Y= 0
I
'5333
I
x
2
.1333
2
.4364
*0545
0.9167
.0833
12 2
10 4 3
O
I
0.1667
- 5000
2
- 3000
3
I0
O
'0333
4 4
0.0714
xi 3 3
0 0.3394
I
•5091
z
3
•1455
•006I
Ix 4 1
I
'1143
•0048
xx
io 5 x
O
I
0.5000
-5000
0.8333
•1667
12 2 2
o
0.6818
I
2
'3030
'0152
0 0.6364
I
'3810
2 -4286
3
4
0
I
I
•3636
4 2
0 0.3818
•5091
x
2
'1091
II 4 3
12
3 i
o
I
0'7500
*2500
12
0
3
2
I
0.5455
.4091
2
-0455
o 0'2121
22
3 3
12
- 5091
I
0'2222
2
•2545
0 0.3818
N R n
N R n
N R n
N R n
12
13
13 5 5
14 3
5 4
Y= 0
I
2
0'0707
3
4
' 1414
•oioi
'3535
'4242
I
.5556
3
•0242
2 '2222
10
5 3
ix 4 4
0 0•1061
O
0.0833
I
2
'4167
I
2
•4167
•0833
4
3
3
'4242
•3818
•0848
•0030
1
.4909
2
*1227
3
.0045
12 4
I
0
0.6667
I
'3333
12 4 2
I0
5 4
O
0'0238
I
•2381
.4762
- 2381
'0238
2
3
4
xo 5 5
0
0'0040
x
-0992.
2
•3968
3
4
5
-3968
.0992
.0040
II I I
O
I
0'0091
'0909
II 2 I
O
1
0.8182
•1818
II 2 2
O
•
0. 6545
.3273
2
'0182
II
3
O 0.7273
I
•2727
II 5 I
0
I
II
0. 5455
•4545
5 2
0
0'2727
I
2
O
I
2
0'4242
•4848
'0909
5 5
12
.5455
O
x
0. 2545
•5091
'1818
2
'2182
3
•.0182
II 5 3
O
0.1212
x
-2210
2
3
'441 9
'2652
4
' 044 2
5
•0013
I
'4545
0 0. 1414
*3636
3
.0606
I
2
*3394
3
4
.0646
.0020
II 5 4
•4525
O
0.0455
I
•3030
12
5 I
0
I
0.5833
2
*4545
3
•1818
4
'0152
12
6 x
12
0
I
0.5000
•5000
6
12
Ix
5 5
O
I
2
0'0130
-1623
'4329
3
4
5
.3247
-0649
•0022
O
I
2
2
0.3182
.5303
.1515
0
I
0 2273
.
'5455
' 2273
6
12
3
0 0'0909
x
.4091
2
409 1
3
.0909
12
6
4
0
0'0303
I
2
3
• 2424
.4545
. 2424
0303
4
0
I
2
6
5
0'0076
•1136
-3788
•3788
.1136
.0076
3
4
5
12 6 6
0
I
2
0'0011
•0390
3
4
5
6
'4329
'2435
.0390
.00 1 1
'2435
13 I I
O
I
0- 9231
-0769
5 3
O
0.8462
O
0'1591
I
'1538
•
'4773
•3182.
.0455
2
3
r = 0
2704
4335 51
I
.2.720
3
13 3 I
I
'2308
13 3
2
0 0. 5769
'3846
I
'0385
2
0 0.4196
x
-47zo
2
'1049
3
.0035
13 4 I
0
0'6923
x
•3077
13 4
0
I
2
2
0 . 4615
•4615
'0769
13 4 3
13 2 I
12
- 0128
2
'4167
5
•2821
2
x3 3 3
12 4 4
2
1
0 0-0265
12
12 4 3
7 = 0 0'7051
4
5
•2176
3 01 51
00
0 0.7692
xo 5
O
2 2
75
13 6 I
O
I
0'5385
O
I
2
2
0'2692
'1923
x3 6
3
O
I
2
0'1224
*4406
'3671
3
-0699
13
O
I
2
3
4
6 4
0.0490
.2937
.4406
-1958
•0210
x3 6
5
I
- 1632
3
2 '4079
I
-4699
2 '3021
3
4
13
O
I
.0503
.00i4
5 i
0.6154
3846
13 5
0
I
2
3
4
13
O
I
2
6 6
0'0041
'0734
- 3059
'4079
'1836
•0245
6 •0006
3
4
5
'0714
'3916
'1119
'0070
0. 7143
- 2857
14
O
I
2
4
0'4945
.4396
•0659
14 4 3
0
I
0.3297
'4945
2
'1648
3
•0110
14 4 4
O
I
2
0'2098
.4795
- 2697
3 •0400
4
•0010
14 5 I
x
0.9286
2
0
I
.3571
5 •0047
I
0.0979
.3916
-0907
•0027
0 0.6429
O
O
I
2
3
•0816
14
13 5 4
0'4533
'4533
4
5128
'1282
2 '2797
.0350
3
O
I
.3263
2
13 5 3
0 0.1958
-4895
I
14 3 3
3
0.3590
•
'0330
.5385
0.0163
0 0- 1762
• 3626
14 4 I
13 6
O
13 4 4
I
2
'4615
0 0'2937
-5035
I
•1888
2
•0140
2
r = 0 0. 6044
14 2 I
O
I
0.8571
•1429
14 2 2
O
I
2
0/253
'2637
'0110
14 3
O 0.7857
'2143
I
14 5 2
O
I
0.3956
2
'1099
'4945
14 5 3
O
0'2308
I
2
3
*4945
'2473
-0275
14 5 4
O
I
2
0.1259
•4196
3
4
.0899
'3596
'0050
14 5 5
O
0'0629
•
2
'3 147
'4196
3
4
'0225
5
.1798
•0005
TABLE 26. HYPERGEOMETRIC PROBABILITIES
N
R n
14
6
1
N
14
R n
7
5
N
15
R n
N
R n
N
R n
N
R n
N
4
15
6
x6
x
16
5
16
3
r= 0
0'5714
7' = 0
0'0105
r= 0
0.3626
I
-4286
x
•1224
I
. 4835
2
'3671
2
3
4
5
•3671
•1224
3
•1451
•0088
•0105
15
14
o
6 2
0.3077
I
•5275
2
.1648
14
6
3
0
14
0
7
6
•1510
I
2
14
7 7
0
0.0003
1
'3357
2
•4196
3
4
•1598
I
•0143
•0150
2
'1285
'2098
2
'4196
3
4
5
'2797
'0599
14
0030
6
6
0
0'0093
I
'1119
IS
•1285
0143
;0003
I
1
•-0667
2
I
0.8667
'1333
2
3497
3
4
'3730
'1399
I
5
'0160
15
6
*0003
0
0.7429
I
'2476
'0095
14
7
1
0
0'5000
I
•5000
14
0
7
2
2
15
2
3
2
1
0
0.8000
I
•2000
I
0.2308
-5385
15
2
'2308
0
0.6286
I
'3429
2
•0286
14
7
3
2
0.0962,
'4038
'4038
3
-0962
0
I
14
7
4
15
0
3
2
3
3
-0022
0'0350
' 2448
15
2
' 4406
' 2448
' 0350
0
j
15
1
0.6667
1
'3333
15
0
5
2
0 .4286
I
•
4762
2
'0952
15
5
3
0
0'2637
X
'4945
2
'2198
3
'0220
15
5
4
0
0'1538
I
'4396
2
'3297
3
4
. 0037
15
0
'0733
5
5
0-0839
I
'3497
2
'3996
3
4
'1499
•0167
5
'0003
15
0
I
15
0
6
1
0.6000
' 4000
6
2
0-3429
I
5143
2
' 1429
0'4835
'4352
I
3
4
3
I
2
0
5
0
x
0.9333
0
15
3569
'3569
0
15
16
'3776
'3357
'4079
. 2448
'0490
•0023
0.0280
6
I
3
4
5
6
I
6
2
'3297
' 0549
5
'0020
'0007
2
6
0
I
•0322
'0490
'2448
14
o
15
16
3
4
I
3
4
5
6
7
3
4
5
'4835
2
00699
-4196
'2398
'0450
'2418
•4615
4
2
r= 0
I
1
0.1538
6
0'0420
*2517
2
I
14
0
4
0
1
0'0023
0
3
4
0'2418
r=
5
'0791
4
I
0.7333
'2667
4
2
15
6
3
0
1
0.1846
2
*2 967
3
x5
o
0.0168
3
4
5
6
15
'1079
•0108
•0002
7
x
0
0 '5333
I
-4667
15
7
0.2667
2
'2000
7
6¢
0
0'0923
I
.3692
0
0'5238
2
x
•4190
'3956
2
'0571
3
4
•0110
'1319
2
I
0'8750
'1250
3
2
0
0•6500
I
' 3 2 50
'0250
3
3
0
05107
I
. 4179
2
•o696
3
.00'8
.0769
0
0•7500
I
.2500
•2872
4308
'2051
3
4
15
0
•0256
7 5
0•0186
I
1632
2
•3916
•3263
3
4
'0932
'0070
5
x5
0
I
7
0.0056
'0783
-2937
-3916
. 1958
2
3
4
5
6
.0336
•0014
7
7
2
0'5500
-4000
2
'0500
0
4 3
0.3929
1
'4714
2
•1286
3
•0071
4
5
5
•3231
'4154
•0002
6
x
16
I
'3750
3
4
5
'0721
x6
6
2
0
01750
I
•5000
2
'1250
16
6
3
o
0'2143
x
2
'4821
•2679
3
'0357
16
6
4
16
•Ixoi
'3304
3
4
5
6
.3671
x6
0
I
.0514
2
* 2313
.1099
3
4
' 2570
3
4
16
•0082
6
5
o
0.0577
x
•2885
2
'4121
3
4
5
.2060
0343
•0014
6
6
$
6
7
16
'3125
16
-1828
16
2
'1964
'0002
3
'01 79
0
1
'0843
.0075
•0001
5
3
0- 2946
49 11
7
8
z
•5000
16
0
8
2
0.2333
'5333
'2333
8
3
0
0'1000
I
2
'4000
'4000
3
•1000
I
0•5625
x6
x
'4375
0
7
'0001
x
0
16
•0664
•0055
0•5000
16
3
4
5
6
3855
0
'3934
'2997
.3807
-3046
.0914
•0087
7
-3709
2
•0833
7
0'0031
•3956
•0005
'4583
0009
I
2
•0264
I
'0236
0- 1154
3
2
'1573
0
4
'0305
6
I
I
0'0012
7
2
2
I
2
3
4
5
6
7
- 0048
0'0105
•1888
2
-2885
0
0•0262
5
.20'9
'4038
x
0.4583
5
0.0288
0
o
7
0
x
'2176
x
'0192
'0126
'4835
5
.1731
2
4
0•6875
4
0'0692
I
0
I
7
0
2
x6
'4500
0.6250
16
x
0.1500
0
0•2720
0
76
x6
0
16
15
4
z
x
x6
6
4
0
x6
z6
16
3
4
3
x
•0625
'3777
' 1259
i fs
2
3
2
'1875
x6
•0604
•0027
3
4
5
2
4
3
4
'2333
' 4308
'3692
7
'3375
'0083
1
0'0513
2
I
0. 123I
0
'4533
•3022
2
x
3
I
'3777
3
7
2
0•1058
0
15
0.1813
R n
r= 0
I
0
o
2
0'8125
16
r=
4
1
2
0.7583
I
2
3
0 '9375
'0625
0
x6
'5333
'4747
'0 440
x6
2
0
x
15
0
z
2
0
0'3000
z
.525o
2
* 1750
8
4
0'0385
I
•2462
2
'4308
3
4
•2462
. 0385
TABLE 26. HYPERGEOMETRIC PROBABILITIES
N R n
x6 8 5
r= o o.0128
I
'1282
2
'3590
'3590
• I282
•oiz8
3
4
5
16 8 6
O
0.0035
I
.0559
2
'2448
3
4
5
-3916
'2448
' 0559
6
'0035
x6 8 7
O
0'0007
x
•0196
.1371
'3427
'3427
' 1371
.0196
•0007
2
3
4
5
6
7
16 8 8
O
I
0'0001
'0050
2
'0609
'2437
'3807
'2437
'0609
3
4
5
6
7
8
'0050
'0001
N R n
17 3 3
r = 0 0'5353
I
O
0'9412
I
•0588
17
I
2
'0618
2
3
'0015
17 4 2
0 0. 5735
I
•3824
2
'0441
17
4 3
O 0.4206
1
'4588
2
' 1147
3
'0059
17 4 4
o 0-3004
x
-48o7
.1966
2
3
.0218
'0004
4
17
5
1
O
0'7059
I
•2941
x7 5
2
O 04853
I
'4412
2
'0735
5 3
O 0.3235
'4853
I
2
3
'1765
•0147
17
5 4
O 0.2080
I
'4622
2 2
2
'2773
O
0'7721
I
'2206
3
4
'0504
'0021
17
2 '0074
17
17
3
O 0.8235
x
-1765
17 3
2
O 0.6691
I
'3088
2
'0221
2
.4853
'1103
17 6 3
17 4 1
o 0.7647
.2353
I
2 I
O 0.8824
I
•1176
17 6
r = 0 04044
'4015
17
17 I I
N R n
5 5
O
I
0'1280
'4000
2
'3555
.1o67
'0097
3
4
5
'0002
17 6 I
o
I
6
0._47r
•3529
o 0.2426
i
'4853
2
3
N R n
N R n
17 7 6
x7 8 7
r = 0 0'0170
I
2
3
4
5
6
'1425
.3563
'3394
' 1273
'0170
•0006
'2426
.0294
17 6 4
17 7 7
0 o•oo62
.0756
I
= 0 0'0019
I
'0346
2
'1814
3
•3628
4
'3023
•1037
5
'0130
6
7 •0004
17 8 8
0 0'0004
'0118
0 0.1387
2
'2721
I
I
'4160
'3466
'3779
.2160
•0486
.0036
•000i
2 '0968
2
3
4
5
6
7
3
'0924
4 •0063
17 6 5
0
0.0747
I
2
'3200
'4000
3
4
5
' 1 778
-oz67
'0010
17 6 6
O
0'0373
x
'2240
'4000
2
•2666
3
4
•o667
5
'0053
6 •000l
17 8 x
0
I
0'5294
'4706
x7 8
0
I
2
0.2647
• 5294
2 '2059
x7 8 3
0 O'1235
I
*4235
2
'3706
3
•o824
17 8 4
17
O
I
7 I
o
0'5882
'4118
x
17 7 2
O
0'3309
I
'5147
2 '1544
17
7 3
O o.1765
•4632
I
'3088
2
.0515
3
17
7 4
O 0.0882
I
'3529
2
'3971
3
4
.14.71
'0147
2
3
4
0 0.0204
I
'1629
2 '3801
3
4
5
3
4
5
'3258
.1o18
'0090
17 8 6
o
x
o-oo68
.0814
.2851
2
'3801
3
'2036
4
5 '0407
6
'0023
7 5
O 0.0407
I
'4235
.2118
'0294
x7 8 5
17
2
0'0529
'2824
'2376
'4072
'2545
•0566
'0034
77
3
.2.903
•3628
4
'1935
5
6
'041 5
'0030
7
8 •0000
TABLE 27. RANDOM SAMPLING NUMBERS
Each digit is an independent sample from a population in which the digits o to 9 are equally likely,
that is, each has a probability of-116.
84
28
64
49
o6
75
09
73
49
64
93
39
89
77
86
95
53
03
65
76
31
97
97
o8
4!
70
44
z6
87
57
37
49
24
65
97
52
46
33
59
68
75
09
93
64
33
12
83
76
z6
29
51
45
70
14
OI
14
04
49
59
50
32
73
8z
98
27
74
00
78
6o
36
88
8o
79
98
77
12
39
04
40
71
92
23
92
z6
69
o8
73
19
16
79
86
16
99
36
17
74
41
05
o8
82
42
72
92
41
6o
25
35
05
44
8o
31
79
46
65
8z
31
96
40
50
44
68
36
78
65
96
81
75
74
o6
96
37
69
17
61
97
01
97
89
13
24
34
8o
31
81
o6
64
6z
77
58
33
57
36
65
92
02
65
63
22
50
4z
87
41
46
56
35
81
69
6o
05
88
34
29
75
98
56
77
39
73
69
96
89
II
8o
49
36
ii
19
03
51
77
22
09
30
77
03
46
65
68
93
61
54
z8
61
68
12
21
76
13
39
73
85
59
68
53
66
04
60
30
10
44
89
55
67
57
21
63
38
8o
13
77
42
76
97
95
85
98
17
46
99
85
50
92
69
51
27
44
40
92
20
40
19
72
89
73
52
12
84
74
69
25
03
59
91
02
30
76
81
6z
49
07
48
46
6z
68
12
16
98
04
61
64
27
86
93
25
00
20
29
96
75
68
48
82
20
14
20
10
73
62
73
19
92
18
85
76
06
52
68
91
86
46
70
67
68
00 76
64 85
26 32
71 57
99 51
81 14
35 II
15 17
24 78
76 49
97 56
11 76
04 47
18 85
z6 04
92 27
28 47
61 o8
89 81
zi
o6
69
53
91
87
22
19
99
97
63
55
38
31
97
56
43
15
4
6o
52
5
2
17 5
09
14
61
09
91
93
19
58
85
24
35
54
14
65
59
28
19
77
12
z8
zi
09
57
74
70
76
46
II
99
85
6o
39
22
8z
72
58
83
54
z6
21
64
40
56
72
0!
90
49
85
88
19
37
37
04
79
64
II
83
4
94
64
67
54
io
88
16
96
27
94
42
16
53
z8
78
36
92
52
12
83
97 II
s
15
3 1 -1
28
94 54
6o
39 16
16 33
33
46
6 3 37
15 89
94
15 97
16
54 32
76
86 21
25
i8 6o
48
64
6i 48
63 10
76
58 38 98
R
17
32 2 3
33
23 89
45 08
44
6o
15
84
(D
7II 44
39
98 68
io
66 69
87
95 87
65
70
5
85
83
12 33
43 24
96 56
97 63
97 17
83 00
6o 65
09 44
77 96
43 40
11 36
44 33
05 40
43 42
91 65
62 83
53 05
20 53
70 52
51 62
2 3 74
76 96
88 83
69 o8
24 6z
95 47
58 6z
35 22
35 72
22 73
91 58
76 56
87 00
6z io
22 06
84 03
83 87
00 87
76 6z
31 65
91 30
71 56
08
3
03
74 8
o
77
40
59 16
0 7 72
96 25
59 35
69 71
3! 20
66
86
09
37
07
97
91
18
92
47
2
02
20
24
22 86
73 49
00 42
27 44
23
83
37
35
37
96
25
34
88
84
00
33
35
30
61
34
35
1
61
9
70
84
83
87
67
67
22
03
17
24
14
6o
8o
42
08
57
24
58
44
33
75
83
17
29
54
05
33
89
43
12
15
78
69
78
10
02
43
10
6o
76
06
64
41
91
14
26
98
34
65
28
37
6o
92
8o
30
9z
91
30
17
41
57
41
46
II
68
77
90
69
93
73
95
32
72
64
87
68
52
52
oit
78
77
63
29
21
26
90
55
27
74
48
41
02
o6
42
87
66
09
87
o8
8
92
67
o8
93
19
72
47
89
29
02
62
99
81
21
17
68
03
12
8z
82
o6
67
40
53
19
8z
41
93
94
16
61
65
33
63
48
52
68
66
89
14
94
45
o6
21
04
07
6o
38
88
73
22
21
40
22
6z
64
03
95
45
of
34
76
52
42
63
8
67
72
79
10
36
33
04
21
o6
14
63
04
97
99
94
13
30
95
II
49
90
86
51
55
90
19
39
67
88
9
98
4
67
39
76
53
38
70
56
90
07
47
04
48
83
90
36
97
56
02
74
55
67
43
72
63
40
03
07
47
02
62
20
19
45
23
8o
99
16
89
52
85
91
00
z8
86
39
33
73
48
14
91
z6
75
58
30
54
52
of
68
33
95
70
02
00
20
41
30
44
70
72
49
7
29
33
48
47
52
88
7
46
5
36
47
21
86
6i
0
8:
89
22
85
47
95
Jo
22
76
01
23
44
65
92
15
17
55
09
83
z6
o8
62
78
39
54
55
19
57
02
z6
44
58
40
o6
59
12
04
17
69
65
31
65
58
01
85
90
74
TABLE 28. RANDOM NORMAL DEVIATES
Each number in this table is an independent sample from the normal distribution with zero mean and unit variance.
0.7691
-0.5256
0.9614
0.3003
1•1853
1.0861
P5109
0-3639
1.7218
-1.7850
0'2411
0'2614
1'0204
1'0167
PI2I1
-0•2628
0'3413
0.8185
-0-1489
0.4711
0.2836
- P7888
-0. 4654
0.7887
0.9046
-0.9189
-0.2884
0.9222
0.0989
-0.8744
-0'1051
-0'2588
-0'7164
0•1110
-0'3172
-0'7442
P9225
0.8966
-0'2911
- P4II9
- p2664
-0'4255
-0'2201
1'5256
0•6820
0 '4547
0.0213
0.8897
0.4814
-0.4014
-0.9607
P258o
1.8266
0.8452
- p4908
0.2071
-0 ' 40 70
-0'5601
-0'2685
-
-0'3797
-0.9013
1.6169
-0.6995
-0.0617
1.2541
P4531
-0.8250
0.8620
24288
0.0450
0.2532
-0.9363
-0.8698
0.4890
-0.1291
0.1939
0.2668
-0.1144
-0.0339
-0.1236
01285
-0.7122
0.5823
0.7836
0.1600
1.0383
0•1671
0.1137
0.4104
-0.3707
-0.4590
-1.1334
0.2755
0.3674
2'1522
0'2133
0'3379
0. 34-44
-0-3912
-0-5941
-0.0768
- V06 37
-0 ' 0948
0. 4198
-1•8812
0. 0778
0.8105
23696
-0.3968
1.1401
-0.1992
0.7894
-0.5056
-0.5669
0.7913
0- 9914
1.7055
-1.9071
-0-3260
0.4862
-0.8312
-1-9095
0.7738
-I-3150
P2719
I•I404
0'1964
-0'2309
0'9651
- P1403
P 6399
0.7164
1.9058
-0'5440
-1'3378
1'5939
0.7378
p4320
0.5276
-2.1245
P6638
-0.0429
-0'1320
-0'2098
-0'1563
-0'8492
-1'8692
-0'5447
0.6299
-0.1507
- P1798
1.3921
I•1049
P2474
P1397
-
P9211
0 '4393
-0'4597
0'7258
I.4880
-0.4037
-1.1855
-0.0251
-0.4311
PI002
0'1176
-0'6501
1'7248
-1'0621
0'9133
-0.456o
0.8729
0.3646
0.1885
-0-9633
0.6117
1.0033
2.1098
0.8366
P2725
1.0492
-1.1969
0.2146
-0'3594
P2193
0'2024
0'4722
-0'2230
-0'4389
- P5498
-I•5027
-16761
0. 1433
0 '7375
-0.5608
-2'6278
0.7064
-0.5104
-0.6110
0'2239
-0. 6841
0'2177
0•5802
0.0556
2'0196
-18613
0.8646
- V0438
0.2533
-0. 4953
0'8258
0.6571
0.0679
0.9970
0.6705
-0.1985
-0'0917
0•9517
0'7272
-0'2223
0.5697
0.4869
-1.3296
-0.1765
-0.2777
-0.7707
0.3990
0. 9649
-1'5953
-0.0382
-0.2558
- P5290
-0.7695
0.0153
-2.6190
-0.3052
- P2785
-0.3767
•o193
- P6919
-0.1825
0.2461
0.8169
0.0790
0.7225
0.9014
-0'4089
0.3838
0.5219
0 '3779
-1'9919
-0.1679
-0.004I
-0.3111
0.7127
0.1428
-0'3749
0.0722
-o-7849
-0'6237
1.5668
0.45 21
0.1263
0.1663
-0.2830
1.2061
-1.4135
P0313
1'8116
- P9062
2'2544
-0'6327
0. 5464
-1.4139
0.1833
-1.6417
-0'4379
-0'3937
-0.0851
-0.2088
0.2889
-0. 1720
0.3946
-0. 0909
0'7673
- P9853
p6459
0.8910
P1387
0.6764
0. 2794
0'8007
-0/296
-0.5887
0.2325
- P0242
- P4929
-0'1122
-
1.2370
0'7125
P1813
-0'8344
-0'8015
0.6259
-1.4433
-0'3211
-1.2630
0•7890
- P2187
-2.6399
-0'7494
-0.9592
- 1.1750
-1.2106
-0-7991
1.0306
-0'6252
0'4124
-
-1.5349
0 '3377
-0'7473
0.4011
-0.2193
-0'5749
-0.4263
-0.1614
0.5114
-0'3337
-0. 5094
0.2414
-0.5231
1.8868
0. 6994
0'2013
P3071
- P2255
-0•6109
-0'7522
0.4899
0.9586
-1' 2 947
0.5067
0.5021
1.3002
0'4787
-11240
- 2'0142
-0/707
0'1095
-0'0994
1'0811
P5762
0/726
-0'8558
1'3333
-0.1512
-0-0416
P1621
-o.2743
v7767
- p8o3I
2.1364
0.1295
0. 2454
1.8961
0.3863
-0.0943
-0.7132
0.248,
V1558
-0-0814
-0'7431
P4721
0.5274
-0'3755
-0-3730
0.1049
-0.1819
I.2704
0.6109
-0.8716
0.36o2
1'3133
0.4986
-12309
0.2453
-i•1675
0•1115
0.6226
-0'2521
-0•5518
-11584
79
-0.6933
-
-1.1761
-0.5156
-0.5671
P2094
1'0500
0'6314
P5742
0.6715
-0'7459
0. 1543
-0.1108
0.5970
0.0842
- P5805
-0. 2542
-0-4762
0.9149
-0.8178
-0.5567
0. 0945
0.7350
- P6734
-1.8237
-1'0210
P6674
-0.8427
0.0398
0.5787
1'5481
0.2236
-0.8393
-0.2523
1.4846
0.8527
0 '5541
- 0 • 7841
-0'9212
-2'4283
0'0046
-0.3490
-0.6525
-10642
-i.o6o6
0 '4431
0 '9379
3'4347
0.7703
-0.6135
P3934
P0013
0'8129
-0.4194
-0.0173
0.5664
-0.7216
-0'3349
0.8688
I.41 I0
-
P40II
0'6272
P2943
P4732
2'1097
0'5529
-0'1047
0.6484
0.9714
0.0982
-0.6845
P6357
P2928
0.1785
-0.2936
-1.1208
0.7254
-1•1351
P4302
0.6226
0.6017
1.1673
-0.0962
-11846
- P4375
po786
0 '5354
0.6161
0.6177
-0.4680
0.5638
0.3650
-0.1867
1.0176
-0.6125
-0.0772
0.9166
p4564
-0.2898
1.0821
1.1889
p5315
-0.7601
-0.2105
-0.6962
0 '4598
- P7724
0.5089
P3761
TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL
PARAMETER
If r is an observation from a binomial distribution
(Table 1) of known index n and unknown parameter p,
then, for an assigned probability C per cent, the pair of
entries gives a C per cent Bayesian confidence interval
for p. That is, there is C per cent probability that p lies
between the values given. The intervals are the shortest
possible, compatible with the requirement on probability.
The tabulation is restricted to r < Zn. If r > Zn replace r
by n - r and take 1 minus the tabulated entries, in reverse
order.
Example 1. r = 7,n = 12. Use n = 12 and r = 5 in the
Table, which at a confidence level of 95 per cent gives
0.1856 and 0•6768, yielding the interval 0.3232 to 0-8144.
The intervals have been calculated using the reference
prior which is uniform over the entire range (0,1) of p.
The entries can be used for any beta prior with density
proportional to pa(1-p)b , where a and b are non-negative
integers, by replacing r with r + a and n with n + a + b.
If r + a is outside the tabulated range, replace r + a with
n - r + b and n with n + a+ b, and take 1 minus the entries,
in reverse order.
Example 2. r = 7, n = 12. If the prior has a = 2, b = 1,
then r +a = 9, n+ a + b = 15 and n - r + b = 6. Use n = 15
posterior
probability
density
of p
,
p
(This shape applies only when 0 < r < n. When r = 0 or
r = n, the intervals are one-sided.)
and r = 6 in the Table, which at a confidence level of
95 per cent gives 0.1909 and 0.6381, yielding the interval
0.3619 to 0.8091.
When n exceeds 30, C per cent limits for p are given
approximately by
± x(P)[i(1 - P)/n]1
where /5 = r/n, P = 1(100 C) and x(P) is the P percentage point of the normal distribution (Table 5).
-
CONFIDENCE LEVEL PER CENT
90
n =1
r=0
95
99
99'9
0.0000
0.6838
0.0000
0.7764
0.0000
0.9000
0.0000
0•9684
0'0000
0'5358
0'0000
0'6316
0. 0000
0'7846
0•0000
0'9000
'1354
.8646
.0943
.9057
-0414
'9586
•0130
•9870
0.0000
0'4377
•7122
0.0000
-0438
0.5271
•7723
0.0000
.0159
0•6838
•8668
0.0000
-0037
0'8222
'0679
r=0
0. 0000
0. 4507
0.0000
0. 6019
0•0000
0. 7488
•0425
•1893
0•3690
•6048
0.0000
I
•0260
•6701
•0083
•7820
•0016
'8788
•8107
•1466
'8534
•0828
•9172
-0375
.9625
0.0000
0. 3187
0.0000
0. 3930
0'0000
•0302
•5253
-0178
•5906
•0052
05358
•7083
0.0000
.0009
0•6838
•8186
•I380
•7
•I 048
•7613
*0567
'8441
•0242
.9133
n
=2
r =0
n
=3
=
n=
0
4
2
n
'9377
=5
=0
2
8o
TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL
PARAMETER
CONFIDENCE LEVEL PER CENT
90
r=0
99
95
n=6
'4641
0.0000
'0133
•22 53
•6317
'7747
•0805
'1841
•5273
-6846
•8159
r= 0
0•0000
0'2505
0.0000
0'3123
0•0000
0 '4377
I
•0185
•0105
•065o
•1488
'4759
•6210
'7459
•0028
.0331
.0934
.5913
•7174
.8227
0.0000
•0004
•0129
.0495
0.5783
•7113
•8115
'8912
0. 0000
I
n=
2
'02 31
'1076
3
0.2803
99'9
0'3482
0'0000
0'4821
0'0000
o'6272
•0037
•0421
•1177
•6452
•7769
•8823
-0006
•0171
•0639
•7625
•8616
•9361
7
2
•0878
3
•1839
'4155
•5677
•7008
r= 0
0'0000
0'2257
0'0000
0'2831
0'0000
0'4005
0.0000
0'5358
I
•0154
-0739
'1549
•3761
'5152
.6388
•0086
'0542
•1245
'4334
•5676
•6854
.0022
-0271
•0769
'5451
'2514
'7486
'2120
'7880
•1461
•6651
•7679
'8539
•0003
-0103
•0400
.0884
•6651
'7645
•8463
•9116
r=0
0.0000
0'2057
0.0000
0.2589
0.0000
0•3690
I
'01 32
•0638
'1337
'3435
'4714
•5863
'0073
•0464
•io68
'3978
•5224
'6332
•0018
•0229
'0652
•5053
•6192
'7184
'2165
•6901
•1816
'7316
'1237
'8039
0.0000
•0003
•0085
'0333
'0739
0.4988
'62 37
•7212
•8032
•8714
0'1889
0•0000
0'2384
0.0000
3
4
'1175
•1899
•0063
•0406
.0934
•1586
•3675
'4837
•5880
•6818
•0016
'0197
•0564
.1071
0'0000
'0002
0'4663
•3160
'4344
'5416
.6393
0•3421
'4706
2
0'0000
'0115
•0560
'5788
-6741
'7578
•0072
•0284
•0632
-5866
•6817
•7627
•8320
5
0'2712
0'7288
0.2338
0'7662
0'1693
0'8307
0'1100
0'8900
r=0
0'0000
•0102
0.1746
•2926
0'0000
I
2
'0499
3
4
•1047
•1691
5
0'2411
=8
n
2
3
4
n
=9
2
3
4
n
= Io
r=0
I
n=
II
0'2209
0'0000
01187
0.0000
.4027
.5030
'5951
-0055
'0360
•0829
'1407
'3415
'4502
'5485
'6377
.0013
•0173
'0497
.0943
'4402
'5431
'6344
'7156
•0002
•0062
.0248
'0551
0'4377
'5534
-6456
•7252
'7943
0.6803
0'2070
0'7191
0'1488
0'7878
0'0958
0'8539
See page 8o for explanation of the use of this table.
81
TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL
PARAMETER
CONFIDENCE LEVEL PER CENT
90
99
95
99'9
rt = 12
r
=0
0•0000
I
•0091
2
'0449
3
4
'0944
•1524
•5564
5
6
0.2169
•2870
I
0.1623
•2724
0.0000
•0049
0.2058
•3188
0.0000
•0012
'3753
'4695
•0323
'0745
•1263
•4210
'5138
•5987
•0154
'0443
•0841
0'6374
'7130
0.1856
.2513
0•6768
'7487
0.0000
•0082
0.1517
•2548
0.0000
•0044
2
-0409
3
4
•1386
'3514
'4400
•5223
•2604
0.0000
0.2983
0.0000
0.4122
'4134
•5113
'5987
•6773
•0002
•0055
.0219
•0488
•5234
•6128
•6905
'7588
0.1326
•1887
0 '7479
•8113
0.0848
•1290
0.8188
•8710
0.1926
•2990
0.0000
. 001 I
0.2803
'3896
0.0000
•0001
0.3895
•0293
•0676
'1146
'3953
•4832
'5639
•0139
•0400
'0759
•4829
•5666
'6424
-0049
•0196
'0437
0'5994
•6717
0-1682
0.6388
0.1195
•7082
•1698
0.7112
'7738
0.0759
'11 54
0'7855
•2274
n = 13
r
=0
5
6
n = 14
r=0
I
.o859
0.1971
'4963
•5828
'6585
'7257
'8384
0'1423
0'0000
0.1810
0.0000
0.2644
0.0000
0.3691
•0075
'2394
.0375
•0788
.3303
'4140
.0040
-0267
•0619
•1271
.4921
'1048
•2814
-3726
'4559
'5329
.0009
-0126
'0364
•0691
•3684
'4574
'5376
•6106
-0001
'0044
•0177
'0396
•4718
'5554
•6290
'6948
o•i8o6
0-1537
'2075
•2659
0.6045
'6715
'7341
0.1087
'1542
'20 51
0.6775
•7388
'7949
0.0687
•1043
'1457
0 '7539
-3000
0•5654
'6346
•7000
r=0
0'0000
0'1340
0.0000
•0009
0.3506
•2257
0.1708
•2658
0.0000
•0069
0.0000
'0037
0.2505
i
2
-0346
-3116
3
4
•0728
'3909
'1173
'4650
•0246
-0570
'0966
'3522
'4315
'5049
'0115
.0334
'0634
'3493
'4344
.5113
•5817
•0001
'0040
•0162
•0361
'4495
'5303
•6017
•6660
0.1666
0'5349
•6012
•6641
0.1415
0'5736
•6381
0'0997
.1413
0.6465
0.0627
. 1909
•7063
.0951
'2442
•6988
•1876
-7615
•1327
0.7242
'7770
•8246
2
3
4
5
6
7
'2383
•8070
'8543
n = 15
5
6
7
•2197
•2762
See page 8o for explanation of the use of this table.
82
TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL
PARAMETER
CONFIDENCE LEVEL PER CENT
90
99
95
99'9
n = 16
=
0
0.0000
0.1267
0.0000
0.1616
0.0000
0.2373
•oo64
•2135
2
•0321
•2949
3
4
•0676
.1090
.3703
•0034
•0228
•0528
'4408
•o895
•2518
'3340
.4095
'4797
•0008
•oio6
•0308
'0585
•3320
'4135
'4874
'5552
0.0000
•0001
.0037
•0148
.0332
0'3339
i
5
6
7
8
0.1546
0.5075
0.1311
-2037
'2558
•3108
'5710
0'5455
•6076
•6314
•6892
•1767
•2258
•2781
•7219
o•o920
•1303
•1728
-2193
o•618o
•6762
.7304
•7807
0.0576
•0873
•1217
•i6o6
0•6964
•7486
•7962
'8394
It = 17
r= 0
0.0000
0.1201
0.0000
i
•006o
-0032
•0492
.0834
0•1533
.2393
.3175
•3897
'4568
o•0000
•0007
'0099
•0286
'0544
0.2257
•3164
'3945
'4656
.5310
o•0000
•0001
.0034
'01 37
.0307
0.3187
•4105
•4860
'5533
'61 44
0.5200
'5798
•6366
•6905
0.0854
•1209
•i6oi
•2030
0.5917
•6483
.7013
'7508
0.0533
•0807
•1124
•1481
0.6703
•7217
-7690
-8124
0.2152
•3021
0.0000
•0001
0.3048
-0031
•0128
•0286
.3934
'4665
.5317
'5912
r
2
•0300
3
4
•0631
•1017
•2025
.2799
•3516
'4189
5
6
7
8
0.1442
0.4827
0.1221
'5435
•0213
•6664
'4292
.5073
.5766
.6393
•1899
•2383
•2893
•6o17
'6574
•1644
-2099
-2583
0.1141
•1926
•2663
0.0000
•0030
. 0199
0.1459
0.0000
2
0.0000
•0056
•0281
•2279
'3026
•0007
•0092
3
4
'0592
'0953
'3348
'3991
.0461
•0781
'3716
-4360
-0267
•0508
'3771
*4455
•5086
5
6
7
8
9
0.1351
0.4602
0.1142
0.4967
0 '0797
0'5674
•1778
.2230
-2705
•3201
•5185
'5745
-1537
•1962
•6282
•2412
•6799
•2886
'5543
•6092
-6615
. 7114
•1127
'1492
•1889
•2316
•6224
'6741
•7228
•7684
0.0496
'0750
.1044
'1374
•1738
0.6459
'6964
'7432
-7864
•8262
r= 0
0.0000
0• I087
0'0000
0'1391
I
-0052
•1836
•0028
•2175
2
•0264
'2540
•0187
•2890
'3551
•4170
0•0000
•0006
•0086
'0251
.0476
0.2057
•2891
•3612
•4271
•4880
o.0000
•0001
•0029
-0119
•0267
0.2920
'3776
'4484
•5117
-5696
0'4754
0.0747
.1056
'1397
0'5449
'5984
•6488
0.0463
•0701
'0974
o•6231
•6726
'7187
•6964
•1281
•7615
'7413
-1619
-8013
n = 18
r=0
I
n = 19
3
4
'0557
-0897
'3194
•3810
.0433
.0734
5
6
7
8
9
0•1271
•1672
•2096
•2540
.3004
0.4396
. 4957
'5495
•6014
•6514
0.1073
.1444
'1841
'5309
•5839
'2261
•6346
•1766
•2704
•6832
•2164
See page 8o for explanation of the use of this table.
83
TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL
PARAMETER
CONFIDENCE LEVEL PER CENT
90
95
99
99'9
n = 20
r=0
0.0000
0. 1039
0.0000
0.1329
0.0000
0'1969
0'0000
0.2803
I
•0049
'1 754
•0026
2
'0249
•0526
•2428
'3055
•0176
'3645
.0409
'0692
•0006
•oo8i
•0236
'0448
•2771
•3466
•4101
'4690
•000i
•0027
•0112
•0251
-3630
'4315
-o847
•2080
•2766
•3401
'3995
'5494
5
6
7
8
9
0'1200
0•4208
0'1012
0 '4557
•I578
.1 977
•2395
-2828
'4747
•5266
.5767
.6253
.1361
•1734
•2129
'2544
'5093
5606
•6097
•6569
0.0703
'0993
•1312
•1659
'2030
0.5241
.576o
•6253
•6717
•7158
0.0435
•0657
.0913
•1199
.1514
0.6016
•6501
'6955
'7379
'7775
io
0'3281
0.6719
0'2978
0•7022
0.2425
0'7575
0.1856
0•8144
0
0.0000
0.0994
0.0000
0.1273
I
-0047
•0025
•0167
'1994
'2652
0.0000
•0006
-3262
'3834
•0223
'0423
0.1889
-2661
'3331
'3944
'4514
0.0000
•0001
•0026
'0105
•0236
0.2695
'3494
. 4159
'4757
•5306
0.0664
'0937
•1238
•1563
'1912
0.5048
'5552
•6030
•6485
'6917
0.0409
•0619
•0859
•1128
.1423
0.5815
-6290
'6735
•7153
'7546
3
4
'4931
n = 21
r=
n=
2
•0236
•1679
'2325
3
4
•0498
•0802
•2926
'3494
•0387
.0655
5
6
7
8
9
0.1136
' 1493
' 1870
0.4035
'4554
0.0957
'1287
'1639
I0
•0076
•2265
'5055
'5539
•2675
•6007
•2402
0.4376
'4894
'5389
•5866
•6324
0.3099
0.6461
0.2809
0•6766
0.2281
0.7328
0.1742
0.7914
0
0.0000
•0044
0'0000
'0005
•0224
'0473
•0762
•2809
'3354
•0621
'2547
'3134
-3686
-0072
3
4
0'0000
'0023
•0158
0367
0'1221
'1914
2
0'0953
•1611
'2231
0.1815
I
'2557
•3206
'3798
'4350
0-0000
•0001
•0024
.0099
•0223
0.2594
'3369
•4014
'4594
•5129
5
6
7
8
9
0.1079
0.3875
0.0908
0.4209
0.0628
-1418
•1 775
'4376
'4859
'4709
.5189
•5651
•6096
•0887
•1171
'1478
•1807
0.4868
'5358
-5823
-6267
-6690
0.0387
•0584
•0811
•1064
-1341
0.5626
•6091
-6528
'6939
•7328
I0
0.7094
'7479
0-1641
'1963
0.7693
•8037
•2011
22
r=
ii
'0211
-0401
•2148
'5327
-2536
-5781
-1220
•1554
'1905
-2274
0.2937
'3351
0.6221
0.2659
0'6526
0.2154
'6649
'3059
'6941
'2521
See page 8o for explanation of the use of this table.
84
TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL
PARAMETER
CONFIDENCE LEVEL PER CENT
90
II =
95
99
99'9
23
r=0
0.0000
I
•0042
2
•0214
0.0915
. 1547
•2144
0.0000
•0022
.0150
0.1173
•1840
•2450
3
4
.0451
-0726
•2700
.0349
3016
0.0000
•0005
•0069
•0200
0.1746
•2465
•3089
•3663
. 3225
.0591
'3548
.0380
*4197
0.0000
•0001
•0023
•0094
•0211
5
6
7
8
9
0.1027
0 '3728
0.0864
0.4053
0.0597
0.4700
0.0367
0'5448
-1349
-1689
•2043
'4211
*4678
•1160
•1475
'4537
•5005
•5131
•1810
.5450
'2411
'5570
•2160
•5883
'0842
-II 11
•1402
•1712
.5176
•5629
•6062
•6476
.0554
•0768
•1007
•1269
•5903
•6331
.6736
.7119
I0
0.6872
•7251
0.1551
•1854
0.7481
0.2791
0.5998
0.2524
0.6302
0.2041
ii
'3183
•6413
•2902
•6707
•2386
/1 = 24
r=0
0'woo
0•0880
0.0000
0.1129
0.0000
I
•0040
•0021
•0143
•1772
•0005
0.1682
'2377
0.0000
•0001
li
0.2505
•3252
•3878
*4442
'4963
•7824
2
•0204
•1489
•2063
•2360
•0065
•2981
•0022
3
4
'0430
•0692
'2599
•3106
.0333
•2906
•0191
'0090
•0564
•3420
•0362
'3537
'4055
0.2414
•3142
'3753
'4300
•0201
•4807
5
6
7
8
9
0.0980
0.3591
0.0824
0'3909
'4377
0.0568
'0799
0.5280
'5725
'6145
'6543
•6920
I()
II
•4828
•1057
'5447
'5374
'1724
'2056
.5263
•5684
'1333
•1627
'5869
•6274
0'0348
•0526
'0729
'0956
•1203
0.2659
0.5789
0.2402
12
•3031
'3414
-6193
•6586
•2760
•3131
0.6092
•6487
-6869
0.1938
-2265
•2607
0•6662
'7035
'7393
0.1471
'1756
•2060
0.7279
•7618
'7940
= 25
r=0
0'0000
0'0848
0.0000
0'1088
0'0000
0' I 623
0.0000
0'2333
'1435
•0020
•1708
•0004
'0195
'1988
•2276
•0062
•0411
•2505
•0318
•2804
•0182
-0662
'2995
'0539
'3301
' 0346
•0001
•0021
•oo85
'0191
•3040
•0137
'2295
•2880
'3419
'3921
5
6
7
8
9
0'0937
0'3464
0.0787
0 '3775
0'0542
0'4395
0.0332
•4228
•0764
•4846
•0501
•ioo8
•1271
'1551
•5276
-5688
-6084
•0694
•0910
-1145
0.5122
'5557
'5969
•6360
•6731
0•1846
•2156
•2480
0.6464
•6830
•7182
0.1398
•1668
'1955
0.7085
•7421
'7741
I
2
3
4
•1287
•1610
'1 948
'2297
'0038
•4058
•4510
'4948
•1106
•1407
-1230
'1 539
-1861
'3916
•1056
'4353
'4778
'1344
'1646
•2194
•5191
•1962
'4665
•5088
'5497
I0
0.2538
0.5594
II
.2893
12
'3257
'5986
.6369
0.2291
•2632
•2983
0.5894
•6279
•6654
0'4543
'5011
See page 8o for explanation of the use of this table.
85
•3631
•4166
•4660
TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL
PARAMETER
CONFIDENCE LEVEL PER CENT
n
90
= 26
r=
99
95
99.9
0
0'0000
0.0817
0.0000
0.1050
o•0000
0.1568
I
-0037
•1384
•0019
•1649
'0004
'2219
•00005
'2944
2
•0187
3
4
.0394
•0634
•1919
•2418
•2892
.0131
•0305
•0516
-2198
•2709
'3190
•0059
.0174
.0331
•2786
•3308
'3796
•0020
•008,
•0183
.3519
'4040
•4522
5
6
7
8
9
o•o898
'1179
"474
0'3345
0.0753
•1011
•1286
0 '4973
'1575
'1877
'4696
•5115
.5517
•2100
'4618
'5019
o•o518
'0731
•0963
.1214
0'0317
'1781
0•3649
•4089
•4513
'4924
0'4257
•3783
•4206
'5322
•1481
'5904
'0478
•o663
•0868
•1091
'5399
•5802
-6185
-6551
10
0•2429
0'5411
0'2190
0'5708
0•1762
o'6276
0.1332
II
'2767
12
-3114
'3470
'5793
•6166
'6530
'2515
•2850
•3195
•6084
•6450
•6805
•2057
•2365
-2686
•6635
•6981
•7314
-1589
-1861
•2148
0.6899
•7232
'7549
•7852
o•0000
•oo18
•0126
'0292
'0495
0.1015
'1594
-2125
•2620
•3086
o•0000
•0004
•0057
•0167
•0317
0.1517
•2148
•2698
•3205
'3679
o•0000
•00005
•0019
'0078
'0175
0.2186
'2854
*3414
'3921
'4391
0'3531
0'0496
0'4127
0.0303
0'4832
-0700
'0922
'1162
.1417
'4554
'4963
'5355
'5733
'0457
•0634
•0829
'1043
'5248
'5643
•6019
'6379
13
0.0000
0.2257
I 1 = 27
r=0
2
•0035
•0179
3
4
'0378
•0609
0.0789
'1337
•1854
'2337
.2796
5
6
7
8
9
0.0861
0.3235
'1131
'1414
•1708
'3658
•4069
0'0723
*0970
' 12 33
-2014
'4469
'4859
'1509
•1798
'3958
•4370
'4770
'5157
10
0.2328
0.5239
0.2098
0'5534
0.1685
0.6098
o.1272
0.6722
•2408
'5900
•1966
'6450
'1516
'7051
-2260
-2565
•6790
•7118
'1775
•2048
'7365
•7665
o•0000
•0004
0•1468
.208 I
0.0000
0.2119
'2769
I
II
0.0000
•2651
•5611
12
'2983
'5974
13
•3323
•6330
•2728
.3057
•6257
•6605
28
r=0
0.0000
0.0763
o•0000
0.0981
I
•0034
'1293
'1793
'0018
'1543
•2057
ii =
2
•0172
3
4
•0364
5
6
7
8
9
0.0828
•1087
-1358
•1641
'1934
'o585
'00004
'0055
'2615
•0018
.0475
'2537
'2989
•o16o
'0304
•3107
'3569
•0075
•0168
'3314
•3809
•4268
0'3131
0'0694
0'3421
0'0476
0'4005
0'0290
0'4698
'3542
'3941
'4329
'4708
'0931
•1184
'1449
'1724
.3836
•4236
•4624
'5004
•o672
•o885
•1114
•1358
'4420
•4819
•5203
'5572
'0438
•0607
'0794
'0998
•51o6
'5493
•5862
•6215
.226 I
'2705
•0121
•0281
See page 8o for explanation of the use of this table.
86
TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL
PARAMETER
CONFIDENCE LEVEL PER CENT
90
95
99
99.9
n=28
I' = IO
0.2235
0.5078
0'2013
'2 545
•2863
•3188
'5439
'5794
'6140
.2310
'2615
-2930
'3520
'6480
'3253
r=0
0.0000
I
0'0739
•1253
0'0000
'0017
"737
•0116
3
4
•0032
•0166
•0350
•0564
5
6
7
8
9
10
II
12
13
14
0'1615
0'5929
0.1217
0.6553
.5727
•6075
.6415
.6747
-1884
•2164
.2455
•2757
-6274
•6608
•6931
'7243
'1450
•1697
'1957
.2229
.6877
•7188
'7486
'7771
0'0950
0'0000
0.1423
0'0000
0.2057
•1494
•1993
•0004
'0052
•2018
'2 537
0.5369
n=29
2
II
•00004
•0017
-0072
-016i
•2689
•3220
'3703
'4151
'2190
•0270
•2458
'0154
'3016
•2621
-0458
•2898
•0292
'3465
0.0797
•1046
•1307
.1579
•186o
0.3034
0.0668
'0896
0.3317
•3720
0.0458
•0645
0.3890
'4295
0.0279
'0421
•1138
•4110
'0850
'4684
'0582
'5349
'4197
•4566
.1393
.1659
'4488
'4855
•1070
.1304
-5058
'5419
'0762
'0957
'5711
.6058
0'2150
0.4926
'5278
0.1935
0-5213
o-155o
0.5769
0.1167
0.6391
'5562
•1807
'6107
'1390
'6710
'5623
'2219
'2512
'5903
'6435
'6752
•7060
•1626
'1873
•2133
•7017
'7312
'7595
0'0000
0.1997
'3433
'3820
0'4572
'4970
12
'2 447
'2752
13
14
•3063
'3382
•5962
'6293
•2814
'3123
'6236
•6560
•2075
'2 354
•2642
0'0000
0.1380
•0004
-0050
.1958
•2463
•00004
-0017
•0338
'0543
0'0000
'0016
'0112
'0260
'1449
3
4
0'0716
'1213
'1683
•2123
0'0921
2
0.0000
•0031
•0160
•2385
'0148
'2929
•0069
•3602
'2541
'0441
•2812
•0281
-3366
-0155
•4040
5
6
7
8
9
0.0768
-ioo8
•1260
0.2942
'3330
0.3219
•3612
'3991
'4359
'4717
0.0441
•0621
•0818
•1030
'1254
0'3780
•4176
'4555
'4921
'5274
0.0268
-0404
'o56o
•0732
'0919
0'4452
n= 3o
r=0
r
'1933
•2613
•3131
.1522
'3707
'4073
'1792
'4432
0.0644
•0863
'1097
.1342
'1597
10
0.2071
0'4782
0'1862
0.5066
0'1490
'2 357
12
'2649
13
14
'2949
•3254
'5126
'5462
'5793
-6116
'2136
'2 417
•2706
-3002
'5407
'5740
-6065
'6383
'1737
-1994
•2261
'2536
0.5616
'5948
•6270
•6582
•6885
0.1120
'1334
•156o
•1797
'2045
0•6236
II
15
0'3566
0•6434
0.3306
0•6694
0.2821
0.7179
0.2304
0.7696
See page 8o for explanation of the use of this table.
87
•4842
-5213
-5568
'5909
-6551
•6854
•7146
'7426
TABLE 30. BAYESIAN CONFIDENCE LIMITS FOR A POISSON MEAN
If xi , x2,... ,
is a random sample of size n from a
Poisson distribution (Table 2) of unknown mean kt, and
r = xi, then, for an assigned probability C per cent,
the pair of entries when divided by n gives a C per cent
Bayesian confidence interval for lt. That is, there is C
per cent probability that /2 lies between the values given.
The intervals are the shortest possible, compatible with
the requirement on probability.
Example. r = 30, n = 10. With a confidence level of 95
per cent, the Table at r = 30 gives 19.66 and 40.91. On
division by n = 10, the required interval is 1.966 to 4-091.
The intervals have been calculated using the reference
prior with density proportional to 12-1, and the posterior
density is such that nit= 1A, (Table 8). The entries can
be used for any gamma prior with density
posterior
probability
density
of p
0
(This shape applies only when r > 2. When r 1, the
intervals are one-sided.)
When r exceeds 45, C per cent limits for it are given
approximately by
exp(-mµ)µs-i ms /(s - 1)!,
r
n
r2
x(P)-
where m and s are non-negative integers, by replacing n
with m + n and r with r + s. No limits are available in the where P = z (100 - C) and x(P) is the P percentage point
of the normal distribution (Table 5).
extreme case r = 0.
CONFIDENCE LEVEL PER CENT
90
99
95
99.9
r =1
0.000
2. 303
0.000
2
0.084
0'441
0.937
3'932
5'479
6.946
0-042
0.304
0.713
2.996
4'765
6.40i
7'948
0.000
0.009
0.132
0.393
4'605
6.638
8 '451
10.15
0.176
I'509
1•207
1.758
2-350
2.974
3.623
9'430
10•86
12.26
13-63
14'98
0.749
2-785
3'46 7
4'171
8.355
9.723
11.06
12.37
13.66
11.77
13.33
14-84
16.32
17.77
0.399
0•691
1.040
1.433
I.862
19.83
21.39
13
14
4'893
5.629
6.378
7.138
7.908
4'94
16.20
1 7. 45
18.69
19.91
4'292
4'979
5-681
6.395
7.122
16.30
17.61
18.91
20.19
19.19
20.60
21.98
23-35
24.71
2.323
2-811
3.321
3.852
4.401
22.93
2 4'44
25.92
27.39
28.84
15
i6
17
i8
19
8.686
9'472
10.26
I1.06
1I.87
21'14
22.35
23.55
2 4'75
25-95
20
21
22
12.68
1 3'49
14.31
15.14
15.96
27.14
28.32
29.5o
30.68
31.85
11.66
12 '44
13.22
14.01
14.81
3
4
5
6
7
8
9
10
II
12
23
24
2.129
P172
1.646
2.158
2.702
3.272
0.000
0.001
0.042
6.908
9.233
1 P24
13• II
14.88
16.58
18.22
2146
3.864
4.476
5.104
5'746
7.858
22.73
6'402
26.05
8.603
9'355
10-12
10.89
23.98
25.23
27.69
7.069
7'747
8.434
9.131
27.38
28.7o
30.01
31.32
4'96 5
5'545
6.137
6'741
7'356
30.27
31.69
33.10
34'50
35-88
28.92
30.14
31•35
32.56
33'77
9.835
10.55
1 P27
11.99
12.72
32.61
33'90
35-18
36 - 45
37'72
7.981
8.616
9-259
9910
10.57
37-25
38.62
39'97
4P32
42.66
26'46
88
TABLE 30. BAYESIAN CONFIDENCE LIMITS FOR A POISSON MEAN
CONFIDENCE LEVEL PER CENT
90
95
r = 25
16.8o
33.02
15.61
26
27
28
17-63
18 '47
19.31
29
20.15
34.18
35'35
36.50
37-66
16.41
17.22
18•03
18.84
30
2P00
2P85
31
32
33
34
99
34'97
36.16
37'35
38'54
39'73
99.9
38.98
40'24
41'49
42'74
43'98
13'46
14.20
14'95
15.70
16'46
11 . 24
i1.91
12-59
13.27
13.96
44'00
45'32
46'64
47'96
49'27
38.81
19'66
40.91
17.22
45' 22
14.66
50.57
39'96
41. II
42.26
43'40
20. 48
22'13
22'96
42 .09
43'27
44'44
45' 61
17.98
18'75
1 9'52
20'30
46'45
47' 68
48'91
50.14
15.36
16•06
16.78
1 7'49
51.87
53.16
54'45
55'74
26'13
44'54
45' 68
46'78
47'94
49' 11
50.27
51'43
2P08
2186
22.65
2 3'43
24'23
51.36
52'57
53'79
55.00
18.21
18-93
19.66
20.39
57.02
58.30
59'57
6o'84
56.21
21'12
62•10
52'58
53'74
54'89
25'02
25'82
26.62
57'41
58.62
59.82
21.86
22.60
23.35
63'37
64.63
65.88
22.70
2 3'55
2 4'41
21•31
35
36
37
38
39
26.99
46.82
27.86
28'72
47'95
49'09
23.79
24'63
25'46
26.30
2 7'14
50.22
51.35
52'48
27'98
28.83
29.68
2 5'27
40
29'59
41
42
30.46
31'33
43
44
32'20
53•60
30'52
56'04
33.08
54'73
31-37
57.19
2 7'42
28.22
61•02
62'21
24'09
2 4'84
68.38
45
33'95
55'85
32.23
58'34
29'03
63'41
25.59
69.63
67'13
TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A
MULTIPLE CORRELATION COEFFICIENT
For a normal distribution of (k + 1) quantities, let p2
be the square of the true multiple correlation coefficient
between the first quantity and the remaining k (sometimes called 'explanatory variables'). p2 is the proportion
of the variance of the first quantity that is accounted for
by the remaining k. If R2 denotes the square of the corresponding sample multiple correlation coefficient from a
random sample of size n (n > k + 1), then, for an assigned
probability C per cent, the pair of entries gives a C per
cent Bayesian confidence interval for p2. That is, there
is C per cent probability that p2 lies between the values
given. The entries have been calculated using a reference
prior which is uniform over the entire range (0, 1) of p2.
The intervals are the shortest possible, compatible with
the requirement on probability. When R2 = 1, both the
upper and lower limits may be taken to be 1.
Interpolation in n and R2 will often be needed. When
n is large, C per cent limits for p2are given approximately
by
R2 ± 2x(P)(1 - R2)(R2I n) 1
posterior
probability
density
of p2
(In some cases this shape does not apply, and the intervals
are one-sided.)
where P = (100
C) and x(P) is the P percentage
point of the normal distribution (Table 5). More accurate
upper limits are found by harmonic interpolation (see
page 96) in the function f (n) = Vh(U (n)
R2), where
U (n) is the upper limit for sample size n. For the lower
limit, L(n), use the function f (n) =
L(n)); in each
case f (oo) = 2x(P)(1 - R2) f122.
89
-
-
TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A
MULTIPLE CORRELATION COEFFICIENT
k
I
CONFIDENCE LEVEL PER CENT
90
95
n =3
R2 = o•oo
0. 0000
•I 0
'20
•30
'0000
'0000
•0000
'40
99
0.7764
•7882
0.0000
•0000
99'9
0•9000
•9060
0•6838
•6985
0.0000
•0000
0.0000
•0000
'0000
•8004
'0000
'9122
•0000
•0000
•8132
•0000
•0000
•7140
'7305
'7482
'0000
•8269
'0000
•9186
•9253
•0000
•0000
0.9683
-9704
•9725
'9746
•9768
0'50
0.0000
0.7676
0-0000
•6o
-0000
'0000
'0000
•0000
0•8416
•8579
0.0000
•0000
0.9325
-9402
'8764
'8987
•0000
'0000
0.0000
•0000
•0000
•0000
-0000
0'9793
•9817
•9845
-9878
-9919
•70
'0000
•8o
-0391
'7892
'8142
'8810
•90
' 11 55
.9644
0512
•9672
•0000
'9592
•9724
0-95
0.1525
0.9887
0.0767
0.9898
0.0118
0.9905
0.0000
0.9948
R2= woo
0.0000
0'3421
• I0
'0000
0'0000
'0000
0'5671
'6621
0'0000
•0000
0'7152
'7864
•0000
'5938
•0000
•0000
'4481
'5202
'5809
0'4200
'5264
•20
•30
0'0000
'0000
'0000
'0000
'6491
'0000
'40
'0538
.6754
.0170
'7139
•0000
-7163
•7591
•7968
•0000
•0000
•0000
•8238
•8525
-8772
0.50
0.1167
•6o
•1950
'2959
'4328
0•7864
•8466
•8982
•6357
'9413
'9756
0.8347
•8877
•9287
•9608
'90
'3471
'5563
0.0029
•0384
-0931
•1867
.3781
'9847
0.0000
•0000
-0124
-0531
'1720
0.8996
•9206
'70
o•o666
•1310
•2189
'8o
0.7518
-8183
'8770
'9275
'9690
0.95
0.7846
0.9860
0.7259
0.9892
0'5735
0'9936
0'3448
0'9964
n = 25
R2 = woo
0.0000
'9489
n = HI
'9491
'9744
'9909
0-1623
0.0000
0'2058
0'0000
0'2983
0'0000
0'4122
•JO
'20
•30
'0000
'3128
'moo
-366o
•4665
•0262
'0817
'4230
'4655
•0000
•0000
'5747
•6512
'40
' 1 544
-5222
•6106
•0090
'0530
•0000
•0000
0'50
0.2439
0.6906
•6o
/0
'3507
'4762
•8o
'6232
*90
0.95
'5536
•5622
'0129
'6339
'0000
'7116
•1158
'6464
'0535
'7091
'0090
'7696
'7637
0.1980
•3009
0.7215
-7890
o•1166
•2058
0.7746
-8320
0.0445
•Io81
0.8248
•8719
•8306
•4270
•8500
•3266
'8921
'5807
'9052
'4881
'8824
'9268
'7958
'9485
'7683
'9551
'7046
0'8935
0'9749
0'8778
0.9782
0.8403
'2101
'9121
'9659
•3678
'6116
'9463
'9754
0.9836
0.7821
0.9883
See page 89 for explanation of the use of this table.
90
TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A
MULTIPLE CORRELATION COEFFICIENT
k =2
CONFIDENCE LEVEL PER CENT
90
95
n =4
122 = woo
o-0000
o•6o19
0.0000
•io
•0000
•6179
•20
'0000
•6352
•30
'40
•0000
•0000
•6541
•6751
•0000
•0000
•0000
•0000
0'50
o'0000
0.6986
•6o
•0000
'7255
99
99'9
o•8415
•85o3
•8595
-8694
•8799
0.0000
•0000
•0000
•0000
•0000
0.9369
'7434
•7611
o'0000
•0000
•0000
•0000
•0000
0.7807
•8028
•8284
•8597
•9I00
0.0000
•0000
•0000
•0000
•0000
0.8914
•9040
'9184
'9353
'9572
o•0000
•0000
•0000
•0000
•0000
0.9586
•9636
.9695
•9765
•9848
0.6983
•7123
•7272
'9408
'9448
'9491
'9535
• 0
•0000
'7573
•8o
•0000
•7969
•90
•0614
-9055
o'0000
•0000
•0000
•0000
•oo85
0.95
o•1189
0'9690
0'0540
0'9707
0.0000
0'9720
0'0000
0'9905
122 = woo
0'0000
0'0000
'0000
0'0000
0'5671
0.0000
0.7152
'0000
0'3421
'4101
0'4200
•IO
'4904
'20
'4741
•30
•0000
'am
'40
•0000
'5354
'5953
•0000
•0000
•0000
•5527
-6098
-6641
•0000
•0000
•0000
•0000
•6331
•6857
•7313
•7728
•0000
•0000
•0000
•0000
•7665
•8040
•8352
•8626
0'50
0'0363
0'6854
0.0000
0'7170
•6o
•1122
•7800
-2107
-8o
'9656
'8779
'9321
'9728
0.8118
•8493
•9062
•9523
•9825
o'0000
•0000
•0000
•0105
•90
'3487
'5639
'8557
.9171
0.0000
•0000
'0307
•1014
-2688
0•8877
•9112
• 0
'0557
' 1351
'2564
'4692
•8073
'0774
'9887
0'95
0.7323
0.9848
0.6562
0.9883
0.4649
0.9929
0.2109
0 '9959
0'0000
0'1623
'2822
0'0000
'0000
0'2058
•3364
0'0000
'0000
0'2983
•4400
0'0000
'0000
0.4122
•3783
•0000
'0242
•0849
'4329
'5334
•6273
•0000
•0000
-0272
•5321
-6079
-6907
•0000
•0000
•0000
0.1667
'2708
•4000
'5589
0.7079
0.0866
0'7628
0'0211
0'8134
•8241
•0786
' 1772
•8653
•9081
'3352
•5866
'9442
'9746
0.9830
0'7664
0 '9879
it = 10
it = 25
R2 = woo
'10
•20
'0000
'moo
•30
'40
.0503
•1225
0'50
0'2130
'3221
•6o
'4938
. 5904
0'6758
•8o
'4514
•6040
'7530
•8234
•8878
•90
'7845
'9465
'7550
'9535
'1742
'2958
•4611
•6865
0.95
0.8873
0 '9739
0•8705
0'9774
o'8298
io
'7794
'8436
-9015
'8774
-9240
-9647
See page 89 for explanation of the use of this table.
91
'9338
•9629
'5525
•6341
•6983
'7532
TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A
MULTIPLE CORRELATION COEFFICIENT
k =3
CONFIDENCE LEVEL PER CENT
90
99
95
n =5
R2 = woo
o•0000
0•5358
•IO
•0000
'5532
0.0000
•0000
0•6316
•6477
99'9
o•0000
•000o
0.7846
•7962
o-0000
•0000
•20
•0000
•5721
'0000
'6652
'0000
•8085
'0000
'30
•000o
'40
•0000
'5931
•6166
•0000
•0000
•6842
•7053
•0000
•0000
•8217
•836o
•0000
•0000
0'50
0.0000
0.6433
•0000
0•0000
•0000
0.7289
•6o
•70
'0000
•0000
'0000
'7874
'8262
0.0000
•0000
•0000
0•8517
-8691
•8890
0.0000
•000o
•0000
•0000
'9125
0•900o
-9061
•9126
•9196
•9266
0 '9344
•8o
'0000
'6743
•7114
'7582
'0000
•9634
•90
•oun
•8337
•0000
•8787
•0000
-9426
•0000
•9770
0'95
0 '0951
0'9446
0.0362
0•9470
0.0000
0•9630
0.0000
0.9858
n = 15
R2 = woo
0.0000
0.0000
•0000
0.3123
•3892
O•0000
•0000
0 4377
•5186
0.5784
•0000
'0000
•6522
•20
•0000
•30
- 0000
0'2505
•3204
•3930
'4662
o•0000
•I0
'40
-0000
'5395
•0000
-0000
•0000
•4630
'5340
•6026
•0000
-0000
-0000
•5879
•6498
•707o
•0000
•0000
•0000
'7093
'7576
•8002
0'50
0.0648
0.6565
•1562
'7522
'70
'2779
•8o
'4409
'3635
-602!
0.7602
•8230
•8901
'9381
0•0000
-0000
.0045
•0628
•6656
0•6885
-7820
-8548
'9138
•9620
0.0000
•0153
-0823
'2093
•90
•8309
'8976
'9539
0.0231
'0973
•2053
0.8388
•6o
'4537
'9739
'2546
'9830
0'95
0'8139
0.9783
0.7720
0.9823
0.6638
0.9882
0.4879
0.9926
R2 = o•oo
•IO
0.0000
0.1623
0.0000
0.2058
o•0000
0.2983
•2580
'0000
•3123
'0000
•20
•0000
•3514
•30
'0190
'40
•0890
'4556
'5667
•0000
-0000
'0533
'4075
'4941
•6038
•000o
•0000
•0033
'4175
•5103
'5895
-6631
o•0000
•0000
-0000
•0000
•0000
0.4122
•0000
0'50
0.1797
0'6592
0-1333
'2909
'7412
'2382
.70
'4242
0.7484
'8152
•8719
0.0014
'0491
•1421
'5827
'90
'7719
'9444
'3704
'5349
'7402
0.0562
'1407
-2623
'8o
.8155
'8830
0.6924
•7688
-8367
0'7947
•6o
'8974
'9517
'4314
•6664
'9209
'9634
'2995
'5586
'9037
'9419
'9737
0.95
0.8804
0'9729
0.8621
0.9766
0.8179
0.9824
0'7484
0'9875
'7558
*9430
•9529
'8745
•9117
•9561
n = 25
See page 89 for explanation of the use of this table.
92
'5330
•6166
-6841
.7421
'8572
TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A
MULTIPLE CORRELATION COEFFICIENT
k= 4
CONFIDENCE LEVEL PER CENT
90
95
n =6
99
99'9
R2 = woo
o-0000
o. 482o
o•0000
0•5751
0.0000
0.7317
•10
•5928
•0000
-6121
•0000
•0000
'7458
'7609
•30
•0000
-0000
'7771
•0000
6334
•6571
•0000
'40
•5011
•5203
•5427
•5681
- 0000
'20
'0000
•0000
•0000
•7948
•0000
•0000
0•50
0.0000
0.5971
•6312
0.0000
•0000
0.8143
•8360
0.0000
•0000
0.9092
•0000
0.0000
•0000
0•6839
•6o
•70
'0000
•0000
.9346
•8o
•0000
'6724
•7249
•90
'0000
0.95
•0000
0.0000
•0000
•0000
o•86io
•8696
•8786
•888o
•8982
•9212
•0000
•7147
•7512
•8610
•0000
'7993
-0000
-0000
•7964
•8581
•0000
•0000
•8907
•9287
•0000
•0000
'9499
•9689
0.0731
0.9167
o•o186
o•9199
0•0000
0'9543
0-0000
0.9810
R2 = woo
0.0000
0'4377
'0000
•30
•0000
'4343
-000o
•3732
•4377
-5047
'40
•0000
'5061
•0000
'5731
•0000
•0000
•0000
•0000
•5032
-5661
•6264
•6846
o-0000
•0000
•0000
•0000
•0000
0.5784
'20
0.0000
•0000
•0000
0.0000
•0000
0.2505
•3050
'3669
0.3123
•10
0.50
•6o
0-0110
0.5916
.7160
0.0000
0•6424
0.0000
0'7406
0.0000
o•8258
.0959
'0442
'7438
•0000
• 0
'2130
'8117
'1409
*8363
•8o
'3792
•2952
.9055
'0347
•1384
•90
'6200
•8882
.9505
'7947
-8689
•9306
'5463
'9592
'3770
'9719
•0000
•0000
•0218
•1662
-8641
-9001
-9461
•9813
0.95
0.7850
0.9769
0'7345
0.9812
0.6035
0'9874
0'3957
0.9921
R2 = woo
o•0000
0.1623
0.0000
0'2058
0'0000
0.2983
0.0000
0'4122
•IO
'20
'0000
•2399
•3263
-2937
•3835
•30
'0000
'4148
•0000
•4709
'40
•0552
•5367
•0229
•5720
•0000
•0000
-0000
•0000
'3993
•4892
•5702
•6438
-0000
•0000
•0000
-0000
•5170
•0000
•0000
•0000
0•50
0.1441
0.0985
•2030
0.0273
0.7289
0.0000
0.7838
•2570
/0
'3942
0.640o
•7282
•8068
0.6741
-60
'7570
•8290
•8930
•8049
•8657
.9175
-9620
-0223
-1059
•2606
•5271
'8456
-8985
'9498
-1054
•2259
•3985
'6436
0 '9757
0.8043
0.9818
0.7278
0.9871
n = 15
6394
•6927
•7408
7849
n=25
•8o
•5591
'90
'7578
'8779
'9422
'3379
.5082
.7235
0•95
o'8726
0.9719
o'8527
See page 89 for explanation of the use of this table.
93
'5993
•6690
•7297
'9393
'9727
TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A
MULTIPLE CORRELATION COEFFICIENT
k=5
CONFIDENCE LEVEL PER CENT
90
n=7
R2 =o-oo
99
95
o•0000
0 '4377
•20
'0000
'0000
•30
•0000
'40
•0000
'4563
'4771
'5011
•5269
0'50
o-0000
0.5576
-6o
•0000
'70
•0000
-8o
•0000
•90
0.0000
•0000
-0000
'0000
'0000
0.5271
99'9
'5459
•5665
'5895
'6151
0.0000
•0000
•0000
•0000
'0000
0.6838
•6998
•0000
o'6444
-6783
'7188
•7695
•8392
0.0000
•0000
•0000
•0000
•0000
0'7795
'5940
'6384
'6955
'7774
0.0000
•0000
'0000
•000o
•0000
0-95
0.0509
0•8858
0.0000
0•8898
n = 20
R2 = woo
0.0000
0.1969
•10
'0000
•20
•30
'0000
'2525
'3190
'40
•0000
'3943
'4755
0.0000
•0000
•0000
•0000
•0000
0'50
0.0445
0'5933
-6o
' 1 433
•70
•2783
-8o
'4557
•7074
-7986
'8758
•90
'6874
0•95
0.0000
-0000
•0000
•0000
•0000
0.8222
•8327
•8052
'8347
•8700
•9154
0.0000
•0000
•0000
•0000
•0000
0•8837
•8991
•9164
-9362
•9606
0.0000
0.9460
0.0000
0.9761
0.2482
•3115
•3824
•4584
•5368
0.0000
'0000
'0000
-0000
•0000
0.3550
'4276
'5011
0.0000
'0000
'0000
-0000
•0000
0.4821
0.6230
'7791
'8620
'9199
0.0000
•0000
•0063
'0897
0.790o
-8374
•8867
'8931
'9514
0-0000
•0104
'0896
'2432
'5121
0.7072
'9427
0.0079
.0879
•2101
'3864
•6351
'9647
'3376
'9416
'9757
0.8310
0'9726
0'7986
0.9769
0.7165
0.9835
0.5824
0.9889
R2= woo
0'0000
'20
•30
•4°
•woo
0'1757
'2576
•3486
'0000
•0610
*5157
0'0000
'0000
•0000
•0000
'0291
0.0000
'0000
-0000
•0000
•0000
0•0000
•0000
•0000
•0000
•0000
0.3596
•0000
0'1381
'2092
•2955
'3879
0.257o
-JO
0'50
0'1556
0.1115
.2230
•3627
0'0391
'1288
-8o
*90
'7721
'9373
'5339
'7430
'8835
'9448
'2594
'4370
'6758
0'7098
'7876
'8520
'9079
'9570
0'0000
.0399
•1424
'3124
'5785
0.7598
'2735
'4139
'5788
o•6210
.7119
0. 6543
•6o
0.95
0.8812
0.9694
0.8647
0'9731
0.8251
0.9792
0.7642
0.9847
•IO
n=
'0000
'7378
'8236
'7174
•7361
•7567
'5718
•6409
'8443
•8566
•8696
'5557
'6222
•6830
•7387
30
/0
'7939
•8687
'4408
•5503
'7399
.8155
'3543
'4481
'5354
-6154
See page 89 for explanation of the use of this table.
94
'4649
'5540
•6314
'6994
•8310
-8858
.9304
•9681
TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A
MULTIPLE CORRELATION COEFFICIENT
k= 6
CONFIDENCE LEVEL PER CENT
90
99
95
n =8
99'9
R2= woo
0.0000
0'4005
0.0000
0'4861
0'0000
0'6406
0'0000
0'7845
•0
'0000
•20
•0000
•0000
•0000
-5056
•5271
•0000
•0000
•6584
•6776
'0000
'0000
'5511
•5783
'0000
'6986
•7972
•8114
•8258
•0000
•7216
•0000
•0000
•0000
•0000
0.6094
•6457
•6896
0 '7473
o•0000
-0000
-0000
-0000
•0000
0•8586
•7763
•81oo
•8504
•9027
•30
'0000
'40
-0000
'4193
'4404
'4642
'4916
0'50
0'0000
0.5234
0.0000
•6o
•0000
• 0
•0000
•8o
-0000
'5614
'6083
'6692
'0000
'0000
'0000
•90
•0000
'7575
•0000
•82I7
o•0000
•0000
•0000
•0000
'0000
0095
0.0278
0'8520
0'0000
o'8778
0'0000
0'9381
0'0000
0'9713
n= 20
R2 = o•oo
0-0000
o.1969
0'0000
•0000
0.0000
0.4821
•0000
0'2482
•3018
03550
•0
0'0000
•0000
'4173
•0000
-0000
-0000
•0000
'4835
•5517
•6206
•0000
•0000
'0000
•0000
•5462
•6o8o
•6691
•7238
0.7769
•8269
'7449
'8415
'8774
•8984
•9227
•9524
•20
'0000
•30
'0000
•2436
•30I3
•3698
'40
-0000
'4477
•0000
•3644
'4350
•5114
0•50
0.0034
'0947
0'5358
•676o
0.0000
•0451
0.5914
•7045
o-0000
•0000
0•6887
-6o
• 0
•2257
•7821
•1568
•8079
.0469
'8444
•80
'4084
'8674
090
'6556
'9393
'3334
'5967
'8857
'9486
' 1834
'4584
'9137
.9627
0.0000
•0000
•0000
'0446
•2671
0095
0'8123
0'9711
0'7748
0'9757
0.6797
0.9827
0'5243
0.9884
n = 30
R2 = o•oo
o-0000
O.1381
'0000
'0000
0.1757
•2459
•3297
0.0000
-0000
•0000
0.2570
•3420
•20
•30
'1984
'2764
'3658
0'0000
•0000
'0000
'0000
•4200
'0000
'40
•0327
'4839
•0047
•5151
•0000
o•0000
•0000
-0000
•0000
'0000
0.3596
•0
0•50
0.0155
.0972
•2269
•4086
0•0000
•0172
•1094
0.7506
•8183
•8806
'2785
'5525
'9277
-9670
0'7481
0.9843
•0000
•0000
0.1239
0.6022
o•o809
0.6357
•6o
'2430
'6993
/0
'3872
•8o
'5581
'7855
•8637
•90
'7600
'9350
' 1913
'3339
'5107
'7289
'7284
•8080
'8791
.9428
0.95
0 '8747
0.9683
0-8569
0'9722
'7550
'4305
'5175
'5998
0.6881
'6569
'7773
'8459
•9045
'9556
o•8142
0.9786
See page 89 for explanation of the use of this table.
95
'8737
'9341
'9742
'4531
'5389
•6169
•6871
A NOTE ON INTERPOLATION
Part of the tabulation of a function f(x) at intervals h of x is
in the form given in the first two columns of the figure :
For x = 0.034, p = (0.034-0.03)/0.01 = 0.4 and the linear
interpolate is
xo J
o
x, 4
x, f,
0 '9790 + 0.4
Ai
The additional term for the quadratic interpolate is
A",
A;1
—0.25 x 0.4 x o•6 x ( — 0'0090 — 0'0087) = 0'001i
A°
x, f,
where ft = f(x) and x,„ = x,H-h. Interpolation of f(x) at
values of x other than those tabulated uses the differences in
the last three columns, where each entry is the value in the
column immediately to the left and below minus the value to
the left and above : thus, Ail =f 2 —fi and A; =
These are usually written in units of the last place of decimals
in f(x). Linear interpolation between x1 and x2 approximates
f(x) by
+PAil
with p = (x—x1)/h. This simple rule uses only the values
within the lines of the figure and is often adequate. Quadratic
interpolation between x1and x2 approximates f(x) by
+pA',4 -1p(i—p) (A';+
and is not negligible, the quadratic interpolate being 0-9709.
This is exact, as is expected since A;1 at 3 is well below 6o.
The quadratic method uses fo ana f, (needed for A; and
A;) and so fails if either is unavailable, for example at the
ends of the range of x or when the interval of tabulation h
changes. Modified quadratic forms between xl and x2 are
+pevil
20,
Example. The F-distribution, P = to, v1 = i (Table 2(a),
page 5o), interpolation in v2, now x.
0'9929
'03
'9790
—
F(P)
0
2'706
120
1/120
2•748
6o 2/12 0
2.791
40 3/120
2.835
43
+3
—8 7
—316
' 05
'9245
CONSTANTS
e = 2.71828
18285
It = 3'141 59 26536
T1E
=
0
44
h--,±0)/(th) =
—90
-9561
I /v2
Notice that the values of v2 chosen for tabulation are such
that the intervals of i/v2 are constsnt, here
The differences show that linear interpolation will be adequate. For
•5 and the linear interv2 = 8o, p = (
polate is 2.748 + o.5 x 0.043 = 2•77o with the possibility of
an error of I in the last place.
139
—229
-04
v2
oo
42
r = 2 (Table t,
page 22), interpolation in p, now x.
(f, missing),
Ocassionally, harmonic interpolation is advisable. To do this
the argument x is replaced by i/x and then linear (or
quadratic) interpolation performed.
2.
Example. The binomial distribution, n =
—p) A;
f1 +0,4-1p(i —p) A; (fo missing).
This is generally adequate provided Ail is less than 6o in
units of the last place of decimals in the tabulation. Notice
that the quadratic interpolate consists of the addition of an
extra term to the linear one, so that a rough assessment of it
will indicate whether the linear form is adequate. The
maximum possible value of ip(t —p) is when p =
0•02
x ( — 0-0229) = 0-9698.
0-39894 22804
logio e = 043429 44819
loge Io = 2.30258 50930
10g,
2n = 0.91893 85332
CAMBRIDGE UNIVERSITY PRESS
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Published in the United States of America by Cambridge University Press, New York
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© Cambridge University Press 1984
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1984
Eighth printing 1994
Second edition 1995
Thirteenth printing 2009
Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this publication is available from the British Library
ISBN 978-0-521-48485-5 paperback