Statistical Tables
Contents
Binomial distribution (example, for n = 9)
2
Normal distribution and inverse
3
χ2 - distribution
4
t-distribution
5
F -distribution (10%)
6
F -distribution (5%)
7
F -distribution (2.5%)
8
F -distribution (1%)
9
F -distribution (0.5%)
10
Wilcoxon signed rank
11
R Commands
For each table the general command is given, with an example which reproduces the
third entry of the first column of that table.
page
2
3i
3ii
4
5
6
7
8
9
10
11
distribution
binomial
normal
normal
χ2
t
F
F
F
F
F
Wilcoxon
general
pbinom(r, n, p)
pnorm(x)
qnorm(p)
qchisq(p, ν, lower=F)
qt(p, ν, lower=F)
qf(0.1, ν1 , ν2 , lower=F)
qf(0.05, ν1 , ν2 , lower=F)
qf(0.025, ν1 , ν2 , lower=F)
qf(0.01, ν1 , ν2 , lower=F)
qf(0.005, ν1 , ν2 , lower=F)
qsignrank(p, n)
example
pbinom(0, 9, 0.05)
pnorm(0.10)
qnorm(0.975)
qchisq(0.1, 3, lower=F)
qt(0.1, 3, lower=F)
qf(0.1, 1, 4, lower=F)
qf(0.05, 1, 4, lower=F)
qf(0.025, 1, 4, lower=F)
qf(0.01, 1, 4, lower=F)
qf(0.005, 1, 4, lower=F)
qsignrank(0.05, 10)
Note the general form of the commands qdist and pdist.
1
Binomial Distribution Function for n = 9
This table gives P(X ≤ r) for X ∼ Bin(9, p)
For p ≥ .5 you may use the result
P (X ≤ r) = 1 − P (Y ≤ n − r − 1)
with Y ∼ Bin(n, 1 − p)
r
p
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0.31
0.33
0.35
0.37
0.39
0.41
0.43
0.45
0.47
0.49
0
0.9135
0.7602
0.6302
0.5204
0.4279
0.3504
0.2855
0.2316
0.1869
0.1501
0.1199
0.0952
0.0751
0.0589
0.0458
0.0355
0.0272
0.0207
0.0156
0.0117
0.0087
0.0064
0.0046
0.0033
0.0023
1
0.9966
0.9718
0.9288
0.8729
0.8088
0.7401
0.6696
0.5995
0.5315
0.4670
0.4066
0.3509
0.3003
0.2548
0.2144
0.1788
0.1478
0.1211
0.0983
0.0790
0.0628
0.0495
0.0385
0.0296
0.0225
2
0.9999
0.9980
0.9916
0.9791
0.9595
0.9328
0.8991
0.8591
0.8139
0.7643
0.7115
0.6566
0.6007
0.5448
0.4898
0.4364
0.3854
0.3373
0.2924
0.2511
0.2134
0.1796
0.1495
0.1231
0.1001
3
4
1.0000
0.9999
0.9994
0.9977
0.9943
0.9883
0.9791
0.9661
0.9488
0.9270
0.9006
0.8696
0.8343
0.7950
0.7522
0.7065
0.6585
0.6089
0.5584
0.5078
0.4576
0.4087
0.3614
0.3164
0.2740
1.0000
1.0000
1.0000
0.9998
0.9995
0.9986
0.9970
0.9944
0.9902
0.9842
0.9760
0.9650
0.9511
0.9338
0.9130
0.8885
0.8602
0.8283
0.7928
0.7540
0.7122
0.6678
0.6214
0.5735
0.5246
2
5
1.0000
1.0000
1.0000
1.0000
1.0000
0.9999
0.9997
0.9994
0.9987
0.9977
0.9960
0.9935
0.9900
0.9851
0.9787
0.9702
0.9596
0.9464
0.9304
0.9114
0.8891
0.8634
0.8342
0.8015
0.7654
6
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9999
0.9998
0.9996
0.9992
0.9987
0.9978
0.9965
0.9947
0.9922
0.9888
0.9843
0.9785
0.9710
0.9617
0.9502
0.9363
0.9196
7
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9999
0.9999
0.9998
0.9997
0.9994
0.9991
0.9986
0.9979
0.9969
0.9954
0.9935
0.9909
0.9875
0.9831
8
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9999
0.9999
0.9998
0.9997
0.9995
0.9992
0.9989
0.9984
Normal Distribution Function Tables
The first table gives
Z
x
1 2
e− 2 t dt
0.4
1
Φ(x) = √
2π
−∞
0.0
0.1
0.2
0.3
and this corresponds to the shaded area in
the figure to the right. Φ(x) is the probability that a random variable, normally distributed with zero mean amd unit variance,
will be less than or equal to x. When x < 0
use Φ(x) = 1 − Φ(−x), as the normal distribution with mean zero is symmetric about
zero. To interpolate, use the formula
−3
−2
−1
0
1
x
2
3
x − x1
(Φ(x2 ) − Φ(x1 ))
Φ(x) ≈ Φ(x1 ) +
x2 − x1
Table 1
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
0.00
0.05
0.10
0.15
0.20
0.5000
0.5199
0.5398
0.5596
0.5793
0.50
0.55
0.60
0.65
0.70
0.6915
0.7088
0.7257
0.7422
0.7580
1.00
1.05
1.10
1.15
1.20
0.8413
0.8531
0.8643
0.8749
0.8849
1.50
1.55
1.60
1.65
1.70
0.9332
0.9394
0.9452
0.9505
0.9554
2.00
2.05
2.10
2.15
2.20
0.9772
0.9798
0.9821
0.9842
0.9861
2.50
2.55
2.60
2.65
2.70
0.9938
0.9946
0.9953
0.9960
0.9965
0.25
0.30
0.35
0.40
0.45
0.5987
0.6179
0.6368
0.6554
0.6736
0.75
0.80
0.85
0.90
0.95
0.7734
0.7881
0.8023
0.8159
0.8289
1.25
1.30
1.35
1.40
1.45
0.8944
0.9032
0.9115
0.9192
0.9265
1.75
1.80
1.85
1.90
1.95
0.9599
0.9641
0.9678
0.9713
0.9744
2.25
2.30
2.35
2.40
2.45
0.9878
0.9893
0.9906
0.9918
0.9929
2.75
2.80
2.85
2.90
2.95
0.9970
0.9974
0.9978
0.9981
0.9984
0.50
0.6915
1.00
0.8413
1.50
0.9332
2.00
0.9772
2.50
0.9938
3.00
0.9987
The inverse function Φ−1 (p) is tabulated below for various values of p.
Table 2
p
Φ−1 (p)
0.900
1.2816
0.950
1.6449
0.975
1.9600
0.990
2.3263
3
0.995
2.5758
0.999
3.0902
0.9995
3.2905
Percentage Points of the χ2-Distribution
This table gives the percentage points χ2ν (P )
for various values of P and degrees of freedom ν, as indicated by the figure to the
right, plotted in the case ν = 3.
If X is a variable distributed as χ2 with
ν degrees of freedom, P/100 is the probability that X ≥ χ2ν (P
√ ).
norFor ν > 100, 2X is approximately
√
mally distributed with mean 2ν − 1 and
unit variance.
P/100
χ2ν (P )
0
Percentage points P
10
2.706
4.605
6.251
7.779
9.236
5
3.841
5.991
7.815
9.488
11.070
2.5
5.024
7.378
9.348
11.143
12.833
1
6.635
9.210
11.345
13.277
15.086
0.5
7.879
10.597
12.838
14.860
16.750
0.1
10.828
13.816
16.266
18.467
20.515
0.05
12.116
15.202
17.730
19.997
22.105
6
7
8
9
10
10.645
12.017
13.362
14.684
15.987
12.592
14.067
15.507
16.919
18.307
14.449
16.013
17.535
19.023
20.483
16.812
18.475
20.090
21.666
23.209
18.548
20.278
21.955
23.589
25.188
22.458
24.322
26.124
27.877
29.588
24.103
26.018
27.868
29.666
31.420
11
12
13
14
15
17.275
18.549
19.812
21.064
22.307
19.675
21.026
22.362
23.685
24.996
21.920
23.337
24.736
26.119
27.488
24.725
26.217
27.688
29.141
30.578
26.757
28.300
29.819
31.319
32.801
31.264
32.909
34.528
36.123
37.697
33.137
34.821
36.478
38.109
39.719
16
17
18
19
20
23.542
24.769
25.989
27.204
28.412
26.296
27.587
28.869
30.144
31.410
28.845
30.191
31.526
32.852
34.170
32.000
33.409
34.805
36.191
37.566
34.267
35.718
37.156
38.582
39.997
39.252
40.790
42.312
43.820
45.315
41.308
42.879
44.434
45.973
47.498
25
30
40
50
80
34.382
40.256
51.805
63.167
96.578
37.652
43.773
55.758
67.505
101.879
40.646
46.979
59.342
71.420
106.629
44.314
50.892
63.691
76.154
112.329
46.928
53.672
66.766
79.490
116.321
52.620
59.703
73.402
86.661
124.839
54.947
62.162
76.095
89.561
128.261
ν
1
2
3
4
5
4
Percentage Points of the t-Distribution
This table gives the percentage points tν (P )
for various values of P and degrees of freedom ν, as indicated by the figure to the
right.
The lower percentage points are given
by symmetry as −tν (P ), and the probability that |t| ≥ tν (P ) is 2P/100.
The limiting distribution of t as ν → ∞
is the normal distribution with zero mean
and unit variance.
P/100
0
tν (P )
Percentage points P
10
3.078
1.886
1.638
1.533
1.476
5
6.314
2.920
2.353
2.132
2.015
2.5
12.706
4.303
3.182
2.776
2.571
1
31.821
6.965
4.541
3.747
3.365
0.5
63.657
9.925
5.841
4.604
4.032
0.1
318.309
22.327
10.215
7.173
5.893
0.05
636.619
31.599
12.924
8.610
6.869
6
7
8
9
10
1.440
1.415
1.397
1.383
1.372
1.943
1.895
1.860
1.833
1.812
2.447
2.365
2.306
2.262
2.228
3.143
2.998
2.896
2.821
2.764
3.707
3.499
3.355
3.250
3.169
5.208
4.785
4.501
4.297
4.144
5.959
5.408
5.041
4.781
4.587
11
12
13
14
15
1.363
1.356
1.350
1.345
1.341
1.796
1.782
1.771
1.761
1.753
2.201
2.179
2.160
2.145
2.131
2.718
2.681
2.650
2.624
2.602
3.106
3.055
3.012
2.977
2.947
4.025
3.930
3.852
3.787
3.733
4.437
4.318
4.221
4.140
4.073
16
18
21
25
30
1.337
1.330
1.323
1.316
1.310
1.746
1.734
1.721
1.708
1.697
2.120
2.101
2.080
2.060
2.042
2.583
2.552
2.518
2.485
2.457
2.921
2.878
2.831
2.787
2.750
3.686
3.610
3.527
3.450
3.385
4.015
3.922
3.819
3.725
3.646
40
50
70
100
∞
1.303
1.299
1.294
1.290
1.282
1.684
1.676
1.667
1.660
1.645
2.021
2.009
1.994
1.984
1.960
2.423
2.403
2.381
2.364
2.326
2.704
2.678
2.648
2.626
2.576
3.307
3.261
3.211
3.174
3.090
3.551
3.496
3.435
3.390
3.291
ν
1
2
3
4
5
5
10 Percent Points of the F -Distribution
This table gives the percentage points
Fν1 ,ν2 (P ) for P = 0.10 and degrees of freedom ν1 , ν2 , as indicated by the figure to the
right.
The lower percentage points, that is the
values Fν′ 1 ,ν2 (P ) such that the probability
that F ≤ Fν′ 1 ,ν2 (P ) is equal to P/100, may
be found using the formula
P/100
Fν′ 1 ,ν2 (P ) = 1/Fν2 ,ν1 (P )
F (P )
0
ν1
1
2
3
4
5
6
12
24
∞
2
3
4
5
8.526
5.538
4.545
4.060
9.000
5.462
4.325
3.780
9.162
5.391
4.191
3.619
9.243
5.343
4.107
3.520
9.293
5.309
4.051
3.453
9.326
5.285
4.010
3.405
9.408
5.216
3.896
3.268
9.450
5.176
3.831
3.191
9.491
5.134
3.761
3.105
6
7
8
9
10
3.776
3.589
3.458
3.360
3.285
3.463
3.257
3.113
3.006
2.924
3.289
3.074
2.924
2.813
2.728
3.181
2.961
2.806
2.693
2.605
3.108
2.883
2.726
2.611
2.522
3.055
2.827
2.668
2.551
2.461
2.905
2.668
2.502
2.379
2.284
2.818
2.575
2.404
2.277
2.178
2.722
2.471
2.293
2.159
2.055
11
12
13
14
15
3.225
3.177
3.136
3.102
3.073
2.860
2.807
2.763
2.726
2.695
2.660
2.606
2.560
2.522
2.490
2.536
2.480
2.434
2.395
2.361
2.451
2.394
2.347
2.307
2.273
2.389
2.331
2.283
2.243
2.208
2.209
2.147
2.097
2.054
2.017
2.100
2.036
1.983
1.938
1.899
1.972
1.904
1.846
1.797
1.755
16
17
18
19
20
3.048
3.026
3.007
2.990
2.975
2.668
2.645
2.624
2.606
2.589
2.462
2.437
2.416
2.397
2.380
2.333
2.308
2.286
2.266
2.249
2.244
2.218
2.196
2.176
2.158
2.178
2.152
2.130
2.109
2.091
1.985
1.958
1.933
1.912
1.892
1.866
1.836
1.810
1.787
1.767
1.718
1.686
1.657
1.631
1.607
25
30
40
50
100
2.918
2.881
2.835
2.809
2.756
2.528
2.489
2.440
2.412
2.356
2.317
2.276
2.226
2.197
2.139
2.184
2.142
2.091
2.061
2.002
2.092
2.049
1.997
1.966
1.906
2.024
1.980
1.927
1.895
1.834
1.820
1.773
1.715
1.680
1.612
1.689
1.638
1.574
1.536
1.460
1.518
1.456
1.377
1.327
1.214
2.706
2.303
2.084
1.945
1.847
1.774
1.546
1.383
1.002
ν2
∞
6
5 Percent Points of the F -Distribution
This table gives the percentage points
Fν1 ,ν2 (P ) for P = 0.05 and degrees of freedom ν1 , ν2 , as indicated by the figure to the
right.
The lower percentage points, that is the
values Fν′ 1 ,ν2 (P ) such that the probability
that F ≤ Fν′ 1 ,ν2 (P ) is equal to P/100, may
be found using the formula
P/100
Fν′ 1 ,ν2 (P ) = 1/Fν2 ,ν1 (P )
F (P )
0
ν1
1
2
3
4
5
6
12
24
∞
2
3
4
5
18.513
10.128
7.709
6.608
19.000
9.552
6.944
5.786
19.164
9.277
6.591
5.409
19.247
9.117
6.388
5.192
19.296
9.013
6.256
5.050
19.330
8.941
6.163
4.950
19.413
8.745
5.912
4.678
19.454
8.639
5.774
4.527
19.496
8.526
5.628
4.365
6
7
8
9
10
5.987
5.591
5.318
5.117
4.965
5.143
4.737
4.459
4.256
4.103
4.757
4.347
4.066
3.863
3.708
4.534
4.120
3.838
3.633
3.478
4.387
3.972
3.687
3.482
3.326
4.284
3.866
3.581
3.374
3.217
4.000
3.575
3.284
3.073
2.913
3.841
3.410
3.115
2.900
2.737
3.669
3.230
2.928
2.707
2.538
11
12
13
14
15
4.844
4.747
4.667
4.600
4.543
3.982
3.885
3.806
3.739
3.682
3.587
3.490
3.411
3.344
3.287
3.357
3.259
3.179
3.112
3.056
3.204
3.106
3.025
2.958
2.901
3.095
2.996
2.915
2.848
2.790
2.788
2.687
2.604
2.534
2.475
2.609
2.505
2.420
2.349
2.288
2.404
2.296
2.206
2.131
2.066
16
17
18
19
20
4.494
4.451
4.414
4.381
4.351
3.634
3.592
3.555
3.522
3.493
3.239
3.197
3.160
3.127
3.098
3.007
2.965
2.928
2.895
2.866
2.852
2.810
2.773
2.740
2.711
2.741
2.699
2.661
2.628
2.599
2.425
2.381
2.342
2.308
2.278
2.235
2.190
2.150
2.114
2.082
2.010
1.960
1.917
1.878
1.843
25
30
40
50
100
4.242
4.171
4.085
4.034
3.936
3.385
3.316
3.232
3.183
3.087
2.991
2.922
2.839
2.790
2.696
2.759
2.690
2.606
2.557
2.463
2.603
2.534
2.449
2.400
2.305
2.490
2.421
2.336
2.286
2.191
2.165
2.092
2.003
1.952
1.850
1.964
1.887
1.793
1.737
1.627
1.711
1.622
1.509
1.438
1.283
3.841
2.996
2.605
2.372
2.214
2.099
1.752
1.517
1.002
ν2
∞
7
2.5 Percent Points of the F -Distribution
This table gives the percentage points
Fν1 ,ν2 (P ) for P = 0.025 and degrees of freedom ν1 , ν2 , as indicated by the figure to the
right.
The lower percentage points, that is the
values Fν′ 1 ,ν2 (P ) such that the probability
that F ≤ Fν′ 1 ,ν2 (P ) is equal to P/100, may
be found using the formula
P/100
Fν′ 1 ,ν2 (P ) = 1/Fν2 ,ν1 (P )
F (P )
0
ν1
1
2
3
4
5
6
12
24
∞
2
3
4
5
38.506
17.443
12.218
10.007
39.000
16.044
10.649
8.434
39.165
15.439
9.979
7.764
39.248
15.101
9.605
7.388
39.298
14.885
9.364
7.146
39.331
14.735
9.197
6.978
39.415
14.337
8.751
6.525
39.456
14.124
8.511
6.278
39.498
13.902
8.257
6.015
6
7
8
9
10
8.813
8.073
7.571
7.209
6.937
7.260
6.542
6.059
5.715
5.456
6.599
5.890
5.416
5.078
4.826
6.227
5.523
5.053
4.718
4.468
5.988
5.285
4.817
4.484
4.236
5.820
5.119
4.652
4.320
4.072
5.366
4.666
4.200
3.868
3.621
5.117
4.415
3.947
3.614
3.365
4.849
4.142
3.670
3.333
3.080
11
12
13
14
15
6.724
6.554
6.414
6.298
6.200
5.256
5.096
4.965
4.857
4.765
4.630
4.474
4.347
4.242
4.153
4.275
4.121
3.996
3.892
3.804
4.044
3.891
3.767
3.663
3.576
3.881
3.728
3.604
3.501
3.415
3.430
3.277
3.153
3.050
2.963
3.173
3.019
2.893
2.789
2.701
2.883
2.725
2.595
2.487
2.395
16
17
18
19
20
6.115
6.042
5.978
5.922
5.871
4.687
4.619
4.560
4.508
4.461
4.077
4.011
3.954
3.903
3.859
3.729
3.665
3.608
3.559
3.515
3.502
3.438
3.382
3.333
3.289
3.341
3.277
3.221
3.172
3.128
2.889
2.825
2.769
2.720
2.676
2.625
2.560
2.503
2.452
2.408
2.316
2.247
2.187
2.133
2.085
25
30
40
50
100
5.686
5.568
5.424
5.340
5.179
4.291
4.182
4.051
3.975
3.828
3.694
3.589
3.463
3.390
3.250
3.353
3.250
3.126
3.054
2.917
3.129
3.026
2.904
2.833
2.696
2.969
2.867
2.744
2.674
2.537
2.515
2.412
2.288
2.216
2.077
2.242
2.136
2.007
1.931
1.784
1.906
1.787
1.637
1.545
1.347
5.024
3.689
3.116
2.786
2.567
2.408
1.945
1.640
1.003
ν2
∞
8
1 Percent Points of the F -Distribution
This table gives the percentage points
Fν1 ,ν2 (P ) for P = 0.01 and degrees of freedom ν1 , ν2 , as indicated by the figure to the
right.
The lower percentage points, that is the
values Fν′ 1 ,ν2 (P ) such that the probability
that F ≤ Fν′ 1 ,ν2 (P ) is equal to P/100, may
be found using the formula
P/100
Fν′ 1 ,ν2 (P ) = 1/Fν2 ,ν1 (P )
F (P )
0
ν1
1
2
3
4
5
6
12
24
∞
2
3
4
5
98.503
34.116
21.198
16.258
99.000
30.817
18.000
13.274
99.166
29.457
16.694
12.060
99.249
28.710
15.977
11.392
99.299
28.237
15.522
10.967
99.333
27.911
15.207
10.672
99.416
27.052
14.374
9.888
99.458
26.598
13.929
9.466
99.499
26.125
13.463
9.020
6
7
8
9
10
13.745
12.246
11.259
10.561
10.044
10.925
9.547
8.649
8.022
7.559
9.780
8.451
7.591
6.992
6.552
9.148
7.847
7.006
6.422
5.994
8.746
7.460
6.632
6.057
5.636
8.466
7.191
6.371
5.802
5.386
7.718
6.469
5.667
5.111
4.706
7.313
6.074
5.279
4.729
4.327
6.880
5.650
4.859
4.311
3.909
11
12
13
14
15
9.646
9.330
9.074
8.862
8.683
7.206
6.927
6.701
6.515
6.359
6.217
5.953
5.739
5.564
5.417
5.668
5.412
5.205
5.035
4.893
5.316
5.064
4.862
4.695
4.556
5.069
4.821
4.620
4.456
4.318
4.397
4.155
3.960
3.800
3.666
4.021
3.780
3.587
3.427
3.294
3.602
3.361
3.165
3.004
2.868
16
17
18
19
20
8.531
8.400
8.285
8.185
8.096
6.226
6.112
6.013
5.926
5.849
5.292
5.185
5.092
5.010
4.938
4.773
4.669
4.579
4.500
4.431
4.437
4.336
4.248
4.171
4.103
4.202
4.102
4.015
3.939
3.871
3.553
3.455
3.371
3.297
3.231
3.181
3.084
2.999
2.925
2.859
2.753
2.653
2.566
2.489
2.421
25
30
40
50
100
7.770
7.562
7.314
7.171
6.895
5.568
5.390
5.179
5.057
4.824
4.675
4.510
4.313
4.199
3.984
4.177
4.018
3.828
3.720
3.513
3.855
3.699
3.514
3.408
3.206
3.627
3.473
3.291
3.186
2.988
2.993
2.843
2.665
2.562
2.368
2.620
2.469
2.288
2.183
1.983
2.169
2.006
1.805
1.683
1.427
6.635
4.605
3.782
3.319
3.017
2.802
2.185
1.791
1.003
ν2
∞
9
0.5 Percent Points of the F -Distribution
This table gives the percentage points
Fν1 ,ν2 (P ) for P = 0.005 and degrees of freedom ν1 , ν2 , as indicated by the figure to the
right.
The lower percentage points, that is the
values Fν′ 1 ,ν2 (P ) such that the probability
that F ≤ Fν′ 1 ,ν2 (P ) is equal to P/100, may
be found using the formula
P/100
Fν′ 1 ,ν2 (P ) = 1/Fν2 ,ν1 (P )
F (P )
0
ν1
1
2
3
4
5
6
12
24
∞
2
3
4
5
198.501
55.552
31.333
22.785
199.000
49.799
26.284
18.314
199.166
47.467
24.259
16.530
199.250
46.195
23.155
15.556
199.300
45.392
22.456
14.940
199.333
44.838
21.975
14.513
199.416
43.387
20.705
13.384
199.458
42.622
20.030
12.780
199.500
41.828
19.325
12.144
6
7
8
9
10
18.635
16.236
14.688
13.614
12.826
14.544
12.404
11.042
10.107
9.427
12.917
10.882
9.596
8.717
8.081
12.028
10.050
8.805
7.956
7.343
11.464
9.522
8.302
7.471
6.872
11.073
9.155
7.952
7.134
6.545
10.034
8.176
7.015
6.227
5.661
9.474
7.645
6.503
5.729
5.173
8.879
7.076
5.951
5.188
4.639
11
12
13
14
15
12.226
11.754
11.374
11.060
10.798
8.912
8.510
8.186
7.922
7.701
7.600
7.226
6.926
6.680
6.476
6.881
6.521
6.233
5.998
5.803
6.422
6.071
5.791
5.562
5.372
6.102
5.757
5.482
5.257
5.071
5.236
4.906
4.643
4.428
4.250
4.756
4.431
4.173
3.961
3.786
4.226
3.904
3.647
3.436
3.260
16
17
18
19
20
10.575
10.384
10.218
10.073
9.944
7.514
7.354
7.215
7.093
6.986
6.303
6.156
6.028
5.916
5.818
5.638
5.497
5.375
5.268
5.174
5.212
5.075
4.956
4.853
4.762
4.913
4.779
4.663
4.561
4.472
4.099
3.971
3.860
3.763
3.678
3.638
3.511
3.402
3.306
3.222
3.112
2.984
2.873
2.776
2.690
25
30
40
50
100
9.475
9.180
8.828
8.626
8.241
6.598
6.355
6.066
5.902
5.589
5.462
5.239
4.976
4.826
4.542
4.835
4.623
4.374
4.232
3.963
4.433
4.228
3.986
3.849
3.589
4.150
3.949
3.713
3.579
3.325
3.370
3.179
2.953
2.825
2.583
2.918
2.727
2.502
2.373
2.128
2.377
2.176
1.932
1.786
1.485
7.879
5.298
4.279
3.715
3.350
3.091
2.358
1.898
1.004
ν2
∞
10
Percentage Points of the Wilcoxon Signed Rank
Distribution
This table gives the 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 which is symmetric about zero. The function
tabulated x(P ) is the largest x such that P (W + < x) ≤ P/100.
n
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
5
6
9
11
14
18
22
26
31
36
42
48
54
61
68
76
84
92
101
111
120
131
141
152
164
176
188
201
214
228
242
257
272
287
303
320
337
2.5
4
6
9
11
14
18
22
26
30
35
41
47
53
59
66
74
82
90
99
108
117
127
138
148
160
171
183
196
209
222
236
250
265
280
295
311
P
1 0.5
2
4
6
8
10
13
16
20
24
28
33
38
44
50
56
63
70
77
85
93
102
111
121
131
141
152
163
174
186
199
212
225
239
253
267
282
1
2
4
6
8
10
13
16
20
24
28
33
38
43
49
55
62
69
76
84
92
101
110
119
129
139
149
160
172
183
195
208
221
234
248
262
0.1
n
0
0
1
2
3
5
7
9
12
15
19
22
27
31
36
41
46
52
59
65
72
80
87
95
104
113
122
132
142
152
163
174
186
198
210
223
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
11
5
337
354
372
390
408
427
447
467
487
508
530
551
574
596
619
643
667
691
716
742
768
794
821
848
876
904
932
961
991
1021
1051
1082
1113
1145
1177
1210
P
2.5
311
328
344
362
379
397
416
435
454
474
495
515
537
558
580
603
626
649
673
698
722
748
773
799
826
853
880
908
937
965
995
1024
1054
1085
1116
1148
1
282
297
313
329
346
363
380
398
417
435
455
474
494
515
536
557
579
601
624
647
670
694
719
743
769
794
820
847
874
902
929
958
987
1016
1045
1076
0.5
262
277
292
308
323
340
356
374
391
409
428
446
466
485
505
526
547
568
590
612
635
658
682
706
730
755
780
806
832
859
885
913
941
969
998
1027
0.1
223
236
250
264
278
293
308
324
340
356
373
390
408
426
444
463
483
502
522
543
564
585
607
629
652
675
698
722
746
771
796
822
848
874
901
928