Download - Appendix B - Statistical Tables
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APPENDIX B Statistical Tables
TABLE B.1 THE UNIT NORMAL TABLE*
*Column A lists z-score values. A vertical line drawn through a normal distribution at a z-score location divides the
distribution into two sections.
Column B identifies the proportion in the larger section, called the body.
Column C identifies the proportion in the smaller section, called the tail.
Column D identifies the proportion between the mean and the z-score.
Note: Because the normal distribution is symmetrical, the proportions for negative z-scores are the same as those for
positive z-scores.
0 +z -z 0 0 z
(A) (B) (C) (D) (A) (B) (C ) (D)
Proportion Proportion Proportion Proportion Proportion Proportion
z in Body in Tail Between Mean and z z in Body in Tail Between Mean and z
0.00 .5000 .5000 .0000 0.25 .5987 .4013 .0987
0.01 .5040 .4960 .0040 0.26 .6026 .3974 .1026
0.02 .5080 .4920 .0080 0.27 .6064 .3936 .1064
0.03 .5120 .4880 .0120 0.28 .6103 .3897 .1103
0.04 .5160 .4840 .0160 0.29 .6141 .3859 .1141
0.05 .5199 .4801 .0199 0.30 .6179 .3821 .1179
0.06 .5239 .4761 .0239 0.31 .6217 .3783 .1217
0.07 .5279 .4721 .0279 0.32 .6255 .3745 .1255
0.08 .5319 .4681 .0319 0.33 .6293 .3707 .1293
0.09 .5359 .4641 .0359 0.34 .6331 .3669 .1331
0.10 .5398 .4602 .0398 0.35 .6368 .3632 .1368
0.11 .5438 .4562 .0438 0.36 .6406 .3594 .1406
0.12 .5478 .4522 .0478 0.37 .6443 .3557 .1443
0.13 .5517 .4483 .0517 0.38 .6480 .3520 .1480
0.14 .5557 .4443 .0557 0.39 .6517 .3483 .1517
0.15 .5596 .4404 .0596 0.40 .6554 .3446 .15540.16 .5636 .4364 .0636 0.41 .6591 .3409 .1591
0.17 .5675 .4325 .0675 0.42 .6628 .3372 .1628
0.18 .5714 .4286 .0714 0.43 .6664 .3336 .1664
0.19 .5753 .4247 .0753 0.44 .6700 .3300 .1700
0.20 .5793 .4207 .0793 0.45 .6736 .3264 .1736
0.21 .5832 .4168 .0832 0.46 .6772 .3228 .1772
0.22 .5871 .4129 .0871 0.47 .6808 .3192 .1808
0.23 .5910 .4090 .0910 0.48 .6844 .3156 .1844
0.24 .5948 .4052 .0948 0.49 .6879 .3121 .1879
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726 APPENDIX B STATISTICALTABLES
(A) (B)
Proportion
z in Body
(C)
Proportion
inTail
(D)
Proportion
Between Mean and z
(A)
z
(B)
Proportion
inBody
(C)
Proportion
inTail
(D)
Proportion
Between Mean and z
0.50 .6915
0.51 .6950
0.52 .6985
0.53 .7019
0.54 .7054
0.55 .7088
0.56 .7123
0.57 .7157
0.58 .7190
0.59 .7224
0.60 .7257
0.61 .7291
0.62 .7324
0.63 .7357
0.64 .7389
0.65 .7422
0.66 .7454
0.67 .74860.68 .7517
0.69 .7549
0.70 .7580
0.71 .7611
0.72 .7642
0.73 .7673
0.74 .7704
0.75 .7734
0.76 .7764
0.77 .7794
0.78 .7823
0.79 .7852
0.80 .7881
0.81 .7910
0.82 .7939
0.83 .7967
0.84 .7995
0.85 .8023
0.86 .8051
0.87 .8078
0.88 .8106
0.89 .8133
0.90 .8159
0.91 .8186
0.92 .8212
0.93 .8238
0.94 .82640.95 .8289
0.96 .8315
0.97 .8340
0.98 .8365
0.99 .8389
.3085
.3050
.3015
.2981
.2946
.2912
.2877
.2843
.2810
.2776
.2743
.2709
.2676
.2643
.2611
.2578
.2546
.2514
.2483
.2451
.2420
.2389
.2358
.2327
.2296
.2266
.2236
.2206
.2177
.2148
.2119
.2090
.2061
.2033
.2005
.1977
.1949
.1922
.1894
.1867
.1841
.1814
.1788
.1762
.1736
.1711
.1685
.1660
.1635
.1611
.1915
.1950
.1985
.2019
.2054
.2088
.2123
.2157
.2190
.2224
.2257
.2291
.2324
.2357
.2389
.2422
.2454
.2486
.2517
.2549
.2580
.2611
.2642
.2673
.2704
.2734
.2764
.2794
.2823
.2852
.2881
.2910
.2939
.2967
.2995
.3023
.3051
.3078
.3106
.3133
.3159
.3186
.3212
.3238
.3264
.3289
.3315
.3340
.3365
.3389
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1 . 1 0
1 . 1 1
1 . 1 2
1 . 1 3
1 . 1 4
1 . 1 5
1 . 1 6
1 . 1 7
1 . 1 8
1 . 1 9
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1 . 3 0
1 . 3 1
1 . 3 2
1 . 3 3
1 . 3 4
1 . 3 5
1 . 3 6
1 . 3 7
1 . 3 8
1 . 3 9
1.40
1.41
1.42
1.43
1.441.45
1.46
1.47
1.48
1.49
.8413
.8438
.8461
.8485
.8508
.8531
.8554
.8577
.8599
.8621
.8643
.8665
.8686
.8708
.8729
.8749
.8770
.8790
.8810
.8830
.8849
.8869
.8888
.8907
.8925
.8944
.8962
.8980
.8997
.9015
.9032
.9049
.9066
.9082
.9099
.9115
.9131
.9147
.9162
.9177
.9192
.9207
.9222
.9236
.9251
.9265
.9279
.9292
.9306
.9319
.1587
.1562
.1539
.1515
.1492
.1469
.1446
.1423
.1401
.1379
.1357
.1335
.1314
.1292
.1271
.1251
.1230
.1210
.1190
.1170
.1151
.1131
.1112
.1093
.1075
.1056
.1038
.1020
.1003
.0985
.0968
.0951
.0934
.0918
.0901
.0885
.0869
.0853
.0838
.0823
.0808
.0793
.0778
.0764
.0749
.0735
.0721
.0708
.0694
.0681
.3413
.3438
.3461
.3485
.3508
.3531
.3554
.3577
.3599
.3621
.3643
.3665
.3686
.3708
.3729
.3749
.3770
.3790
.3810
.3830
.3849
.3869
.3888
.3907
.3925
.3944
.3962
.3980
.3997
.4015
.4032
.4049
.4066
.4082
.4099
.4115
.4131
.4147
.4162
.4177
.4192
.4207
.4222
.4236
.4251
.4265
.4279
.4292
.4306
.4319
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APPEND IX B STAT IST ICAL TABLES
(A) (B)
Proportion
z in Body
1.50 .9332
1.51 .9345
1.52 .9357
1.53 .9370
1.54 .9382
1.55 .9394
1.56 .9406
1.57 .9418
1.58 .9429
1.59 .9441
1.60 .9452
1.61 .9463
1.62 .9474
1.63 .9484
1.64 .9495
1.65 .9505
1.66 .9515
1.67 .95251.68 .9535
1.69 .9545
1.70 .9554
1.71 .9564
1.72 .9573
1.73 .9582
1.74 .9591
1.75 .9599
1.76 .9608
1.77 .9616
1.78 .9625
1.79 .9633
1.80 .9641
1.81 .9649
1.82 .9656
1.83 .9664
1.84 .9671
1.85 .9678
1.86 .9686
1.87 .9693
1.88 .9699
1.89 .9706
1.90 .9713
1.91 .9719
1.92 .9726
1.93 .9732
1.94 .9738
1.95 .9744
1.96 .9750
1.97 .9756
1.98 .9761
1.99 .9767
(C)
Proportion
inTail
.0668
.0655
.0643
.0630
.0618
.0606
.0594
.0582
.0571
.0559
.0548
.0537
.0526
.0516
.0505
.0495
.0485
.0475
.0465
.0455
.0446
.0436
.0427
.0418
.0409
.0401
.0392
.0384
.0375
.0367
.0359
.0351
.0344
.0336
.0329
.0322
.0314
.0307
.0301
.0294
.0287
.0281
.0274
.0268
.0262
.0256
.0250
.0244
.0239
.0233
(D)
Proportion
Between Mean and z
.4332
.4345
.4357
.4370
.4382
.4394
.4406
.4418
.4429
.4441
.4452
.4463
.4474
.4484
.4495
.4505
.4515
.4525
.4535
.4545
.4554
.4564
.4573
.4582
.4591
.4599
.4608
.4616
.4625
.4633
.4641
.4649
.4656
.4664
.4671
.4678
.4686
.4693
.4699
.4706
.4713
.4719
.4726
.4732
.4738
.4744
.4750
.4756
.4761
.4767
(A) (B)
Proportion
z inBody
2.00 .9772
2.01 .9778
2.02 .9783
2.03 .9788
2.04 .9793
2.05 .9798
2.06 .9803
2.07 .9808
2.08 .9812
2.09 .9817
2.10 .9821
2.11 .9826
2.12 .9830
2.13 .9834
2.14 .9838
2.15 .9842
2.16 .9846
2.17 .98502.18 .9854
2.19 .9857
2.20 .9861
2.21 .9864
2.22 .9868
2.23 .9871
2.24 .9875
2.25 .9878
2.26 .9881
2.27 .9884
2.28 .9887
2.29 .9890
2.30 .9893
2.31 .9896
2.32 .9898
2.33 .9901
2.34 .9904
2.35 .9906
2.36 .9909
2.37 .9911
2.38 .9913
2.39 .9916
2.40 .9918
2.41 .9920
2.42 .9922
2.43 .9925
2.44 .9927
2.45 .9929
2.46 .9931
2.47 .9932
2.48 .9934
2.49 .9936
(C)
Proportion
inTail
(D)
Proportion
Between Mean and z
.0228
.0222
.0217
.0212
.0207
.0202
.0197
.0192
.0188
.0183
.0179
.0174
.0170
.0166
.0162
.0158
.0154
.0150
.0146
.0143
.0139
.0136
.0132
.0129
.0125
.0122
.0119
.0116
.0113
.0110
.0107
.0104
.0102
.0099
.0096
.0094
.0091
.0089
.0087
.0084
.0082
.0080
.0078
.0075
.0073
.0071
.0069
.0068
.0066
.0064
.4772
.4778
.4783
.4788
.4793
.4798
.4803
.4808
.4812
.4817
.4821
.4826
.4830
.4834
.4838
.4842
.4846
.4850
.4854
.4857
.4861
.4864
.4868
.4871
.4875
.4878
.4881
.4884
.4887
.4890
.4893
.4896
.4898
.4901
.4904
.4906
.4909
.4911
.4913
.4916
.4918
.4920
.4922
.4925
.4927
.4929
.4931
.4932
.4934
.4936
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728 APPENDIX B STATISTICALTABLES
(A) (B)
Proportion
z in Body
(C)
Proportion
inTail
(D)
Proportion
Between Mean and z
(A)
z
(B)
Proportion
inBody
(C)
Proportion
inTail
(D)
Proportion
Between Mean and z
2.50 .9938
2.51 .9940
2.52 .9941
2.53 .9943
2.54 .9945
2.55 .9946
2.56 .9948
2.57 .9949
2.58 .9951
2.59 .9952
2.60 .9953
2.61 .9955
2.62 .9956
2.63 .9957
2.64 .9959
2.65 .9960
2.66 .9961
2.67 .99622.68 .9963
2.69 .9964
2.70 .9965
2.71 .9966
2.72 .9967
2.73 .9968
2.74 .9969
2.75 .9970
2.76 .9971
2.77 .9972
2.78 .9973
2.79 .9974
2.80 .9974
2.81 .9975
2.82 .9976
2.83 .9977
2.84 .9977
2.85 .9978
2.86 .9979
2.87 .9979
2.88 .9980
2.89
2.90
2.91
2.92
2.93
2.94
.9981
.9981
.9982
.9982
.9983
.9984
.0062
.0060
.0059
.0057
.0055
.0054
.0052
.0051
.0049
.0048
.0047
.0045
.0044
.0043
.0041
.0040
.0039
.0038
.0037
.0036
.0035
.0034
.0033
.0032
.0031
.0030
.0029
.0028
.0027
.0026
.0026
.0025
.0024
.0023
.0023
.0022
.0021
.0021
.0020
.0019
.0019
.0018
.0018
.0017
.0016
.4938
.4940
.4941
.4943
.4945
.4946
.4948
.4949
.4951
.4952
.4953
.4955
.4956
.4957
.4959
.4960
.4961
.4962
.4963
.4964
.4965
.4966
.4967
.4968
.4969
.4970
.4971
.4972
.4973
.4974
.4974
.4975
.4976
.4977
.4977
.4978
.4979
.4979
.4980
.4981
.4981
.4982
.4982
.4983
.4984
2.95
2.96
2.97
2.98
2.99
3.00
3.01
3.02
3.03
3.04
3.05
3.06
3.07
3.08
3.09
3.10
3.11
3.123.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.30
3.40
3.50
3.60
3.70
3.80
3.90
4.00
.9984
.9985
.9985
.9986
.9986
.9987
.9987
.9987
.9988
.9988
.9989
.9989
.9989
.9990
.9990
.9990
.9991
.9991
.9991
.9992
.9992
.9992
.9992
.9993
.9993
.9993
.9993
.9994
.9994
.9994
.9995
.9997
.9998
.9998
.9999
.99993
.99995
.99997
.0016
.0015
.0015
.0014
.0014
.0013
.0013
.0013
.0012
.0012
.0011
.0011
.0011
.0010
.0010
.0010
.0009
.0009
.0009
.0008
.0008
.0008
.0008
.0007
.0007
.0007
.0007
.0006
.0006
.0006
.0005
.0003
.0002
.0002
.0001
.00007
.00005
.00003
.4984
.4985
.4985
.4986
.4986
.4987
.4987
.4987
.4988
.4988
.4989
.4989
.4989
.4990
.4990
.4990
.4991
.4991
.4991
.4992
.4992
.4992
.4992
.4993
.4993
.4993
.4993
.4994
.4994
.4994
.4995
.4997
.4998
.4998
.4999
.49993
.49995
.49997
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APPEND IX B STAT IST ICAL TABLES
TABLE B .2 THE t DISTRIBUTION
Table entries are values of t corresponding to proportions in one tailor in two tails combined.
One tail
(either right or left)
Two tails
combined
Proportion in One Tail
0.25 0.10 0.05 0.025 0.01 0.005
Proportion in Two Tails Combined
df 0.50 0.20 0.10 0.05 0.02 0.01
1.000 3.078 6.314 12.706 31.821 63.657
2 0.816 1.886 2.920 4.303 6.965 9.925
3 0.765 1.638 2.353 3.182 4.541 5.841
4 0.741 1.533 2.132 2.776 3.747 4.604
5 0.727 1.476 2.015 2.571 3.365 4.032
6 0.718 1.440 1.943 2.447 3.143 3.707
7 0.711 1.415 1.895 2.365 2.998 3.499
8 0.706 1.397 1.860 2.306 2.896 3.355
9 0.703 1.383 1.833 2.262 2.821 3.250
10 0.700 1.372 1.812 2.228 2.764 3.169
11 0.697 1.363 1.796 2.201 2.718 3.106
12 0.695 1.356 1.782 2.179 2.681 3.055
13 0.694 1.350 1.771 2.160 2.650 3.012
14 0.692 1.345 1.761 2.145 2.624 2.97715 0.691 1.341 1.753 2.131 2.602 2.947
16 0.690 1.337 1.746 2.120 2.583 2.921
17 0.689 1.333 1.740 2.110 2.567 2.898
18 0.688 1.330 1.734 2.101 2.552 2.878
19 0.688 1.328 1.729 2.093 2.539 2.861
20 0.687 1.325 1.725 2.086 2.528 2.845
21 0.686 1.323 1.721 2.080 2.518 2.831
22 0.686 1.321 1.717 2.074 2.508 2.819
23 0.685 1.319 1.714 2.069 2.500 2.807
24 0.685 1.318 1.711 2.064 2.492 2.797
25 0.684 1.316 1.708 2.060 2.485 2.787
26 0.684 1.315 1.706 2.056 2.479 2.779
27 0.684 1.314 1.703 2.052 2.473 2.771
28 0.683 1.313 1.701 2.048 2.467 2.76329 0.683 1.311 1.699 2.045 2.462 2.756
30 0.683 1.310 1.697 2.042 2.457 2.750
40 0.681 1.303 1.684 2.021 2.423 2.704
60 0.679 1.296 1.671 2.000 2.390 2.660
120 0.677 1.289 1.658 1.980 2.358 2.617
00 0.674 1.282 1.645 1.960 2.326 2.576
Table III of R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research, 6th ed. London: Longman Group Lt
1974 (previously published by Oliver and Boyd Ltd., Edinburgh). Copyright ©1963 R. A. Fisher and F. Yates. Adapted and reprinted with
permission of Pearson Education Limited.
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730 A PPE ND IX B ST ATIST IC AL TA BLE S
TABLE B.3 CRITICAL VALUES FOR THE F-MAX STATISTIC*
*The critical values for ex= .05 are in lightface type, and for ex= .01, they are in boldface type.
k =Number of Samples
n - 1 2 3 4 5 6 7 8 9 10 11 12
4 9.60 15.5 20.6 25.2 29.5 33.6 37.5 41.4 44.6 48.0 51.4
23.2 37. 49. 59. 69. 79. 89. 97. 106. 113. 120.
5 7.15 10.8 13.7 16.3 18.7 20.8 22.9 24.7 26.5 28.2 29.9
14.9 22. 28. 33. 38. 42. 46. 50. 54. 57. 60.
6 5.82 8.38 10.4 12.1 13.7 15.0 16.3 17.5 18.6 19.7 20.7
11.1 15.5 19.1 22. 25. 27. 30. 32. 34. 36. 37.
7 4.99 6.94 8.44 9.70 10.8 11.8 12.7 13.5 14.3 15.1 15.8
8.89 12.1 14.5 16.5 18.4 20. 22. 23. 24. 26. 27.
8 4.43 6.00 7.18 8.12 9.03 9.78 10.5 11.1 11.7 12.2 12.7
7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 19.8 21.
9 4.03 5.34 6.31 7.11 7.80 8.41 8.95 9.45 9.91 10.3 10.7
6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 16.0 16.6
10 3.72 4.85 5.67 6.34 6.92 7.42 7.87 8.28 8.66 9.01 9.34
5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 13.4 13.9
12 3.28 4.16 4.79 5.30 5.72 6.09 6.42 6.72 7.00 7.25 7.48
4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 10.2 10.6
15 2.86 3.54 4.01 4.37 4.68 4.95 5.19 5.40 5.59 5.77 5.93
4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 7.8 8.0
20 2.46 2.95 3.29 3.54 3.76 3.94 4.10 4.24 4.37 4.49 4.59
3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 5.8 5.9
30 2.07 2.40 2.61 2.78 2.91 3.02 3.12 3.21 3.29 3.36 3.39
2.63 3.0 3.3 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2
60 1.67 1.85 1.96 2.04 2.11 2.17 2.22 2.26 2.30 2.33 2.36
1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 2.7 2.7
Table 31 ofE. Pearson and H. O. Hartley, Biometrika Tablesfor Statisticians, 2nd ed. New York: Cambridge University Press, 1958. Adapted and
reprinted with permission of the Biometrika trustees.
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TABLE B .4 THE F DISTRIBUTION*
APPEND IX B STAT IST ICAL TABLES
*Table entries in lightface type are critical values for the .05 level of significance.
Boldface type values are for the .01 level of significance.
Critical
F
Degrees of
Freedom:
Denominator 2 3 4 5 6
Degrees of Freedom: Numerator
7 8 9 10 11 12 14 16 2
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
161 200 2164052 4999 5403
2255625
230 2345764 5859
2375928
18.51 19.00 19.16 19.25 19.30 19.33 19.36
98.49 99.00 99.17 99.25 99.30 99.33 99.34
10.13 9.55 9.28 9.12 9.01 8.94 8.88
34.12 30,92 29.46 28.71 28.24 27.91 27.67
2395981
19.37 19.38
99.36 99.38
8.84 8.81
27.49 27.34
6.04
14.80
4.82
10.27
4.15
8.10
3.73
6.84
3.44
6.03
3.23
5.47
3.07
5.06
2.95
4.74
2.85
4.50
2.77
4.30
2.70
4.14
2.64
4.00
2.59
3.89
2.55
3.79
2.51
3.71
6.00
14.66
4.78
10.15
4.10
7.98
3.68
6.71
3.39
5.91
3.18
5.35
3.02
4.95
2.90
4.63
2.80
4.39
2.72
4.19
2.65
4.03
2.59
3.89
2.54
3.78
2.50
3.68
2.46
3.60
2416022
2426056
7.71 6.94 6.59 6.39 6.26 6.16
21.20 18.00 16.69 15.98 15.52 15.21
6.61 5.79 5.41 5.19 5.05 4.95
16.26 13.27 12.06 11.39 10.97 10.67
5.99 5.14 4.76 4.53 4.39 4.28
13.74 10,92 9.78 9.15 8.75 8.47
5.59 4.74 4.35 4.12 3.97 3.87
12.25 9.55 8.45 7.85 7.46 7.19
5.32 4.46 4.07 3.84 3.69 3.58
11.26 8.65 7.59 7.01 6.63 6.37
5.12 4.26 3.86 3.63 3.48 3.37
10.56 8.02 6.99 6.42 6.06 5.80
4.96 4.10 3.71 3.48 3.33 3.22
10.04 7.56 6.55 5.99 5.64 5.39
4.84 3.98 3.59 3.36 3.20 3.09
9.65 7.20 6.22 5.67 5.32 5.07
4.75 3.88 3.49 3.26 3.11 3.00
9.33 6.93 5.95 5.41 5.06 4.82
4.67 3.80 3.41 3.18 3.02 2.92
9.07 6.70 5.74 5.20 4.86 4.62
4.60 3.74 3.34 3.11 2.96 2.85
8.86 6.51 5.56 5.03 4.69 4.46
4.54 3.68 3.29 3.06 2.90 2.79
8.68 6.36 5.42 4.89 4.56 4.32
4.49 3.63 3.24 3.01 2.85 2.74
8.53 6.23 5.29 4.77 4.44 4.20
4.45 3.59 3.20 2.96 2.81 2.70
8.40 6.11 5.18 4.67 4.34 4.10
4.41 3.55 3.16 2.93 2.77 2.66
8.28 6.01 5.09 4.58 4.25 4.01
6.09
14.98
4.88
10.45
4.21
8.26
3.79
7.00
3.50
6.19
3.29
5.62
3.14
5.21
3.01
4.88
2.92
4.65
2.84
4.44
2.77
4.28
2.70
4.14
2.66
4.03
2.62
3.93
2.58
3.85
2436082
244 245 246 26106 6142 6169 62
19.39 19.40 19.41 19.42 19.43 19
99.40 99.41 99.42 99.43 99.44 99
8.78 8.76 8.74 8.71 8.69 8
27.23 27.13 27.05 26.92 26.83 26
5.96 5.93 5.91 5.87 5.84 5
14.54 14.45 14.37 14.24 14.15 14
4.74 4.70 4.68 4.64 4.60 4
10.05 9.96 9.89 9.77 9.68 9
4.06 4.03 4.00 3.96 3.92 3
7.87 7.79 7.72 7.60 7.52 7
3.63 3.60 3.57 3.52 3.49 3
6.62 6.54 6.47 6.35 6.27 6
3.34 3.31 3.28 3.23 3.20 3
5.82 5.74 5.67 5.56 5.48 5
3.13 3.10 3.07 3.02 2.98 2
5.26 5.18 5.11 5.00 4.92 4
2.97 2.94 2.91 2.86 2.82 2
4.85 4.78 4.71 4.60 4.52 4
2.86 2.82 2.79 2.74 2.70 2
4.54 4.46 4.40 4.29 4.21 4
2.76 2.72 2.69 2.64 2.60 2
4.30 4.22 4.16 4.05 3.98 3
2.67 2.63 2.60 2.55 2.51 2
4.10 4.02 3.96 3.85 3.78 3
2.60 2.56 2.53 2.48 2.44 2
3.94 3.86 3.80 3.70 3.62 3
2.55 2.51 2.48 2.43 2.39 2
3.80 3.73 3.67 3.56 3.48 3
2.49 2.45 2.42 2.37 2.33 2
3.69 3.61 3.55 3.45 3.37 3
2.45 2.41 2.38 2.33 2.29 2
3.59 3.52 3.45 3.35 3.27 3
2.41 2.37 2.34 2.29 2.25 2
3.51 3.44 3.37 3.27 3.19 3
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732 APPENDIX B STATISTICALTABLES
TABLE B .4 (continued)
Degrees of
Freedom:
Denominator 2 3 4 5 6
Degrees of Freedom: Numerator
98 10 11 12 14 16 20
19
20
21
22
23
24
25
26
27
28
29
30
32
34
36
38
40
42
4.38
8.18
4.35
8.10
4.32
8.02
4.30
7.94
4.28
7.88
4.26
7.82
4.24
7.77
4.227.72
4.21
7.68
4.20
7.64
4.18
7.60
4.17
7.56
4.15
7.50
4.13
7.44
4.11
7.39
4.10
7.35
4.08
7.31
4.07
7.27
3.52 3.13
5.93 5.01
3.49 3.10
5.85 4.94
3.47 3.07
5.78 4.87
3.44 3.05
5.72 4.82
3.42 3.03
5.66 4.76
3.40 3.01
5.61 4.72
3.38 2.99
5.57 4.68
3.37 2.985.53 4.64
3.35 2.96
5.49 4.60
3.34 2.95
5.45 4.57
3.33 2.93
5.42 4.54
3.32 2.92
5.39 4.51
3.30 2.90
5.34 4.46
3.28 2.88
5.29 4.42
3.26 2.86
5.25 4.38
3.25 2.85
5.21 4.34
3.23 2.84
5.18 4.31
3.22 2.83
5.15 4.29
2.90
4.50
2.87
4.43
2.84
4.37
2.82
4.31
2.80
4.26
2.78
4.22
2.76
4.18
2.744.14
2.73
4.11
2.71
4.07
2.70
4.04
2.69
4.02
2.67
3.97
2.65
3.93
2.63
3.89
2.62
3.86
2.61
3.83
2.59
3.80
2.74
4.17
2.71
4.10
2.68
4.04
2.66
3.99
2.64
3.94
2.62
3.90
2.60
3.86
2.593.82
2.57
3.79
2.56
3.76
2.54
3.73
2.53
3.70
2.51
3.66
2.49
3.61
2.48
3.58
2.46
3.54
2.45
3.51
2.44
3.49
2.63
3.94
2.60
3.87
2.57
3.81
2.55
3.76
2.53
3.71
2.51
3.67
2.49
3.63
2.473.59
2.46
3.56
2.44
3.53
2.43
3.50
2.42
3.47
2.40
3.42
2.38
3.38
2.36
3.35
2.35
3.32
2.34
3.29
2.32
3.26
2.55
3.77
2.52
3.71
2.49
3.65
2.47
3.59
2.45
3.54
2.43
3.50
2.41
3.46
2.393.42
2.37
3.39
2.36
3.36
2.35
3.33
2.34
3.30
2.32
3.25
2.30
3.21
2.28
3.18
2.26
3.15
2.25
3.12
2.24
3.10
2.48
3.63
2.45
3.56
2.42
3.51
2.40
3.45
2.38
3.41
2.36
3.36
2.34
3.32
2.323.29
2.30
3.26
2.29
3.23
2.28
3.20
2.27
3.17
2.25
3.12
2.23
3.08
2.21
3.04
2.19
3.02
2.18
2.99
2.17
2.96
2.43
3.52
2.40
3.45
2.37
3.40
2.35
3.35
2.32
3.30
2.30
3.25
2.28
3.21
2.273.17
2.25
3.14
2.24
3.11
2.22
3.08
2.21
3.06
2.19
3.01
2.17
2.97
2.15
2.94
2.14
2.91
2.12
2.88
2.11
2.86
2.38
3.43
2.35
3.37
2.32
3.31
2.30
3.26
2.28
3.21
2.26
3.17
2.24
3.13
2.223.09
2.20
3.06
2.19
3.03
2.18
3.00
2.16
2.98
2.14
2.94
2.12
2.89
2.10
2.86
2.09
2.82
2.07
2.80
2.06
2.77
2.34
3.36
2.31
3.30
2.28
3.24
2.26
3.18
2.24
3.14
2.22
3.09
2.20
3.05
2.183.02
2.16
2.98
2.15
2.95
2.14
2.92
2.12
2.90
2.10
2.86
2.08
2.82
2.06
2.78
2.05
2.75
2.04
2.73
2.02
2.70
2.31
3.30
2.28
3.23
2.25
3.17
2.23
3.12
2.20
3.07
2.18
3.03
2.16
2.99
2.152.96
2.13
2.93
2.12
2.90
2.10
2.87
2.09
2.84
2.07
2.80
2.05
2.76
2.03
2.72
2.02
2.69
2.00
2.66
1.99
2.64
2.26 2.21 2.15
3.19 3.12 3.00
2.23 2.18 2.12
3.13 3.05 2.94
2.20 2.15 2.09
3.07 2.99 2.88
2.18 2.13 2.07
3.02 2.94 2.83
2.14 2.10 2.04
2.97 2.89 2.78
2.13 2.09 2.02
2.93 2.85 2.74
2.11 2.06 2.00
2.89 2.81 2.70
2.10 2.05 1.992.86 2.77 2.66
2.08 2.03 1.97
2.83 2.74 2.63
2.06 2.02 1.96
2.80 2.71 2.60
2.05 2.00 1.94
2.77 2.68 2.57
2.04 1.99 1.93
2.74 2.66 2.55
2.02 1.97 1.91
2.70 2.62 2.51
2.00 1.95 1.89
2.66 2.58 2.47
1.98 1.93 1.87
2.62 2.54 2.43
1.96 1.92 1.85
2.59 2.51 2.40
1.95 1.90 1.84
2.56 2.49 2.37
1.94 1.89 1.82
2.54 2.46 2.35
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TABLE B .4 (continued)
12
APPEND IX B STAT IST ICAL TABLES
6
Degrees of Freedom: Numerator
9
2.23
3.07
2.22
3.05
2.21
3.04
2.20
3.02
2.18
2.98
2.17
2.95
2.15
2.93
2.142.91
2.12
2.87
2.10
2.82
2.08
2.79
2.07
2.76
2.05
2.73
2.03
2.69
2.02
2.66
2.01
2.64
8
2.16
2.94
2.14
2.92
2.14
2.90
2.13
2.88
2.11
2.85
2.10
2.82
2.08
2.79
2.072.77
2.05
2.74
2.03
2.69
2.01
2.65
2.00
2.62
1.98
2.60
1.96
2.55
1.95
2.53
1.94
2.51
2.10
2.84
2.09
2.82
2.08
2.80
2.07
2.78
2.05
2.75
2.04
2.72
2.02
2.70
2.012.67
1.99
2.64
1.97
2.59
1.95
2.56
1.94
2.53
1.92
2.50
1.90
2.46
1.89
2.43
1.88
2.41
10
2.05
2.75
2.04
2.73
2.03
2.71
2.02
2.70
2.00
2.66
1.99
2.63
1.98
2.61
1.972.59
1.95
2.55
1.92
2.51
1.90
2.47
1.89
2.44
1.87
2.41
1.85
2.37
1.84
2.34
1.83
2.32
11
2.01
2.68
2.00
2.66
1.99
2.64
1.98
2.62
1.97
2.59
1.95
2.56
1.94
2.54
1.932.51
1.91
2.48
1.88
2.43
1.86
2.40
1.85
2.37
1.83
2.34
1.81
2.29
1.80
2.26
1.79
2.24
44
46
48
50
55
60
65
70
80
100
125
150
200
400
2
4.06 3.21
7.24 5.12 4.26
4.05 3.20 2.81
7.21 5.10 4.24
4.04 3.19 2.80
7.19 5.08 4.22
4.03 3.18 2.79
7.17 5.06 4.20
4.02 3.17 2.78
7.12 5.01 4.16
4.00 3.15 2.76
7.08 4.98 4.13
3.99 3.14 2.75
7.04 4.95 4.10
3.98 3.13 2.747.01 4.92 4.08
3.96 3.11 2.72
6.96 4.88 4.04
3.94 3.09 2.70
6.90 4.82 3.98
3.92 3.07 2.68
6.84 4.78 3.94
3.91 3.06 2.67
6.81 4.75 3.91
3.89 3.04 2.65
6.76 4.71 3.88
3.86 3.02 2.62
6.70 4.66 3.83
3.85 3.00 2.61
6.66 4.62 3.80
3.84 2.99 2.60
6.64 4.60 3.78
Degrees of
Freedom:
Denominator 3 4 5 14 16 2
1000
2.82 2.58
3.78
2.57
3.76
2.56
3.74
2.56
3.72
2.54
3.68
2.52
3.65
2.51
3.62
2.503.60
2.48
3.56
2.46
3.51
2.44
3.47
2.43
3.44
2.41
3.41
2.39
3.36
2.38
3.34
2.37
3.32
2.43
3.46
2.42
3.44
2.41
3.42
2.40
3.41
2.38
3.37
2.37
3.34
2.36
3.31
2.353.29
2.33
3.25
2.30
3.20
2.29
3.17
2.27
3.14
2.26
3.11
2.23
3.06
2.22
3.04
2.21
3.02
2.31
3.24
2.30
3.22
2.30
3.20
2.29
3.18
2.27
3.15
2.25
3.12
2.24
3.09
2.233.07
2.21
3.04
2.19
2.99
2.17
2.95
2.16
2.92
2.14
2.90
2.12
2.85
2.10
2.82
2.09
2.80
1.98
2.62
1.97
2.60
1.96
2.58
1.95
2.56
1.93
2.53
1.92
2.50
1.90
2.47
1.892.45
1.88
2.41
1.85
2.36
1.83
2.33
1.82
2.30
1.80
2.28
1.78
2.23
1.76
2.20
1.75
2.18
1.92 1.88 1
2.52 2.44 2
1.91 1.87 1
2.50 2.42 2
1.90 1.86 1
2.48 2.40 2
1.90 1.85 1
2.46 2.39 2
1.88 1.83 1
2.43 2.35 2
1.86 1.81 1
2.40 2.32 2
1.85 1.80 1
2.37 2.30 2
1.84 1.79 12.35 2.28 2
1.82 1.77 1
2.32 2.24 2
1.79 1.75 1
2.26 2.19 2
1.77 1.72 1
2.23 2.15 2
1.76 1.71 1
2.20 2.12 2
1.74 1.69 1
2.17 2.09 1
1.72 1.67 1
2.12 2.04 1
1.70 1.65 1
2.09 2.01 1
1.69 1.64 1
2.07 1.99 1
Table A14 of Statistical Methods, 7th 00. by George W. Snedecor and Will iam G. Cochran. Copyright © 1980 by the Iowa State University Pres
2121 South State Avenue, Ames, Iowa 50010. Reprinted with permission of the Iowa State University Press.
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734 APPEND IX B STAT IST ICAL TABLES
TABLE B .5 TH E STU DE NTIZE D R AN GE STA TISTIC (q)*
*The critical values for q corresponding to ex= .05 (lightface type) and ex= .01 (boldface type).
dffm:
ErrorTenn 2 3 4 5 6
k =Number of Treatments
8 910 11 12
120
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
24
30
40
60
00
3.64
5.70
3.46
5.24
3.34
4.95
3.26
4.75
3.20
4.60
3.15
4.483.11
4.39
3.08
4.32
3.06
4.26
3.03
4.21
3.01
4.17
3.00
4.13
2.98
4.10
2.97
4.07
2.96
4.05
2.95
4.02
2.92
3.96
2.89
3.89
2.863.82
2.83
3.76
2.80
3.70
2.77
3.64
4.60
6.98
4.34
6.33
4.16
5.92
4.04
5.64
3.95
5.43
3.88
5.273.82
5.15
3.77
5.05
3.73
4.96
3.70
4.89
3.67
4.84
3.65
4.79
3.63
4.74
3.61
4.70
3.59
4.67
3.58
4.64
3.53
4.55
3.49
4.45
3.444.37
3.40
4.28
3.36
4.20
3.31
4.12
5.22
7.80
4.90
7.03
4.68
6.54
4.53
6.20
4.41
5.96
4.33
5.774.26
5.62
4.20
5.50
4.15
5.40
4.11
5.32
4.08
5.25
4.05
5.19
4.02
5.14
4.00
5.09
3.98
5.05
3.96
5.02
3.90
4.91
3.85
4.80
3.794.70
3.74
4.59
3.68
4.50
3.63
4.40
5.67
8.42
5.30
7.56
5.06
7.01
4.89
6.62
4.76
6.35
4.65
6.144.57
5.97
4.51
5.84
4.45
5.73
4.41
5.63
4.37
5.56
4.33
5.49
4.30
5.43
4.28
5.38
4.25
5.33
4.23
5.29
4.17
5.17
4.10
5.05
4.044.93
3.98
4.82
3.92
4.71
3.86
4.60
6.03
8.91
5.63
7.97
5.36
7.37
5.17
6.96
5.02
6.66
4.91
6.434.82
6.25
4.75
6.10
4.69
5.98
4.64
5.88
4.59
5.80
4.56
5.72
4.52
5.66
4.49
5.60
4.47
5.55
4.45
5.51
4.37
5.37
4.30
5.24
4.235.11
4.16
4.99
4.10
4.87
4.03
4.76
6.33
9.32
5.90
8.32
5.61
7.68
5.40
7.24
5.24
6.91
5.12
6.675.03
6.48
4.95
6.32
4.88
6.19
4.83
6.08
4.78
5.99
4.74
5.92
4.70
5.85
4.67
5.79
4.65
5.73
4.62
5.69
4.54
5.54
4.46
5.40
4.395.26
4.31
5.13
4.24
5.01
4.17
4.88
6.58
9.67
6.12
8.61
5.82
7.94
5.60
7.47
5.43
7.13
5.30
6.875.20
6.67
5.12
6.51
5.05
6.37
4.99
6.26
4.94
6.16
4.90
6.08
4.86
6.01
4.82
5.94
4.79
5.89
4.77
5.84
4.68
5.69
4.60
5.54
4.525.39
4.44
5.25
4.36
5.12
4.28
4.99
6.80
9.97
6.32
8.87
6.00
8.17
5.77
7.68
5.59
7.33
5.46
7.055.35
6.84
5.27
6.67
5.19
6.53
5.13
6.41
5.08
6.31
5.03
6.22
4.99
6.15
4.96
6.08
4.92
6.02
4.90
5.97
4.81
5.81
4.72
5.65
4.635.50
4.55
5.36
4.47
5.21
4.39
5.08
6.99
10.24
6.49
9.10
6.16
8.37
5.92
7.86
5.74
7.49
5.60
7.215.49
6.99
5.39
6.81
5.32
6.67
5.25
6.54
5.20
6.44
5.15
6.35
5.11
6.27
5.07
6.20
5.04
6.14
5.01
6.09
4.92
5.92
4.82
5.76
4.735.60
4.65
5.45
4.56
5.30
4.47
5.16
7.17
10.48
6.65
9.30
6.30
8.55
6.05
8.03
5.87
7.65
5.72
7.365.61
7.13
5.51
6.94
5.43
6.79
5.36
6.66
5.31
6.55
5.26
6.46
5.21
6.38
5.17
6.31
5.14
6.25
5.11
6.19
5.01
6.02
4.92
5.85
4.825.69
4.73
5.53
4.64
5.37
4.55
5.23
7.32
10.70
6.79
9.48
6.43
8.71
6.18
8.18
5.98
7.78
5.83
7.495.71
7.25
5.61
7.06
5.53
6.90
5.46
6.77
5.40
6.66
5.35
6.56
5.31
6.48
5.27
6.41
5.23
6.34
5.20
6.28
5.10
6.11
5.00
5.93
4.905.76
4.81
5.60
4.71
5.44
4.62
5.29
Table 29 of E. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, 2nd ed. New York: Cambridge University Press, 1958. Adapted and
reprinted with permission of the Biometrika trustees.
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APPEND IX B STAT IST ICAL TABLES
TABLE B .6 CRITICAL VALUES FOR THE PEARSON CORRELATION*
*To be significant, the sample correlation, r, must be greater than or equal to the critical value in the table.
Level of Significance for
One-Tailed Test
.05 .025 .01 .005
Level of Significance for
Two-Tailed Test
df= n - 2 .10 .05 .02 .01
.988 .997 .9995 .9999
2 .900 .950 .980 .990
3 .805 .878 .934 .959
4 .729 .811 .882 .917
5 .669 .754 .833 .874
6 .622 .707 .789 .834
7 .582 .666 .750 .798
8 .549 .632 .716 .765
9 .521 .602 .685 .73510 .497 .576 .658 .708
11 .476 .553 .634 .684
12 .458 .532 .612 .661
13 .441 .514 .592 .641
14 .426 .497 .574 .623
15 .412 .482 .558 .606
16 .400 .468 .542 .590
17 .389 .456 .528 .575
18 .378 .444 .516 .561
19 .369 .433 .503 .549
20 .360 .423 .492 .537
21 .352 .413 .482 .526
22 .344 .404 .472 .515
23 .337 .396 .462 .505
24 .330 .388 .453 .496
25 .323 .381 .445 .487
26 .317 .374 .437 .479
27 .311 .367 .430 .471
28 .306 .361 .423 .463
29 .301 .355 .416 .456
30 .296 .349 .409 .449
35 .275 .325 .381 .418
40 .257 .304 .358 .393
45 .243 .288 .338 .372
50 .231 .273 .322 .354
60 .211 .250 .295 .325
70 .195 .232 .274 .30280 .183 .217 .256 .283
90 .173 .205 .242 .267
100 .164 .195 .230 .254
Table VI of R. A. Fisher and F. Yates, Statistical Tables for Biological,
Agricultural and Medical Research, 6th ed. London: Longman Group Ltd.,
1974 (previously published by Oliver and Boyd Ltd., Edinburgh). Copyright
©1963 R A. Fisher and F. Yates. Adapted and reprinted with permission of
Pearson Education Limited.
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736 APPENDIX B STATISTICALTABLES
TABLE B .7 CRITICAL VALUES FOR THE SPEARMAN CORRELATION *
*To be significant, the sample correlation, rs, must be greater than or equal to the critical value in the table.
Level of Significance for
One-Tailed Test
.05 .025 .01 .005
Level of Significance for
Two-Tailed Test
n .10 .05 .02 .01
4 1.000
5 0.900 1.000 1.000
6 0.829 0.886 0.943 1.000
7 0.714 0.786 0.893 0.929
8 0.643 0.738 0.833 0.881
9 0.600 0.700 0.783 0.833
10 0.564 0.648 0.745 0.794
11 0.536 0.618 0.709 0.755
12 0.503 0.587 0.671 0.727
13 0.484 0.560 0.648 0.703
14 0.464 0.538 0.622 0.675
15 0.443 0.521 0.604 0.654
16 0.429 0.503 0.582 0.635
17 0.414 0.485 0.566 0.615
18 0.401 0.472 0.550 0.600
19 0.391 0.460 0.535 0.584
20 0.380 0.447 0.520 0.570
21 0.370 0.435 0.508 0.556
22 0.361 0.425 0.496 0.544
23 0.353 0.415 0.486 0.532
24 0.344 0.406 0.476 0.521
25 0.337 0.398 0.466 0.511
26 0.331 0.390 0.457 0.501
27 0.324 0.382 0.448 0.491
28 0.317 0.375 0.440 0.483
29 0.312 0.368 0.433 0.475
30 0.306 0.362 0.425 0.467
35 0.283 0.335 0.394 0.433
40 0.264 0.313 0.368 0.405
45 0.248 0.294 0.347 0.382
50 0.235 0.279 0.329 0.363
60 0.214 0.255 0.300 0.331
70 0.190 0.235 0.278 0.307
80 0.185 0.220 0.260 0.287
90 0.174 0.207 0.245 0.271
100 0.165 0.197 0.233 0.257
Zar, J. H. (1972). Signif icance test ing of the Spearman rank correlation
coefficient. Journal of the American Statistical Association, 67,578.
Reprinted with permission from the Journal of the American Statistical
Association. Copyright © 1972 by the American Statistical Association. All
rights reserved.
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APPEND IX B STAT IST ICAL TABLES
TABLE B .8 THE CHI-SQUARE DISTRIBUTION*
*The table entries are critical values of X 2•
Critical
x'
Proportion inCritical Region
df 0.10 0.05 0.025 0.01 0.005
1 2.71 3.84 5.02 6.63 7.88
2 4.61 5.99 7.38 9.21 10.60
3 6.25 7.81 9.35 11.34 12.84
4 7.78 9.49 11.14 13.28 14.86
5 9.24 11.07 12.83 15.09 16.75
6 10.64 12.59 14.45 16.81 18.55
7 12.02 14.07 16.01 18.48 20.28
8 13.36 15.51 17.53 20.09 21.96
9 14.68 16.92 19.02 21.67 23.59
10 15.99 18.31 20.48 23.21 25.19
11 17.28 19.68 21.92 24.72 26.76
12 18.55 21.03 23.34 26.22 28.30
13 19.81 22.36 24.74 27.69 29.82
14 21.06 23.68 26.12 29.14 31.32
15 22.31 25.00 27.49 30.58 32.80
16 23.54 26.30 28.85 32.00 34.27
17 24.77 27.59 30.19 33.41 35.72
18 25.99 28.87 31.53 34.81 37.16
19 27.20 30.14 32.85 36.19 38.5820 28.41 31.41 34.17 37.57 40.00
21 29.62 32.67 35.48 38.93 41.40
22 30.81 33.92 36.78 40.29 42.80
23 32.01 35.17 38.08 41.64 44.18
24 33.20 36.42 39.36 42.98 45.56
25 34.38 37.65 40.65 44.31 46.93
26 35.56 38.89 41.92 45.64 48.29
27 36.74 40.11 43.19 46.96 49.64
28 37.92 41.34 44.46 48.28 50.99
29 39.09 42.56 45.72 49.59 52.34
30 40.26 43.77 46.98 50.89 53.67
40 51.81 55.76 59.34 63.69 66.77
50 63.17 67.50 71.42 76.15 79.49
60 74.40 79.08 83.30 88.38 91.95
70 85.53 90.53 95.02 100.42 104.22
80 96.58 101.88 106.63 112.33 116.32
90 107.56 113.14 118.14 124.12 128.30
100 118.50 124.34 129.56 135.81 140.17
Table 8 of E. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, 3rd ed. New York:
Cambridge University Press, 1966. Adapted and reprinted with permission of the Biometrika
trustees.
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738 APPENDIX B STATISTICAL TABLES
TABLE B.9A CRITICAL VALUES OF THE MANN-WHITNEY U FOR a = .05*
*Critical values are provided for a one-tailed test at u = .05 (lightface type) and for a two-tailed test at u = .05
(boldface type). To be significant for any given nA and nB, the obtained U must be equal to or less than the critical
value in the table. Dashes (-) in the body of the table indicate that no decision is possible at the stated level of
significance and values of nA and nB.
2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 20
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19 0
20 0
o
o
o 0 2
o 1
o 1 2 3
o 1 2
1 2 4 5o 1 2 3
2 3 5 7
1 2 3 5
2 4 6 8
1 3 5 6
3 5 8 10
2 4 6 8
3 6 9 12
2 4 7 10
4 7 11 14
3 5 8 11
5 8 12 16
3 6 9 13
5 9 13 17
4 7 11 14
6 10 15 19
4 8 12 16
7 11 16 21
5 9 13 17
7 12 18 23
5 10 14 19
8 14 19 25
6 11 15 21
o o 2 2 2
1
7
5
11
9
1613
21
17
26
22
31
26
36
31
41
36
46
40
51
45
56
50
61
55
66
59
71
64
3
1
7
5
12
10
1814
23
19
28
24
33
29
39
34
44
39
50
44
55
49
61
54
66
59
72
64
77
70
o 0
3
1
8
6
14
11
1915
25
21
30
26
36
31
42
37
48
42
54
47
60
53
65
59
71
64
77
70
83
75
3 4 4 4
2 2 2 2
9 9 10 11
6 7 7 8
15 16 17 18
11 12 13 13
20 22 23 2517 18 19 20
26 28 30 32
22 24 25 27
33 35 37 39
28 30 32 34
39 41 44 47
34 36 38 41
45 48 51 54
39 42 45 48
51 55 58 62
45 48 52 55
57 61 65 69
51 55 58 62
64 68 72 77
57 61 65 69
79 75 80 84
63 67 72 76
77 82 87 92
67 74 78 83
83 88 94 100
75 80 85 90
89 95 101 107
81 86 92 98
o
o
1
o
1
o
1
o
o
2
1
2
1
2
1
3
1
3
1
3 9 15 20 26
2 6 11 17 22
4 9 16 22 28
2 7 12 18 24
4 10 17 23 30
2 7 13 19 25
4 11 18 25 32
2 8 13 20 27
o 0 0 011
2334556
1223344
4 5 6 7 8 9 10
3445678
6 8 9 11 12 13 155 6 7 8 9 11 12
8 10 12 14 16 17 19
6 8 10 11 13 14 16
11 13 15 17 19 21 24
8 10 12 14 16 18 20
13 15 18 20 23 26 28
10 13 15 17 19 22 24
15 18 21 24 27 30 33
12 15 17 20 23 26 28
17 20 24 27 31 34 37
14 17 20 23 26 29 33
19 23 27 31 34 38 42
16 19 23 26 30 33 37
21 26 30 34 38 42 47
18 22 26 29 33 37 41
24 28 33 37 42 47 51
20 24 28 33 37 41 45
26 31 36 41 46 51 56
22 26 31 36 40 45 50
28 33 39 44 50 55 61
24 29 34 39 44 49 54
30 36 42 48 54 60 65
26 31 37 42 47 53 59
33 39 45 51 57 64 70
28 34 39 45 51 57 63
35 41 48 55 61 68 75
30 36 42 48 55 61 67
37 44 51 58 65 72 80
32 38 45 52 58 65 72
39 47 54 62 69 77 84
34 41 48 55 62 69 76
77 83 89 96 102 109 115
67 75 81 87 93 99 105
82 88 95 102 109 116 123
74 80 86 93 99 106 112
87 94 101 109 116 123 130
78 85 92 99 106 113 119
92 100 107 115 123 130 138
83 90 98 105 112 119 127
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TABLE B.9B
19 2
CRITICAL VALUES OF THE MANN-WHITNEY U FOR a = .01*
APPENDIX B STATISTICALTABLES
*Critical values are provided for a one-tailed test at u = .01 (lightface type) and for a two-tailed test at u =
(boldface type). To be significant for any given nA and nB. the obtained U must be equal to or less than the critical va
in the table. Dashes (-) in the body of the table indicate that no decision is possible at the stated level of significan
and values of nA and nB.
2
2
o
o
o
o
o
o
o
1
o
3 4 5
o
1
o
2
1
3
1
4
2
5
3
6
4
6 8 9 10 11 12
2
1
5
3
8
6
11
9
14
12
17
15
21
18
24
21
13
o
2
1
5
3
9
7
12
10
16
13
20
17
23
20
27
24
31
27
35
31
39
34
43
38
47
42
51
45
55
49
59
53
63
56
67
60
14
o
2
1
6
4
10
7
13
11
17
15
22
18
26
22
30
26
34
30
38
34
43
38
47
42
51
46
56
50
60
54
65
58
69
63
73
67
15
o
3
2
7
5
11
8
15
12
19
16
24
20
28
24
33
29
37
33
42
37
47
42
51
46
56
51
61
55
66
60
70
64
75
69
80
73
44 47 50
39 42 45 4
49 53 56
44 47 51
55 59 63
49 53 56
60 65 69 7
54 58 63
66 70 75
60 64 69 7
71 76 82
65 70 74 7
77 82 88
70 75 81
82 88 94 1
75 81 87
88 94 101 1
81 87 93
93 100 107 1
86 92 99 1
16 17 18
1
o
4
3
9
7
15
12
20
17
26
22 2
32
28 3
38 4
33 3
44 4
39 4
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
o
o
1
o
1
o
2
1
3
1
3
2
o
1
o
o
1
o
2
1
2
1
2
1
3
2
3
2
4
2
42
4
3
5
3
4 7
2 5
5 8
3 6
5 9
3 7
6 10
4 7
7 11
5 8
7 12
5 9
8 13
6 10
9 14
6 11
9 15
7 12
10 16
8 13
7
o o 1
o
3
1
5
3
7
5
9
7
11
9
14
11
16
13
1
o
2
1
3
2
4
3
6
4
7
5
8
6
9
7
11
9
12
10
13
11
15
12
16
13
18
15
19
16
20
17
22
18
1
o
3
1
4
3
6
4
7
6
9
7
11
9
2
1
4
2
6
4
7
6
9
7
11
9
13
11
12 15 18 22
10 13 16 18
14 17 21 24
12 15 18 21
16 20 23 27
13 17 20 24
17 22 26 30
15 18 22 26
19 24 28 33
16 20 24 29
21 26 31 36
18 22 27 31
23 28 33 38
19 24 29 34
24 30 36 41
21 26 31 37
26 32 38 44
22 28 33 39
28 34 40 47
24 30 36 42
1
o
3
2
6
4
8
6
11
9
13
11
16
13
19
16
1
o
4
2
7
5
9
7
12
10
15
13
18
16
22
18
25 28
21 24
28 31
24 27
31 35
27 31
34 38
30 34
37 42
33 37
41 46
36 41
44 49
39 44
47 53
42 47
50 56
45 51
53 60
48 54
o o 0
3
2
7
5
12
9
16
13
21
18
26
22
31
27
36
31
41
36
46
41
51
45
56
50
61
55
66
60
71
65
76
70
82
74
87
79
4 4
2 2
8 9
6 6
13 14
10 11
18 19
15 16
23 24
19 21
28 30
24 26
33 36
29 31
38 41
34 37
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740 APPENDIX B STATISTICALTABLES
TABLE B.10 CRITICAL VALUES OF T FOR THE WILCOXON SIGNED-RANKS TEST*
*To be significant, the obtained T must be equal to or less than the critical value. Dashes (-) in the columns indicate
that no decision is possible for the stated exand n.
L e v e l o f S i g n if i c a n ce o r
O n e - T a i l e d T e s t
. 0 5 . 0 2 5 . 0 1 . 0 0 5
L e v e l o f S i g n i fi c a n c eo r
T w o - T a i l ed Te s t
n . 1 0 . 0 5 . 0 2 . 0 1
5 0 - - -6 2 0 - -7 3 2 0 -8 5 3 1 0
9 8 5 3 1
1 0 1 0 8 5 3
1 1 1 3 1 0 7 5
1 2 1 7 1 3 9 7
1 3 2 1 1 7 1 2 9
1 4 2 5 2 1 1 5 1 2
1 5 3 0 2 5 1 9 1 5
1 6 3 5 2 9 2 3 1 9
1 7 4 1 3 4 2 7 2 3
1 8 4 7 4 0 3 2 2 7
1 9 5 3 4 6 3 7 3 2
2 0 6 0 5 2 4 3 3 7
2 1 6 7 5 8 4 9 4 2
2 2 7 5 6 5 5 5 4 8
2 3 8 3 7 3 6 2 5 4
2 4 9 1 8 1 6 9 6 1
2 5 1 0 0 8 9 7 6 6 8
2 6 1 1 0 9 8 8 4 7 5
2 7 1 1 9 1 0 7 9 2 8 3
L e v e l o f S i g n i fi c a n c eo r
O n e - T a i l e d T e s t
. 0 5 . 0 2 5 . 0 1 . 0 0 5
L e v e l o f S i g n if i c a nc e o r
T w o - T ai l e d T e s t
n . 1 0 . 0 5 . 0 2 . 0 1
2 8 1 3 0 1 1 6 1 0 1 9 1
2 9 1 4 0 1 2 6 1 1 0 1 0 0
3 0 1 5 1 1 3 7 1 2 0 1 0 9
3 1 1 6 3 1 4 7 1 3 0 1 1 8
3 2 1 7 5 1 5 9 1 4 0 1 2 8
3 3 1 8 7 1 7 0 1 5 1 1 3 8
3 4 2 0 0 1 8 2 1 6 2 1 4 8
3 5 2 1 3 1 9 5 1 7 3 1 5 9
3 6 2 2 7 2 0 8 1 8 5 1 7 1
3 7 2 4 1 2 2 1 1 9 8 1 8 2
3 8 2 5 6 2 3 5 2 1 1 1 9 4
3 9 2 7 1 2 4 9 2 2 4 2 0 7
4 0 2 8 6 2 6 4 2 3 8 2 2 0
4 1 3 0 2 2 7 9 2 5 2 2 3 3
4 2 3 1 9 2 9 4 2 6 6 2 4 7
4 3 3 3 6 3 1 0 2 8 1 2 6 1
4 4 3 5 3 3 2 7 2 9 6 2 7 6
4 5 3 7 1 3 4 3 3 1 2 2 9 1
4 6 3 8 9 3 6 1 3 2 8 3 0 7
4 7 4 0 7 3 7 8 3 4 5 3 2 2
4 8 4 2 6 3 9 6 3 6 2 3 3 9
4 9 4 4 6 4 1 5 3 7 9 3 5 5
5 0 4 6 6 4 3 4 3 9 7 3 7 3
A d a p t ed f r o m F. W i l c ox o n , S . K. Katti, a n d R. A. W i l c o x , Critical Values and Probability Levels of the
Wilcoxon Rank-Swn Test and the Wilcoxon Signed-Ranks Test. W a y n e , N J : A m e r i c a n C y a n a m i d C o m p a n y ,
1 9 6 3 . A d a p t e d and r e p r in t e dw i t h p e r m i s s i o no f t h e A m e r ic a n C y a n am i d C o m p a ny .