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Tables

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Appendix B:

Tables

TABLE 1

CUMULATIVE PROBABILITIES FOR THE STANDARD NORMAL
DISTRIBUTION

Entries in the table give the area under the curve to the left of the z value. For example, for z = –.85, the cumulative probability is .1977.

Cumulative probability z

0

z

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

Ϫ3.0

.0013

.0013

.0013

.0012

.0012

.0011

.0011

.0011

.0010

.0010

Ϫ2.9
Ϫ2.8
Ϫ2.7
Ϫ2.6
Ϫ2.5

.0019
.0026
.0035
.0047
.0062

.0018
.0025
.0034
.0045
.0060

.0018
.0024
.0033
.0044
.0059

.0017
.0023
.0032
.0043
.0057

.0016
.0023
.0031
.0041
.0055

.0016
.0022
.0030
.0040
.0054

.0015
.0021
.0029
.0039
.0052

.0015
.0021
.0028
.0038
.0051

.0014
.0020
.0027
.0037
.0049

.0014
.0019
.0026
.0036
.0048

Ϫ2.4
Ϫ2.3
Ϫ2.2
Ϫ2.1
Ϫ2.0

.0082
.0107
.0139
.0179
.0228

.0080
.0104
.0136
.0174
.0222

.0078
.0102
.0132
.0170
.0217

.0075
.0099
.0129
.0166
.0212

.0073
.0096
.0125
.0162
.0207

.0071
.0094
.0122
.0158
.0202

.0069
.0091
.0119
.0154
.0197

.0068
.0089
.0116
.0150
.0192

.0066
.0087
.0113
.0146
.0188

.0064
.0084
.0110
.0143
.0183

Ϫ1.9
Ϫ1.8
Ϫ1.7
Ϫ1.6
Ϫ1.5

.0287
.0359
.0446
.0548
.0668

.0281
.0351
.0436
.0537
.0655

.0274
.0344
.0427
.0526
.0643

.0268
.0336
.0418
.0516
.0630

.0262
.0329
.0409
.0505
.0618

.0256
.0322
.0401
.0495
.0606

.0250
.0314
.0392
.0485
.0594

.0244
.0307
.0384
.0475
.0582

.0239
.0301
.0375
.0465
.0571

.0233
.0294
.0367
.0455
.0559

Ϫ1.4
Ϫ1.3
Ϫ1.2
Ϫ1.1
Ϫ1.0

.0808
.0968
.1151
.1357
.1587

.0793
.0951
.1131
.1335
.1562

.0778
.0934
.1112
.1314
.1539

.0764
.0918
.1093
.1292
.1515

.0749
.0901
.1075
.1271
.1492

.0735
.0885
.1056
.1251
.1469

.0721
.0869
.1038
.1230
.1446

.0708
.0853
.1020
.1210
.1423

.0694
.0838
.1003
.1190
.1401

.0681
.0823
.0985
.1170
.1379

Ϫ.9
Ϫ.8
Ϫ.7
Ϫ.6
Ϫ.5

.1841
.2119
.2420
.2743
.3085

.1814
.2090
.2389
.2709
.3050

.1788
.2061
.2358
.2676
.3015

.1762
.2033
.2327
.2643
.2981

.1736
.2005
.2296
.2611
.2946

.1711
.1977
.2266
.2578
.2912

.1685
.1949
.2236
.2546
.2877

.1660
.1922
.2206
.2514
.2843

.1635
.1894
.2177
.2483
.2810

.1611
.1867
.2148
.2451
.2776

Ϫ.4
Ϫ.3
Ϫ.2
Ϫ.1
Ϫ.0

.3446
.3821
.4207
.4602
.5000

.3409
.3783
.4168
.4562
.4960

.3372
.3745
.4129
.4522
.4920

.3336
.3707
.4090
.4483
.4880

.3300
.3669
.4052
.4443
.4840

.3264
.3632
.4013
.4404
.4801

.3228
.3594
.3974
.4364
.4761

.3192
.3557
.3936
.4325
.4721

.3156
.3520
.3897
.4286
.4681

.3121
.3483
.3859
.4247
.4641

Appendix B

TABLE 1

649

Tables

CUMULATIVE PROBABILITIES FOR THE STANDARD NORMAL
DISTRIBUTION (Continued)

Cumulative probability 0

Entries in the table give the area under the curve to the left of the z value. For example, for z = 1.25, the cumulative probability is .8944.

z

z

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

.0
.1
.2
.3
.4

.5000
.5398
.5793
.6179
.6554

.5040
.5438
.5832
.6217
.6591

.5080
.5478
.5871
.6255
.6628

.5120
.5517
.5910
.6293
.6664

.5160
.5557
.5948
.6331
.6700

.5199
.5596
.5987
.6368
.6736

.5239
.5636
.6026
.6406
.6772

.5279
.5675
.6064
.6443
.6808

.5319
.5714
.6103
.6480
.6844

.5359
.5753
.6141
.6517
.6879

.5
.6
.7
.8
.9

.6915
.7257
.7580
.7881
.8159

.6950
.7291
.7611
.7910
.8186

.6985
.7324
.7642
.7939
.8212

.7019
.7357
.7673
.7967
.8238

.7054
.7389
.7704
.7995
.8264

.7088
.7422
.7734
.8023
.8289

.7123
.7454
.7764
.8051
.8315

.7157
.7486
.7794
.8078
.8340

.7190
.7517
.7823
.8106
.8365

.7224
.7549
.7852
.8133
.8389

1.0
1.1
1.2
1.3
1.4

.8413
.8643
.8849
.9032
.9192

.8438
.8665
.8869
.9049
.9207

.8461
.8686
.8888
.9066
.9222

.8485
.8708
.8907
.9082
.9236

.8508
.8729
.8925
.9099
.9251

.8531
.8749
.8944
.9115
.9265

.8554
.8770
.8962
.9131
.9279

.8577
.8790
.8980
.9147
.9292

.8599
.8810
.8997
.9162
.9306

.8621
.8830
.9015
.9177
.9319

1.5
1.6
1.7
1.8
1.9

.9332
.9452
.9554
.9641
.9713

.9345
.9463
.9564
.9649
.9719

.9357
.9474
.9573
.9656
.9726

.9370
.9484
.9582
.9664
.9732

.9382
.9495
.9591
.9671
.9738

.9394
.9505
.9599
.9678
.9744

.9406
.9515
.9608
.9686
.9750

.9418
.9525
.9616
.9693
.9756

.9429
.9535
.9625
.9699
.9761

.9441
.9545
.9633
.9706
.9767

2.0
2.1
2.2
2.3
2.4

.9772
.9821
.9861
.9893
.9918

.9778
.9826
.9864
.9896
.9920

.9783
.9830
.9868
.9898
.9922

.9788
.9834
.9871
.9901
.9925

.9793
.9838
.9875
.9904
.9927

.9798
.9842
.9878
.9906
.9929

.9803
.9846
.9881
.9909
.9931

.9808
.9850
.9884
.9911
.9932

.9812
.9854
.9887
.9913
.9934

.9817
.9857
.9890
.9913
.9936

2.5
2.6
2.7
2.8
2.9

.9938
.9953
.9965
.9974
.9981

.9940
.9955
.9966
.9975
.9982

.9941
.9956
.9967
.9976
.9982

.9943
.9957
.9968
.9977
.9983

.9945
.9959
.9969
.9977
.9984

.9946
.9960
.9970
.9978
.9984

.9948
.9961
.9971
.9979
.9985

.9949
.9962
.9972
.9979
.9985

.9951
.9963
.9973
.9980
.9986

.9952
.9964
.9974
.9981
.9986

3.0

.9986

.9987

.9987

.9988

.9988

.9989

.9989

.9989

.9990

.9990

650

Appendix B

TABLE 2

Tables

t DISTRIBUTION

Entries in the table give the t values for an area or probability in the upper
Area or tail of the t distribution. For probability example, with 10 degrees of freedom and a .05 area in the upper tail, t.05 = 1.812.
0

t
Area in Upper Tail

Degrees of Freedom

.20

.10

.05

.025

.01

.005

1
2
3
4

1.376
1.061
.978
.941

3.078
1.886
1.638
1.533

6.314
2.920
2.353
2.132

12.706
4.303
3.182
2.776

31.821
6.965
4.541
3.747

63.656
9.925
5.841
4.604

5
6
7
8
9

.920
.906
.896
.889
.883

1.476
1.440
1.415
1.397
1.383

2.015
1.943
1.895
1.860
1.833

2.571
2.447
2.365
2.306
2.262

3.365
3.143
2.998
2.896
2.821

4.032
3.707
3.499
3.355
3.250

10
11
12
13
14

.879
.876
.873
.870
.868

1.372
1.363
1.356
1.350
1.345

1.812
1.796
1.782
1.771
1.761

2.228
2.201
2.179
2.160
2.145

2.764
2.718
2.681
2.650
2.624

3.169
3.106
3.055
3.012
2.977

15
16
17
18
19

.866
.865
.863
.862
.861

1.341
1.337
1.333
1.330
1.328

1.753
1.746
1.740
1.734
1.729

2.131
2.120
2.110
2.101
2.093

2.602
2.583
2.567
2.552
2.539

2.947
2.921
2.898
2.878
2.861

20
21
22
23
24

.860
.859
.858
.858
.857

1.325
1.323
1.321
1.319
1.318

1.725
1.721
1.717
1.714
1.711

2.086
2.080
2.074
2.069
2.064

2.528
2.518
2.508
2.500
2.492

2.845
2.831
2.819
2.807
2.797

25
26
27
28
29

.856
.856
.855
.855
.854

1.316
1.315
1.314
1.313
1.311

1.708
1.706
1.703
1.701
1.699

2.060
2.056
2.052
2.048
2.045

2.485
2.479
2.473
2.467
2.462

2.787
2.779
2.771
2.763
2.756

30
31
32
33
34

.854
.853
.853
.853
.852

1.310
1.309
1.309
1.308
1.307

1.697
1.696
1.694
1.692
1.691

2.042
2.040
2.037
2.035
2.032

2.457
2.453
2.449
2.445
2.441

2.750
2.744
2.738
2.733
2.728

Appendix B

TABLE 2

651

Tables

t DISTRIBUTION (Continued)
Area in Upper Tail

Degrees of Freedom

.20

.10

.05

.025

.01

.005

35
36
37
38
39

.852
.852
.851
.851
.851

1.306
1.306
1.305
1.304
1.304

1.690
1.688
1.687
1.686
1.685

2.030
2.028
2.026
2.024
2.023

2.438
2.434
2.431
2.429
2.426

2.724
2.719
2.715
2.712
2.708

40
41
42
43
44

.851
.850
.850
.850
.850

1.303
1.303
1.302
1.302
1.301

1.684
1.683
1.682
1.681
1.680

2.021
2.020
2.018
2.017
2.015

2.423
2.421
2.418
2.416
2.414

2.704
2.701
2.698
2.695
2.692

45
46
47
48
49

.850
.850
.849
.849
.849

1.301
1.300
1.300
1.299
1.299

1.679
1.679
1.678
1.677
1.677

2.014
2.013
2.012
2.011
2.010

2.412
2.410
2.408
2.407
2.405

2.690
2.687
2.685
2.682
2.680

50
51
52
53
54

.849
.849
.849
.848
.848

1.299
1.298
1.298
1.298
1.297

1.676
1.675
1.675
1.674
1.674

2.009
2.008
2.007
2.006
2.005

2.403
2.402
2.400
2.399
2.397

2.678
2.676
2.674
2.672
2.670

55
56
57
58
59

.848
.848
.848
.848
.848

1.297
1.297
1.297
1.296
1.296

1.673
1.673
1.672
1.672
1.671

2.004
2.003
2.002
2.002
2.001

2.396
2.395
2.394
2.392
2.391

2.668
2.667
2.665
2.663
2.662

60
61
62
63
64

.848
.848
.847
.847
.847

1.296
1.296
1.295
1.295
1.295

1.671
1.670
1.670
1.669
1.669

2.000
2.000
1.999
1.998
1.998

2.390
2.389
2.388
2.387
2.386

2.660
2.659
2.657
2.656
2.655

65
66
67
68
69

.847
.847
.847
.847
.847

1.295
1.295
1.294
1.294
1.294

1.669
1.668
1.668
1.668
1.667

1.997
1.997
1.996
1.995
1.995

2.385
2.384
2.383
2.382
2.382

2.654
2.652
2.651
2.650
2.649

70
71
72
73
74

.847
.847
.847
.847
.847

1.294
1.294
1.293
1.293
1.293

1.667
1.667
1.666
1.666
1.666

1.994
1.994
1.993
1.993
1.993

2.381
2.380
2.379
2.379
2.378

2.648
2.647
2.646
2.645
2.644

75
76
77
78
79

.846
.846
.846
.846
.846

1.293
1.293
1.293
1.292
1.292

1.665
1.665
1.665
1.665
1.664

1.992
1.992
1.991
1.991
1.990

2.377
2.376
2.376
2.375
2.374

2.643
2.642
2.641
2.640
2.639
(Continued)

652

Appendix B

TABLE 2

Tables

t DISTRIBUTION (Continued)
Area in Upper Tail

Degrees of Freedom

.20

.10

.05

.025

.01

.005

80
81
82
83
84

.846
.846
.846
.846
.846

1.292
1.292
1.292
1.292
1.292

1.664
1.664
1.664
1.663
1.663

1.990
1.990
1.989
1.989
1.989

2.374
2.373
2.373
2.372
2.372

2.639
2.638
2.637
2.636
2.636

85
86
87
88
89

.846
.846
.846
.846
.846

1.292
1.291
1.291
1.291
1.291

1.663
1.663
1.663
1.662
1.662

1.988
1.988
1.988
1.987
1.987

2.371
2.370
2.370
2.369
2.369

2.635
2.634
2.634
2.633
2.632

90
91
92
93
94

.846
.846
.846
.846
.845

1.291
1.291
1.291
1.291
1.291

1.662
1.662
1.662
1.661
1.661

1.987
1.986
1.986
1.986
1.986

2.368
2.368
2.368
2.367
2.367

2.632
2.631
2.630
2.630
2.629

95
96
97
98
99
100
ϱ

.845
.845
.845
.845
.845
.845
.842

1.291
1.290
1.290
1.290
1.290
1.290
1.282

1.661
1.661
1.661
1.661
1.660
1.660
1.645

1.985
1.985
1.985
1.984
1.984
1.984
1.960

2.366
2.366
2.365
2.365
2.364
2.364
2.326

2.629
2.628
2.627
2.627
2.626
2.626
2.576

Appendix B

TABLE 3

653

Tables

CHI-SQUARE DISTRIBUTION

Area or probability 2 χα Degrees of Freedom

Entries in the table give χ 2 values, where α is α the area or probability in the upper tail of the chi-square distribution.
For example, with
10 degrees of freedom and a .01 area in the upper tail, χ 2 ϭ 23.209.
.01

Area in Upper Tail
.995

.99

.975

.95

.90

.10

.05

.025

.01

.005

1
2
3
4
5

.000
.010
.072
.207
.412

.000
.020
.115
.297
.554

.001
.051
.216
.484
.831

.004
.103
.352
.711
1.145

.016
.211
.584
1.064
1.610

2.706
4.605
6.251
7.779
9.236

3.841
5.991
7.815
9.488
11.070

5.024
7.378
9.348
11.143
12.832

6.635
9.210
11.345
13.277
15.086

7.879
10.597
12.838
14.860
16.750

6
7
8
9
10

.676
.989
1.344
1.735
2.156

.872
1.239
1.647
2.088
2.558

1.237
1.690
2.180
2.700
3.247

1.635
2.167
2.733
3.325
3.940

2.204
2.833
3.490
4.168
4.865

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

11
12
13
14
15

2.603
3.074
3.565
4.075
4.601

3.053
3.571
4.107
4.660
5.229

3.816
4.404
5.009
5.629
6.262

4.575
5.226
5.892
6.571
7.261

5.578
6.304
7.041
7.790
8.547

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

16
17
18
19
20

5.142
5.697
6.265
6.844
7.434

5.812
6.408
7.015
7.633
8.260

6.908
7.564
8.231
8.907
9.591

7.962
8.672
9.390
10.117
10.851

9.312
10.085
10.865
11.651
12.443

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

21
22
23
24

8.034
8.643
9.260
9.886

8.897
9.542
10.196
10.856

10.283
10.982
11.689
12.401

11.591
12.338
13.091
13.848

13.240
14.041
14.848
15.659

29.615
30.813
32.007
33.196

32.671
33.924
35.172
36.415

35.479
36.781
38.076
39.364

38.932
40.289
41.638
42.980

41.401
42.796
44.181
45.558

25
26
27
28
29

10.520
11.160
11.808
12.461
13.121

11.524
12.198
12.878
13.565
14.256

13.120
13.844
14.573
15.308
16.047

14.611
15.379
16.151
16.928
17.708

16.473
17.292
18.114
18.939
19.768

34.382
35.563
36.741
37.916
39.087

37.652
38.885
40.113
41.337
42.557

40.646
41.923
43.195
44.461
45.722

44.314
45.642
46.963
48.278
49.588

46.928
48.290
49.645
50.994
52.335

(Continued)

654

Appendix B

TABLE 3

Tables

CHI-SQUARE DISTRIBUTION (Continued)
Area in Upper Tail

Degrees of Freedom

.995

.99

.975

.95

.90

.10

.05

.025

.01

.005

30
35
40
45
50

13.787
17.192
20.707
24.311
27.991

14.953
18.509
22.164
25.901
29.707

16.791
20.569
24.433
28.366
32.357

18.493
22.465
26.509
30.612
34.764

20.599
24.797
29.051
33.350
37.689

40.256
46.059
51.805
57.505
63.167

43.773
49.802
55.758
61.656
67.505

46.979
53.203
59.342
65.410
71.420

50.892
57.342
63.691
69.957
76.154

53.672
60.275
66.766
73.166
79.490

55
60
65
70
75

31.735
35.534
39.383
43.275
47.206

33.571
37.485
41.444
45.442
49.475

36.398
40.482
44.603
48.758
52.942

38.958
43.188
47.450
51.739
56.054

42.060
46.459
50.883
55.329
59.795

68.796
74.397
79.973
85.527
91.061

73.311
79.082
84.821
90.531
96.217

77.380
83.298
89.177
95.023
100.839

82.292
88.379
94.422
100.425
106.393

85.749
91.952
98.105
104.215
110.285

80
85
90
95
100

51.172
55.170
59.196
63.250
67.328

53.540
57.634
61.754
65.898
70.065

57.153
61.389
65.647
69.925
74.222

60.391
64.749
69.126
73.520
77.929

64.278
68.777
73.291
77.818
82.358

96.578
102.079
107.565
113.038
118.498

101.879
107.522
113.145
118.752
124.342

106.629
112.393
118.136
123.858
129.561

112.329
118.236
124.116
129.973
135.807

116.321
122.324
128.299
134.247
140.170

TABLE 4

F DISTRIBUTION

Entries in the table give
Fα values, where α is the area or probability in the upper tail of the F distribution.
For example, with
4 numerator degrees of freedom, 8 denominator degrees of freedom, and a .05 area in the upper tail, F.05 ϭ 3.84.

Area or probability 0



Denominator
Degrees
of Freedom

Area in
Upper
Tail

1

2

3

4

5

6

7

8

9

10

15

20

25

30

40

60

100

1000

1

.10
.05
.025
.01

39.86
161.45
647.79
4052.18

49.50
199.50
799.48
4999.34

53.59
215.71
864.15
5403.53

55.83
224.58
899.60
5624.26

57.24
230.16
921.83
5763.96

58.20
233.99
937.11
5858.95

58.91
236.77
948.20
5928.33

59.44
238.88
956.64
5980.95

59.86
240.54
963.28
6022.40

60.19
241.88
968.63
6055.93

61.22
245.95
984.87
6156.97

61.74
248.02
993.08
6208.66

62.05
249.26
998.09
6239.86

62.26
250.10
1001.40
6260.35

62.53
251.14
1005.60
6286.43

62.79
252.20
1009.79
6312.97

63.01
253.04
1013.16
6333.92

63.30
254.19
1017.76
6362.80

2

.10
.05
.025
.01

8.53
18.51
38.51
98.50

9.00
19.00
39.00
99.00

9.16
19.16
39.17
99.16

9.24
19.25
39.25
99.25

9.29
19.30
39.30
99.30

9.33
19.33
39.33
99.33

9.35
19.35
39.36
99.36

9.37
19.37
39.37
99.38

9.38
19.38
39.39
99.39

9.39
19.40
39.40
99.40

9.42
19.43
39.43
99.43

9.44
19.45
39.45
99.45

9.45
19.46
39.46
99.46

9.46
19.46
39.46
99.47

9.47
19.47
39.47
99.48

9.47
19.48
39.48
99.48

9.48
19.49
39.49
99.49

9.49
19.49
39.50
99.50

3

.10
.05
.025
.01

5.54
10.13
17.44
34.12

5.46
9.55
16.04
30.82

5.39
9.28
15.44
29.46

5.34
9.12
15.10
28.71

5.31
9.01
14.88
28.24

5.28
8.94
14.73
27.91

5.27
8.89
14.62
27.67

5.25
8.85
14.54
27.49

5.24
8.81
14.47
27.34

5.23
8.79
14.42
27.23

5.20
8.70
14.25
26.87

5.18
8.66
14.17
26.69

5.17
8.63
14.12
26.58

5.17
8.62
14.08
26.50

5.16
8.59
14.04
26.41

5.15
8.57
13.99
26.32

5.14
8.55
13.96
26.24

5.13
8.53
13.91
26.14

4

.10
.05
.025
.01

4.54
7.71
12.22
21.20

4.32
6.94
10.65
18.00

4.19
6.59
9.98
16.69

4.11
6.39
9.60
15.98

4.05
6.26
9.36
15.52

4.01
6.16
9.20
15.21

3.98
6.09
9.07
14.98

3.95
6.04
8.98
14.80

3.94
6.00
8.90
14.66

3.92
5.96
8.84
14.55

3.87
5.86
8.66
14.20

3.84
5.80
8.56
14.02

3.83
5.77
8.50
13.91

3.82
5.75
8.46
13.84

3.80
5.72
8.41
13.75

3.79
5.69
8.36
13.65

3.78
5.66
8.32
13.58

3.76
5.63
8.26
13.47

5

.10
.05
.025
.01

4.06
6.61
10.01
16.26

3.78
5.79
8.43
13.27

3.62
5.41
7.76
12.06

3.52
5.19
7.39
11.39

3.45
5.05
7.15
10.97

3.40
4.95
6.98
10.67

3.37
4.88
6.85
10.46

3.34
4.82
6.76
10.29

3.32
4.77
6.68
10.16

3.30
4.74
6.62
10.05

3.21
4.56
6.33
9.55

3.19
4.52
6.27
9.45

3.17
4.50
6.23
9.38

3.16
4.46
6.18
9.29

3.14
4.43
6.12
9.20

3.13
4.41
6.08
9.13

3.11
4.37
6.02
9.03

Numerator Degrees of Freedom

3.324
4.62
6.43
9.72

(Continued)

655

656

TABLE 4

F DISTRIBUTION (Continued)

Denominator
Degrees
of Freedom

Area in
Upper
Tail

1

2

3

4

5

6

7

8

9

10

15

20

25

30

40

60

100

1000

6

.10
.05
.025
.01

3.78
5.99
8.81
13.75

3.46
5.14
7.26
10.92

3.29
4.76
6.60
9.78

3.18
4.53
6.23
9.15

3.11
4.39
5.99
8.75

3.05
4.28
5.82
8.47

3.01
4.21
5.70
8.26

2.98
4.15
5.60
8.10

2.96
4.10
5.52
7.98

2.94
4.06
5.46
7.87

2.87
3.94
5.27
7.56

2.84
3.87
5.17
7.40

2.81
3.83
5.11
7.30

2.80
3.81
5.07
7.23

2.78
3.77
5.01
7.14

2.76
3.74
4.96
7.06

2.75
3.71
4.92
6.99

2.72
3.67
4.86
6.89

7

.10
.05
.025
.01

3.59
5.59
8.07
12.25

3.26
4.74
6.54
9.55

3.07
4.35
5.89
8.45

2.96
4.12
5.52
7.85

2.88
3.97
5.29
7.46

2.83
3.87
5.12
7.19

2.78
3.79
4.99
6.99

2.75
3.73
4.90
6.84

2.72
3.68
4.82
6.72

2.70
3.64
4.76
6.62

2.63
3.51
4.57
6.31

2.59
3.44
4.47
6.16

2.57
3.40
4.40
6.06

2.56
3.38
4.36
5.99

2.54
3.34
4.31
5.91

2.51
3.30
4.25
5.82

2.50
3.27
4.21
5.75

2.47
3.23
4.15
5.66

8

.10
.05
.025
.01

3.46
5.32
7.57
11.26

3.11
4.46
6.06
8.65

2.92
4.07
5.42
7.59

2.81
3.84
5.05
7.01

2.73
3.69
4.82
6.63

2.67
3.58
4.65
6.37

2.62
3.50
4.53
6.18

2.59
3.44
4.43
6.03

2.56
3.39
4.36
5.91

2.54
3.35
4.30
5.81

2.46
3.22
4.10
5.52

2.42
3.15
4.00
5.36

2.40
3.11
3.94
5.26

2.38
3.08
3.89
5.20

2.36
3.04
3.84
5.12

2.34
3.01
3.78
5.03

2.32
2.97
3.74
4.96

2.30
2.93
3.68
4.87

9

.10
.05
.025
.01

3.36
5.12
7.21
10.56

3.01
4.26
5.71
8.02

2.81
3.86
5.08
6.99

2.69
3.63
4.72
6.42

2.61
3.48
4.48
6.06

2.55
3.37
4.32
5.80

2.51
3.29
4.20
5.61

2.47
3.23
4.10
5.47

2.44
3.18
4.03
5.35

2.42
3.14
3.96
5.26

2.34
3.01
3.77
4.96

2.30
2.94
3.67
4.81

2.27
2.89
3.60
4.71

2.25
2.86
3.56
4.65

2.23
2.83
3.51
4.57

2.21
2.79
3.45
4.48

2.19
2.76
3.40
4.41

2.16
2.71
3.34
4.32

10

.10
.05
.025
.01

3.29
4.96
6.94
10.04

2.92
4.10
5.46
7.56

2.73
3.71
4.83
6.55

2.61
3.48
4.47
5.99

2.52
3.33
4.24
5.64

2.46
3.22
4.07
5.39

2.41
3.14
3.95
5.20

2.38
3.07
3.85
5.06

2.35
3.02
3.78
4.94

2.32
2.98
3.72
4.85

2.24
2.85
3.52
4.56

2.20
2.77
3.42
4.41

2.17
2.73
3.35
4.31

2.16
2.70
3.31
4.25

2.13
2.66
3.26
4.17

2.11
2.62
3.20
4.08

2.09
2.59
3.15
4.01

2.06
2.54
3.09
3.92

11

.10
.05
.025
.01

3.23
4.84
6.72
9.65

2.86
3.98
5.26
7.21

2.66
3.59
4.63
6.22

2.54
3.36
4.28
5.67

2.45
3.20
4.04
5.32

2.39
3.09
3.88
5.07

2.34
3.01
3.76
4.89

2.30
2.95
3.66
4.74

2.27
2.90
3.59
4.63

2.25
2.85
3.53
4.54

2.17
2.72
3.33
4.25

2.12
2.65
3.23
4.10

2.10
2.60
3.16
4.01

2.08
2.57
3.12
3.94

2.05
2.53
3.06
3.86

2.03
2.49
3.00
3.78

2.01
2.46
2.96
3.71

1.98
2.41
2.89
3.61

12

.10
.05
.025
.01

3.18
4.75
6.55
9.33

2.81
3.89
5.10
6.93

2.61
3.49
4.47
5.95

2.48
3.26
4.12
5.41

2.39
3.11
3.89
5.06

2.33
3.00
3.73
4.82

2.28
2.91
3.61
4.64

2.24
2.85
3.51
4.50

2.21
2.80
3.44
4.39

2.19
2.75
3.37
4.30

2.10
2.62
3.18
4.01

2.06
2.54
3.07
3.86

2.03
2.50
3.01
3.76

2.01
2.47
2.96
3.70

1.99
2.43
2.91
3.62

1.96
2.38
2.85
3.54

1.94
2.35
2.80
3.47

1.91
2.30
2.73
3.37

13

.10
.05
.025
.01

3.14
4.67
6.41
9.07

2.76
3.81
4.97
6.70

2.56
3.41
4.35
5.74

2.43
3.18
4.00
5.21

2.35
3.03
3.77
4.86

2.28
2.92
3.60
4.62

2.23
2.83
3.48
4.44

2.20
2.77
3.39
4.30

2.16
2.71
3.31
4.19

2.14
2.67
3.25
4.10

2.05
2.53
3.05
3.82

2.01
2.46
2.95
3.66

1.98
2.41
2.88
3.57

1.96
2.38
2.84
3.51

1.93
2.34
2.78
3.43

1.90
2.30
2.72
3.34

1.88
2.26
2.67
3.27

1.85
2.21
2.60
3.18

14

.10
.05
.025
.01

3.10
4.60
6.30
8.86

2.73
3.74
4.86
6.51

2.52
3.34
4.24
5.56

2.39
3.11
3.89
5.04

2.31
2.96
3.66
4.69

2.24
2.85
3.50
4.46

2.19
2.76
3.38
4.28

2.15
2.70
3.29
4.14

2.12
2.65
3.21
4.03

2.10
2.60
3.15
3.94

2.01
2.46
2.95
3.66

1.96
2.39
2.84
3.51

1.93
2.34
2.78
3.41

1.99
2.31
2.73
3.35

1.89
2.27
2.67
3.27

1.86
2.22
2.61
3.18

1.83
2.19
2.56
3.11

1.80
2.14
2.50
3.02

15

.10
.05
.025
.01

3.07
4.54
6.20
8.68

2.70
3.68
4.77
6.36

2.49
3.29
4.15
5.42

2.36
3.06
3.80
4.89

2.27
2.90
3.58
4.56

2.21
2.79
3.41
4.32

2.16
2.71
3.29
4.14

2.12
2.64
3.20
4.00

2.09
2.59
3.12
3.89

2.06
2.54
3.06
3.80

1.97
2.40
2.86
3.52

1.92
2.33
2.76
3.37

1.89
2.28
2.69
3.28

1.87
2.25
2.64
3.21

1.85
2.20
2.59
3.13

1.82
2.16
2.52
3.05

1.79
2.12
2.47
2.98

1.76
2.07
2.40
2.88

Numerator Degrees of Freedom

TABLE 4

F DISTRIBUTION (Continued)

Denominator
Degrees
of Freedom

Area in
Upper
Tail

1

2

3

4

5

6

7

8

9

10

15

20

25

30

40

60

100

1000

16

.10
.05
.025
.01

3.05
4.49
6.12
8.53

2.67
3.63
4.69
6.23

2.46
3.24
4.08
5.29

2.33
3.01
3.73
4.77

2.24
2.85
3.50
4.44

2.18
2.74
3.34
4.20

2.13
2.66
3.22
4.03

2.09
2.59
3.12
3.89

2.06
2.54
3.05
3.78

2.03
2.49
2.99
3.69

1.94
2.35
2.79
3.41

1.89
2.28
2.68
3.26

1.86
2.23
2.61
3.16

1.84
2.19
2.57
3.10

1.81
2.15
2.51
3.02

1.78
2.11
2.45
2.93

1.76
2.07
2.40
2.86

1.72
2.02
2.32
2.76

17

.10
.05
.025
.01

3.03
4.45
6.04
8.40

2.64
3.59
4.62
6.11

2.44
3.20
4.01
5.19

2.31
2.96
3.66
4.67

2.22
2.81
3.44
4.34

2.15
2.70
3.28
4.10

2.10
2.61
3.16
3.93

2.06
2.55
3.06
3.79

2.03
2.49
2.98
3.68

2.00
2.45
2.92
3.59

1.91
2.31
2.72
3.31

1.86
2.23
2.62
3.16

1.83
2.18
2.55
3.07

1.81
2.15
2.50
3.00

1.78
2.10
2.44
2.92

1.75
2.06
2.38
2.83

1.73
2.02
2.33
2.76

1.69
1.97
2.26
2.66

18

.10
.05
.025
.01

3.01
4.41
5.98
8.29

2.62
3.55
4.56
6.01

2.42
3.16
3.95
5.09

2.29
2.93
3.61
4.58

2.20
2.77
3.38
4.25

2.13
2.66
3.22
4.01

2.08
2.58
3.10
3.84

2.04
2.51
3.01
3.71

2.00
2.46
2.93
3.60

1.98
2.41
2.87
3.51

1.89
2.27
2.67
3.23

1.84
2.19
2.56
3.08

1.80
2.14
2.49
2.98

1.78
2.11
2.44
2.92

1.75
2.06
2.38
2.84

1.72
2.02
2.32
2.75

1.70
1.98
2.27
2.68

1.66
1.92
2.20
2.58

19

.10
.05
.025
.01

2.99
4.38
5.92
8.18

2.61
3.52
4.51
5.93

2.40
3.13
3.90
5.01

2.27
2.90
3.56
4.50

2.18
2.74
3.33
4.17

2.11
2.63
3.17
3.94

2.06
2.54
3.05
3.77

2.02
2.48
2.96
3.63

1.98
2.42
2.88
3.52

1.96
2.38
2.82
3.43

1.86
2.23
2.62
3.15

1.81
2.16
2.51
3.00

1.78
2.11
2.44
2.91

1.76
2.07
2.39
2.84

1.73
2.03
2.33
2.76

1.70
1.98
2.27
2.67

1.67
1.94
2.22
2.60

1.64
1.88
2.14
2.50

20

.10
.05
.025
.01

2.97
4.35
5.87
8.10

2.59
3.49
4.46
5.85

2.38
3.10
3.86
4.94

2.25
2.87
3.51
4.43

2.16
2.71
3.29
4.10

2.09
2.60
3.13
3.87

2.04
2.51
3.01
3.70

2.00
2.45
2.91
3.56

1.96
2.39
2.84
3.46

1.94
2.35
2.77
3.37

1.84
2.20
2.57
3.09

1.79
2.12
2.46
2.94

1.76
2.07
2.40
2.84

1.74
2.04
2.35
2.78

1.71
1.99
2.29
2.69

1.68
1.95
2.22
2.61

1.65
1.91
2.17
2.54

1.61
1.85
2.09
2.43

21

.10
.05
.025
.01

2.96
4.32
5.83
8.02

2.57
3.47
4.42
5.78

2.36
3.07
3.82
4.87

2.23
2.84
3.48
4.37

2.14
2.68
3.25
4.04

2.08
2.57
3.09
3.81

2.02
2.49
2.97
3.64

1.98
2.42
2.87
3.51

1.95
2.37
2.80
3.40

1.92
2.32
2.73
3.31

1.83
2.18
2.53
3.03

1.78
2.10
2.42
2.88

1.74
2.05
2.36
2.79

1.72
2.01
2.31
2.72

1.69
1.96
2.25
2.64

1.66
1.92
2.18
2.55

1.63
1.88
2.13
2.48

1.59
1.82
2.05
2.37

22

.10
.05
.025
.01

2.95
4.30
5.79
7.95

2.56
3.44
4.38
5.72

2.35
3.05
3.78
4.82

2.22
2.82
3.44
4.31

2.13
2.66
3.22
3.99

2.06
2.55
3.05
3.76

2.01
2.46
2.93
3.59

1.97
2.40
2.84
3.45

1.93
2.34
2.76
3.35

1.90
2.30
2.70
3.26

1.81
2.15
2.50
2.98

1.76
2.07
2.39
2.83

1.73
2.02
2.32
2.73

1.70
1.98
2.27
2.67

1.67
1.94
2.21
2.58

1.64
1.89
2.14
2.50

1.61
1.85
2.09
2.42

1.57
1.79
2.01
2.32

23

.10
.05
.025
.01

2.94
4.28
5.75
7.88

2.55
3.42
4.35
5.66

2.34
3.03
3.75
4.76

2.21
2.80
3.41
4.26

2.11
2.64
3.18
3.94

2.05
2.53
3.02
3.71

1.99
2.44
2.90
3.54

1.95
2.37
2.81
3.41

1.92
2.32
2.73
3.30

1.89
2.27
2.67
3.21

1.80
2.13
2.47
2.93

1.74
2.05
2.36
2.78

1.71
2.00
2.29
2.69

1.69
1.96
2.24
2.62

1.66
1.91
2.18
2.54

1.62
1.86
2.11
2.45

1.59
1.82
2.06
2.37

1.55
1.76
1.98
2.27

24

.10
.05
.025
.01

2.93
4.26
5.72
7.82

2.54
3.40
4.32
5.61

2.33
3.01
3.72
4.72

2.19
2.78
3.38
4.22

2.10
2.62
3.15
3.90

2.04
2.51
2.99
3.67

1.98
2.42
2.87
3.50

1.94
2.36
2.78
3.36

1.91
2.30
2.70
3.26

1.88
2.25
2.64
3.17

1.78
2.11
2.44
2.89

1.73
2.03
2.33
2.74

1.70
1.97
2.26
2.64

1.67
1.94
2.21
2.58

1.64
1.89
2.15
2.49

1.61
1.84
2.08
2.40

1.58
1.80
2.02
2.33

1.54
1.74
1.94
2.22

Numerator Degrees of Freedom

657

(Continued)

658

TABLE 4

F DISTRIBUTION (Continued)

Denominator
Degrees
of Freedom

Area in
Upper
Tail

1

2

3

4

5

6

7

8

9

10

15

20

25

30

40

60

100

1000

25

.10
.05
.025
.01

2.92
4.24
5.69
7.77

2.53
3.39
4.29
5.57

2.32
2.99
3.69
4.68

2.18
2.76
3.35
4.18

2.09
2.60
3.13
3.85

2.02
2.49
2.97
3.63

1.97
2.40
2.85
3.46

1.93
2.34
2.75
3.32

1.89
2.28
2.68
3.22

1.87
2.24
2.61
3.13

1.77
2.09
2.41
2.85

1.72
2.01
2.30
2.70

1.68
1.96
2.23
2.60

1.66
1.92
2.18
2.54

1.63
1.87
2.12
2.45

1.59
1.82
2.05
2.36

1.56
1.78
2.00
2.29

1.52
1.72
1.91
2.18

26

.10
.05
.025
.01

2.91
4.23
5.66
7.72

2.52
3.37
4.27
5.53

2.31
2.98
3.67
4.64

2.17
2.74
3.33
4.14

2.08
2.59
3.10
3.82

2.01
2.47
2.94
3.59

1.96
2.39
2.82
3.42

1.92
2.32
2.73
3.29

1.88
2.27
2.65
3.18

1.86
2.22
2.59
3.09

1.76
2.07
2.39
2.81

1.71
1.99
2.28
2.66

1.67
1.94
2.21
2.57

1.65
1.90
2.16
2.50

1.61
1.85
2.09
2.42

1.58
1.80
2.03
2.33

1.55
1.76
1.97
2.25

1.51
1.70
1.89
2.14

27

.10
.05
.025
.01

2.90
4.21
5.63
7.68

2.51
3.35
4.24
5.49

2.30
2.96
3.65
4.60

2.17
2.73
3.31
4.11

2.07
2.57
3.08
3.78

2.00
2.46
2.92
3.56

1.95
2.37
2.80
3.39

1.91
2.31
2.71
3.26

1.87
2.25
2.63
3.15

1.85
2.20
2.57
3.06

1.75
2.06
2.36
2.78

1.70
1.97
2.25
2.63

1.66
1.92
2.18
2.54

1.64
1.88
2.13
2.47

1.60
1.84
2.07
2.38

1.57
1.79
2.00
2.29

1.54
1.74
1.94
2.22

1.50
1.68
1.86
2.11

28

.10
.05
.025
.01

2.89
4.20
5.61
7.64

2.50
3.34
4.22
5.45

2.29
2.95
3.63
4.57

2.16
2.71
3.29
4.07

2.06
2.56
3.06
3.75

2.00
2.45
2.90
3.53

1.94
2.36
2.78
3.36

1.90
2.29
2.69
3.23

1.87
2.24
2.61
3.12

1.84
2.19
2.55
3.03

1.74
2.04
2.34
2.75

1.69
1.96
2.23
2.60

1.65
1.91
2.16
2.51

1.63
1.87
2.11
2.44

1.59
1.82
2.05
2.35

1.56
1.77
1.98
2.26

1.53
1.73
1.92
2.19

1.48
1.66
1.84
2.08

29

.10
.05
.025
.01

2.89
4.18
5.59
7.60

2.50
3.33
4.20
5.42

2.28
2.93
3.61
4.54

2.15
2.70
3.27
4.04

2.06
2.55
3.04
3.73

1.99
2.43
2.88
3.50

1.93
2.35
2.76
3.33

1.89
2.28
2.67
3.20

1.86
2.22
2.59
3.09

1.83
2.18
2.53
3.00

1.73
2.03
2.32
2.73

1.68
1.94
2.21
2.57

1.64
1.89
2.14
2.48

1.62
1.85
2.09
2.41

1.58
1.81
2.03
2.33

1.55
1.75
1.96
2.23

1.52
1.71
1.90
2.16

1.47
1.65
1.82
2.05

30

.10
.05
.025
.01

2.88
4.17
5.57
7.56

2.49
3.32
4.18
5.39

2.28
2.92
3.59
4.51

2.14
2.69
3.25
4.02

2.05
2.53
3.03
3.70

1.98
2.42
2.87
3.47

1.93
2.33
2.75
3.30

1.88
2.27
2.65
3.17

1.85
2.21
2.57
3.07

1.82
2.16
2.51
2.98

1.72
2.01
2.31
2.70

1.67
1.93
2.20
2.55

1.63
1.88
2.12
2.45

1.61
1.84
2.07
2.39

1.57
1.79
2.01
2.30

1.54
1.74
1.94
2.21

1.51
1.70
1.88
2.13

1.46
1.63
1.80
2.02

40

.10
.05
.025
.01

2.84
4.08
5.42
7.31

2.44
3.23
4.05
5.18

2.23
2.84
3.46
4.31

2.09
2.61
3.13
3.83

2.00
2.45
2.90
3.51

1.93
2.34
2.74
3.29

1.87
2.25
2.62
3.12

1.83
2.18
2.53
2.99

1.79
2.12
2.45
2.89

1.76
2.08
2.39
2.80

1.66
1.92
2.18
2.52

1.61
1.84
2.07
2.37

1.57
1.78
1.99
2.27

1.54
1.74
1.94
2.20

1.51
1.69
1.88
2.11

1.47
1.64
1.80
2.02

1.43
1.59
1.74
1.94

1.38
1.52
1.65
1.82

60

.10
.05
.025
.01

2.79
4.00
5.29
7.08

2.39
3.15
3.93
4.98

2.18
2.76
3.34
4.13

2.04
2.53
3.01
3.65

1.95
2.37
2.79
3.34

1.87
2.25
2.63
3.12

1.82
2.17
2.51
2.95

1.77
2.10
2.41
2.82

1.74
2.04
2.33
2.72

1.71
1.99
2.27
2.63

1.60
1.84
2.06
2.35

1.54
1.75
1.94
2.20

1.50
1.69
1.87
2.10

1.48
1.65
1.82
2.03

1.44
1.59
1.74
1.94

1.40
1.53
1.67
1.84

1.36
1.48
1.60
1.75

1.30
1.40
1.49
1.62

100

.10
.05
.025
.01

2.76
3.94
5.18
6.90

2.36
3.09
3.83
4.82

2.14
2.70
3.25
3.98

2.00
2.46
2.92
3.51

1.91
2.31
2.70
3.21

1.83
2.19
2.54
2.99

1.78
2.10
2.42
2.82

1.73
2.03
2.32
2.69

1.69
1.97
2.24
2.59

1.66
1.93
2.18
2.50

1.56
1.77
1.97
2.22

1.49
1.68
1.85
2.07

1.45
1.62
1.77
1.97

1.42
1.57
1.71
1.89

1.38
1.52
1.64
1.80

1.34
1.45
1.56
1.69

1.29
1.39
1.48
1.60

1.22
1.30
1.36
1.45

1000

.10
.05
.025
.01

2.71
3.85
5.04
6.66

2.31
3.00
3.70
4.63

2.09
2.61
3.13
3.80

1.95
2.38
2.80
3.34

1.85
2.22
2.58
3.04

1.78
2.11
2.42
2.82

1.72
2.02
2.30
2.66

1.68
1.95
2.20
2.53

1.64
1.89
2.13
2.43

1.61
1.84
2.06
2.34

1.49
1.68
1.85
2.06

1.43
1.58
1.72
1.90

1.38
1.52
1.64
1.79

1.35
1.47
1.58
1.72

1.30
1.41
1.50
1.61

1.25
1.33
1.41
1.50

1.20
1.26
1.32
1.38

1.08
1.11
1.13
1.16

Numerator Degrees of Freedom

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