ST2238 Introductory Biostatistics
Critical values of the f-distribution: p = 10%
10%
0
fcrit
These tables give critical values for the f-distribution. The tabulated values are the values of the critical value,
fcrit , such that a random variable with a f-distribution with df1 and df2 degrees of freedom has probability p of
being greater than fcrit .
1
2
3
4
5
6
7
8
9
10
11
12
24
∞
39.86
8.526
5.538
4.545
49.5
9
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
58.2
9.326
5.285
4.01
58.91
9.349
5.266
3.979
59.44
9.367
5.252
3.955
59.86
9.381
5.24
3.936
60.19
9.392
5.23
3.92
60.47
9.401
5.222
3.907
60.71
9.408
5.216
3.896
62
9.45
5.176
3.831
63.33
9.491
5.134
3.761
5
6
7
8
9
4.06
3.776
3.589
3.458
3.36
3.78
3.463
3.257
3.113
3.006
3.619
3.289
3.074
2.924
2.813
3.52
3.181
2.961
2.806
2.693
3.453
3.108
2.883
2.726
2.611
3.405
3.055
2.827
2.668
2.551
3.368
3.014
2.785
2.624
2.505
3.339
2.983
2.752
2.589
2.469
3.316
2.958
2.725
2.561
2.44
3.297
2.937
2.703
2.538
2.416
3.282
2.92
2.684
2.519
2.396
3.268
2.905
2.668
2.502
2.379
3.191
2.818
2.575
2.404
2.277
3.105
2.722
2.471
2.293
2.159
10
11
12
13
14
3.285
3.225
3.177
3.136
3.102
2.924
2.86
2.807
2.763
2.726
2.728
2.66
2.606
2.56
2.522
2.605
2.536
2.48
2.434
2.395
2.522
2.451
2.394
2.347
2.307
2.461
2.389
2.331
2.283
2.243
2.414
2.342
2.283
2.234
2.193
2.377
2.304
2.245
2.195
2.154
2.347
2.274
2.214
2.164
2.122
2.323
2.248
2.188
2.138
2.095
2.302
2.227
2.166
2.116
2.073
2.284
2.209
2.147
2.097
2.054
2.178
2.1
2.036
1.983
1.938
2.055
1.972
1.904
1.846
1.797
15
16
17
18
19
3.073
3.048
3.026
3.007
2.99
2.695
2.668
2.645
2.624
2.606
2.49
2.462
2.437
2.416
2.397
2.361
2.333
2.308
2.286
2.266
2.273
2.244
2.218
2.196
2.176
2.208
2.178
2.152
2.13
2.109
2.158
2.128
2.102
2.079
2.058
2.119
2.088
2.061
2.038
2.017
2.086
2.055
2.028
2.005
1.984
2.059
2.028
2.001
1.977
1.956
2.037
2.005
1.978
1.954
1.932
2.017
1.985
1.958
1.933
1.912
1.899
1.866
1.836
1.81
1.787
1.755
1.718
1.686
1.657
1.631
20
21
22
23
24
2.975
2.961
2.949
2.937
2.927
2.589
2.575
2.561
2.549
2.538
2.38
2.365
2.351
2.339
2.327
2.249
2.233
2.219
2.207
2.195
2.158
2.142
2.128
2.115
2.103
2.091
2.075
2.06
2.047
2.035
2.04
2.023
2.008
1.995
1.983
1.999
1.982
1.967
1.953
1.941
1.965
1.948
1.933
1.919
1.906
1.937
1.92
1.904
1.89
1.877
1.913
1.896
1.88
1.866
1.853
1.892
1.875
1.859
1.845
1.832
1.767
1.748
1.731
1.716
1.702
1.607
1.586
1.567
1.549
1.533
25
26
27
28
29
2.918
2.909
2.901
2.894
2.887
2.528
2.519
2.511
2.503
2.495
2.317
2.307
2.299
2.291
2.283
2.184
2.174
2.165
2.157
2.149
2.092
2.082
2.073
2.064
2.057
2.024
2.014
2.005
1.996
1.988
1.971
1.961
1.952
1.943
1.935
1.929
1.919
1.909
1.9
1.892
1.895
1.884
1.874
1.865
1.857
1.866
1.855
1.845
1.836
1.827
1.841
1.83
1.82
1.811
1.802
1.82
1.809
1.799
1.79
1.781
1.689
1.677
1.666
1.656
1.647
1.518
1.504
1.491
1.478
1.467
30
32
34
36
38
2.881
2.869
2.859
2.85
2.842
2.489
2.477
2.466
2.456
2.448
2.276
2.263
2.252
2.243
2.234
2.142
2.129
2.118
2.108
2.099
2.049
2.036
2.024
2.014
2.005
1.98
1.967
1.955
1.945
1.935
1.927
1.913
1.901
1.891
1.881
1.884
1.87
1.858
1.847
1.838
1.849
1.835
1.822
1.811
1.802
1.819
1.805
1.793
1.781
1.772
1.794
1.78
1.767
1.756
1.746
1.773
1.758
1.745
1.734
1.724
1.638
1.622
1.608
1.595
1.584
1.456
1.437
1.419
1.404
1.39
40
60
120
∞
2.835
2.791
2.748
2.706
2.44
2.393
2.347
2.303
2.226
2.177
2.13
2.084
2.091
2.041
1.992
1.945
1.997
1.946
1.896
1.847
1.927
1.875
1.824
1.774
1.873
1.819
1.767
1.717
1.829
1.775
1.722
1.67
1.793
1.738
1.684
1.632
1.763
1.707
1.652
1.599
1.737
1.68
1.625
1.57
1.715
1.657
1.601
1.546
1.574
1.511
1.447
1.383
1.377
1.291
1.193
1
df1
df2
1
2
3
4
Created by Alex Cook, National University of Singapore
Page 1
ST2238 Introductory Biostatistics
Critical values of the f-distribution: p = 5%
5%
0
fcrit
The tables only give upper critical values. If the value of your test statistic fstat is less than one, look up 1/fstat
and swap the degrees of freedom.
All these tables are one-sided. If the alternative hypothesis is two-sided (as is usually the case), the p-values
should be doubled.
1
2
3
4
5
6
7
8
9
10
11
12
24
∞
161.4
18.51
10.13
7.709
199.5
19
9.552
6.944
215.7
19.16
9.277
6.591
224.6
19.25
9.117
6.388
230.2
19.3
9.013
6.256
234
19.33
8.941
6.163
236.8
19.35
8.887
6.094
238.9
19.37
8.845
6.041
240.5
19.38
8.812
5.999
241.9
19.4
8.786
5.964
243
19.4
8.763
5.936
243.9
19.41
8.745
5.912
249.1
19.45
8.639
5.774
254.3
19.5
8.526
5.628
5
6
7
8
9
6.608
5.987
5.591
5.318
5.117
5.786
5.143
4.737
4.459
4.256
5.409
4.757
4.347
4.066
3.863
5.192
4.534
4.12
3.838
3.633
5.05
4.387
3.972
3.687
3.482
4.95
4.284
3.866
3.581
3.374
4.876
4.207
3.787
3.5
3.293
4.818
4.147
3.726
3.438
3.23
4.772
4.099
3.677
3.388
3.179
4.735
4.06
3.637
3.347
3.137
4.704
4.027
3.603
3.313
3.102
4.678
4
3.575
3.284
3.073
4.527
3.841
3.41
3.115
2.9
4.365
3.669
3.23
2.928
2.707
10
11
12
13
14
4.965
4.844
4.747
4.667
4.6
4.103
3.982
3.885
3.806
3.739
3.708
3.587
3.49
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
3.02
2.896
2.796
2.714
2.646
2.978
2.854
2.753
2.671
2.602
2.943
2.818
2.717
2.635
2.565
2.913
2.788
2.687
2.604
2.534
2.737
2.609
2.505
2.42
2.349
2.538
2.404
2.296
2.206
2.131
15
16
17
18
19
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
3.16
3.127
3.056
3.007
2.965
2.928
2.895
2.901
2.852
2.81
2.773
2.74
2.79
2.741
2.699
2.661
2.628
2.707
2.657
2.614
2.577
2.544
2.641
2.591
2.548
2.51
2.477
2.588
2.538
2.494
2.456
2.423
2.544
2.494
2.45
2.412
2.378
2.507
2.456
2.413
2.374
2.34
2.475
2.425
2.381
2.342
2.308
2.288
2.235
2.19
2.15
2.114
2.066
2.01
1.96
1.917
1.878
20
21
22
23
24
4.351
4.325
4.301
4.279
4.26
3.493
3.467
3.443
3.422
3.403
3.098
3.072
3.049
3.028
3.009
2.866
2.84
2.817
2.796
2.776
2.711
2.685
2.661
2.64
2.621
2.599
2.573
2.549
2.528
2.508
2.514
2.488
2.464
2.442
2.423
2.447
2.42
2.397
2.375
2.355
2.393
2.366
2.342
2.32
2.3
2.348
2.321
2.297
2.275
2.255
2.31
2.283
2.259
2.236
2.216
2.278
2.25
2.226
2.204
2.183
2.082
2.054
2.028
2.005
1.984
1.843
1.812
1.783
1.757
1.733
25
26
27
28
29
4.242
4.225
4.21
4.196
4.183
3.385
3.369
3.354
3.34
3.328
2.991
2.975
2.96
2.947
2.934
2.759
2.743
2.728
2.714
2.701
2.603
2.587
2.572
2.558
2.545
2.49
2.474
2.459
2.445
2.432
2.405
2.388
2.373
2.359
2.346
2.337
2.321
2.305
2.291
2.278
2.282
2.265
2.25
2.236
2.223
2.236
2.22
2.204
2.19
2.177
2.198
2.181
2.166
2.151
2.138
2.165
2.148
2.132
2.118
2.104
1.964
1.946
1.93
1.915
1.901
1.711
1.691
1.672
1.654
1.638
30
32
34
36
38
4.171
4.149
4.13
4.113
4.098
3.316
3.295
3.276
3.259
3.245
2.922
2.901
2.883
2.866
2.852
2.69
2.668
2.65
2.634
2.619
2.534
2.512
2.494
2.477
2.463
2.421
2.399
2.38
2.364
2.349
2.334
2.313
2.294
2.277
2.262
2.266
2.244
2.225
2.209
2.194
2.211
2.189
2.17
2.153
2.138
2.165
2.142
2.123
2.106
2.091
2.126
2.103
2.084
2.067
2.051
2.092
2.07
2.05
2.033
2.017
1.887
1.864
1.843
1.824
1.808
1.622
1.594
1.569
1.547
1.527
40
60
120
∞
4.085
4.001
3.92
3.841
3.232
3.15
3.072
2.996
2.839
2.758
2.68
2.605
2.606
2.525
2.447
2.372
2.449
2.368
2.29
2.214
2.336
2.254
2.175
2.099
2.249
2.167
2.087
2.01
2.18
2.097
2.016
1.938
2.124
2.04
1.959
1.88
2.077
1.993
1.91
1.831
2.038
1.952
1.869
1.789
2.003
1.917
1.834
1.752
1.793
1.7
1.608
1.517
1.509
1.389
1.254
1
df1
df2
1
2
3
4
Created by Alex Cook, National University of Singapore
Page 2
ST2238 Introductory Biostatistics
Critical values of the f-distribution: p = 2.5%
2.5%
0
fcrit
1
2
3
4
5
6
7
8
9
10
11
12
24
∞
647.8
38.51
17.44
12.22
799.5
39
16.04
10.65
864.2
39.17
15.44
9.979
899.6
39.25
15.1
9.605
921.8
39.3
14.88
9.364
937.1
39.33
14.73
9.197
948.2
39.36
14.62
9.074
956.7
39.37
14.54
8.98
963.3
39.39
14.47
8.905
968.6
39.4
14.42
8.844
973
39.41
14.37
8.794
976.7
39.41
14.34
8.751
997.2
39.46
14.12
8.511
1018
39.5
13.9
8.257
5
6
7
8
9
10.01
8.813
8.073
7.571
7.209
8.434
7.26
6.542
6.059
5.715
7.764
6.599
5.89
5.416
5.078
7.388
6.227
5.523
5.053
4.718
7.146
5.988
5.285
4.817
4.484
6.978
5.82
5.119
4.652
4.32
6.853
5.695
4.995
4.529
4.197
6.757
5.6
4.899
4.433
4.102
6.681
5.523
4.823
4.357
4.026
6.619
5.461
4.761
4.295
3.964
6.568
5.41
4.709
4.243
3.912
6.525
5.366
4.666
4.2
3.868
6.278
5.117
4.415
3.947
3.614
6.015
4.849
4.142
3.67
3.333
10
11
12
13
14
6.937
6.724
6.554
6.414
6.298
5.456
5.256
5.096
4.965
4.857
4.826
4.63
4.474
4.347
4.242
4.468
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
3.95
3.759
3.607
3.483
3.38
3.855
3.664
3.512
3.388
3.285
3.779
3.588
3.436
3.312
3.209
3.717
3.526
3.374
3.25
3.147
3.665
3.474
3.321
3.197
3.095
3.621
3.43
3.277
3.153
3.05
3.365
3.173
3.019
2.893
2.789
3.08
2.883
2.725
2.595
2.487
15
16
17
18
19
6.2
6.115
6.042
5.978
5.922
4.765
4.687
4.619
4.56
4.508
4.153
4.077
4.011
3.954
3.903
3.804
3.729
3.665
3.608
3.559
3.576
3.502
3.438
3.382
3.333
3.415
3.341
3.277
3.221
3.172
3.293
3.219
3.156
3.1
3.051
3.199
3.125
3.061
3.005
2.956
3.123
3.049
2.985
2.929
2.88
3.06
2.986
2.922
2.866
2.817
3.008
2.934
2.87
2.814
2.765
2.963
2.889
2.825
2.769
2.72
2.701
2.625
2.56
2.503
2.452
2.395
2.316
2.247
2.187
2.133
20
21
22
23
24
5.871
5.827
5.786
5.75
5.717
4.461
4.42
4.383
4.349
4.319
3.859
3.819
3.783
3.75
3.721
3.515
3.475
3.44
3.408
3.379
3.289
3.25
3.215
3.183
3.155
3.128
3.09
3.055
3.023
2.995
3.007
2.969
2.934
2.902
2.874
2.913
2.874
2.839
2.808
2.779
2.837
2.798
2.763
2.731
2.703
2.774
2.735
2.7
2.668
2.64
2.721
2.682
2.647
2.615
2.586
2.676
2.637
2.602
2.57
2.541
2.408
2.368
2.331
2.299
2.269
2.085
2.042
2.003
1.968
1.935
25
26
27
28
29
5.686
5.659
5.633
5.61
5.588
4.291
4.265
4.242
4.221
4.201
3.694
3.67
3.647
3.626
3.607
3.353
3.329
3.307
3.286
3.267
3.129
3.105
3.083
3.063
3.044
2.969
2.945
2.923
2.903
2.884
2.848
2.824
2.802
2.782
2.763
2.753
2.729
2.707
2.687
2.669
2.677
2.653
2.631
2.611
2.592
2.613
2.59
2.568
2.547
2.529
2.56
2.536
2.514
2.494
2.475
2.515
2.491
2.469
2.448
2.43
2.242
2.217
2.195
2.174
2.154
1.906
1.878
1.853
1.829
1.807
30
32
34
36
38
5.568
5.531
5.499
5.471
5.446
4.182
4.149
4.12
4.094
4.071
3.589
3.557
3.529
3.505
3.483
3.25
3.218
3.191
3.167
3.145
3.026
2.995
2.968
2.944
2.923
2.867
2.836
2.808
2.785
2.763
2.746
2.715
2.688
2.664
2.643
2.651
2.62
2.593
2.569
2.548
2.575
2.543
2.516
2.492
2.471
2.511
2.48
2.453
2.429
2.407
2.458
2.426
2.399
2.375
2.353
2.412
2.381
2.353
2.329
2.307
2.136
2.103
2.075
2.049
2.027
1.787
1.75
1.717
1.687
1.661
40
60
120
∞
5.424
5.286
5.152
5.024
4.051
3.925
3.805
3.689
3.463
3.343
3.227
3.116
3.126
3.008
2.894
2.786
2.904
2.786
2.674
2.567
2.744
2.627
2.515
2.408
2.624
2.507
2.395
2.288
2.529
2.412
2.299
2.192
2.452
2.334
2.222
2.114
2.388
2.27
2.157
2.048
2.334
2.216
2.102
1.993
2.288
2.169
2.055
1.945
2.007
1.882
1.76
1.64
1.637
1.482
1.31
1
df1
df2
1
2
3
4
Created by Alex Cook, National University of Singapore
Page 3
ST2238 Introductory Biostatistics
Critical values of the f-distribution: p = 1%
1%
0
fcrit
1
2
3
4
5
6
7
8
9
10
11
12
24
∞
4052
98.5
34.12
21.2
4999
99
30.82
18
5403
99.17
29.46
16.69
5625
99.25
28.71
15.98
5764
99.3
28.24
15.52
5859
99.33
27.91
15.21
5928
99.36
27.67
14.98
5981
99.37
27.49
14.8
6022
99.39
27.35
14.66
6056
99.4
27.23
14.55
6083
99.41
27.13
14.45
6106
99.42
27.05
14.37
6235
99.46
26.6
13.93
6366
99.5
26.13
13.46
5
6
7
8
9
16.26
13.75
12.25
11.26
10.56
13.27
10.92
9.547
8.649
8.022
12.06
9.78
8.451
7.591
6.992
11.39
9.148
7.847
7.006
6.422
10.97
8.746
7.46
6.632
6.057
10.67
8.466
7.191
6.371
5.802
10.46
8.26
6.993
6.178
5.613
10.29
8.102
6.84
6.029
5.467
10.16
7.976
6.719
5.911
5.351
10.05
7.874
6.62
5.814
5.257
9.963
7.79
6.538
5.734
5.178
9.888
7.718
6.469
5.667
5.111
9.466
7.313
6.074
5.279
4.729
9.02
6.88
5.65
4.859
4.311
10
11
12
13
14
10.04
9.646
9.33
9.074
8.862
7.559
7.206
6.927
6.701
6.515
6.552
6.217
5.953
5.739
5.564
5.994
5.668
5.412
5.205
5.035
5.636
5.316
5.064
4.862
4.695
5.386
5.069
4.821
4.62
4.456
5.2
4.886
4.64
4.441
4.278
5.057
4.744
4.499
4.302
4.14
4.942
4.632
4.388
4.191
4.03
4.849
4.539
4.296
4.1
3.939
4.772
4.462
4.22
4.025
3.864
4.706
4.397
4.155
3.96
3.8
4.327
4.021
3.78
3.587
3.427
3.909
3.602
3.361
3.165
3.004
15
16
17
18
19
8.683
8.531
8.4
8.285
8.185
6.359
6.226
6.112
6.013
5.926
5.417
5.292
5.185
5.092
5.01
4.893
4.773
4.669
4.579
4.5
4.556
4.437
4.336
4.248
4.171
4.318
4.202
4.102
4.015
3.939
4.142
4.026
3.927
3.841
3.765
4.004
3.89
3.791
3.705
3.631
3.895
3.78
3.682
3.597
3.523
3.805
3.691
3.593
3.508
3.434
3.73
3.616
3.519
3.434
3.36
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
20
21
22
23
24
8.096
8.017
7.945
7.881
7.823
5.849
5.78
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
3.71
3.667
3.699
3.64
3.587
3.539
3.496
3.564
3.506
3.453
3.406
3.363
3.457
3.398
3.346
3.299
3.256
3.368
3.31
3.258
3.211
3.168
3.294
3.236
3.184
3.137
3.094
3.231
3.173
3.121
3.074
3.032
2.859
2.801
2.749
2.702
2.659
2.421
2.36
2.305
2.256
2.211
25
26
27
28
29
7.77
7.721
7.677
7.636
7.598
5.568
5.526
5.488
5.453
5.42
4.675
4.637
4.601
4.568
4.538
4.177
4.14
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
3.33
3.324
3.288
3.256
3.226
3.198
3.217
3.182
3.149
3.12
3.092
3.129
3.094
3.062
3.032
3.005
3.056
3.021
2.988
2.959
2.931
2.993
2.958
2.926
2.896
2.868
2.62
2.585
2.552
2.522
2.495
2.169
2.131
2.097
2.064
2.034
30
32
34
36
38
7.562
7.499
7.444
7.396
7.353
5.39
5.336
5.289
5.248
5.211
4.51
4.459
4.416
4.377
4.343
4.018
3.969
3.927
3.89
3.858
3.699
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
3.067
3.021
2.981
2.946
2.915
2.979
2.934
2.894
2.859
2.828
2.906
2.86
2.821
2.786
2.755
2.843
2.798
2.758
2.723
2.692
2.469
2.423
2.383
2.347
2.316
2.006
1.956
1.911
1.872
1.837
40
60
120
∞
7.314
7.077
6.851
6.635
5.179
4.977
4.787
4.605
4.313
4.126
3.949
3.782
3.828
3.649
3.48
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
2.888
2.718
2.559
2.407
2.801
2.632
2.472
2.321
2.727
2.559
2.399
2.248
2.665
2.496
2.336
2.185
2.288
2.115
1.95
1.791
1.805
1.601
1.381
1
df1
df2
1
2
3
4
Created by Alex Cook, National University of Singapore
Page 4