Different Distribution Tables
The Standard Normal Z-table
The Standard Normal Distribution is used in various hypothesis testing including tests on single means, tests on proportion and the difference between two means. The Z-table has a mean of 0 and a standard deviation of 1.
As shown in the illustration below, the values inside the
given table represent the areas under the standard normal curve for values
between 0 and relative z-score.
For example:
1. Determine the area under the curve between 0 and 2.36,
first, look in the intersecting cell for the row labeled 2.30 and the column
labeled 0.06.
The area under the curve is 0.4909.
To determine the area between 0 and a negative value, look
in the intersecting cell of the row and column which sums to the absolute value
of the number in question.
For example:
2. The area under the curve between -1.3 and 0 is equal to
the area under the curve between 1.3 and 0, so look at the cell on the 1.3 row
and the 0.00 column (the area is 0.4032).
|
0.00
|
0.01
|
0.02
|
0.03
|
0.04
|
0.05
|
0.06
|
0.07
|
0.08
|
0.09
|
|
|
0.0
|
0.0000
|
0.0040
|
0.0080
|
0.0120
|
0.0160
|
0.0199
|
0.0239
|
0.0279
|
0.0319
|
0.0359
|
|
0.1
|
0.0398
|
0.0438
|
0.0478
|
0.0517
|
0.0557
|
0.0596
|
0.0636
|
0.0675
|
0.0714
|
0.0753
|
|
0.2
|
0.0793
|
0.0832
|
0.0871
|
0.0910
|
0.0948
|
0.0987
|
0.1026
|
0.1064
|
0.1103
|
0.1141
|
|
0.3
|
0.1179
|
0.1217
|
0.1255
|
0.1293
|
0.1331
|
0.1368
|
0.1406
|
0.1443
|
0.1480
|
0.1517
|
|
0.4
|
0.1554
|
0.1591
|
0.1628
|
0.1664
|
0.1700
|
0.1736
|
0.1772
|
0.1808
|
0.1844
|
0.1879
|
|
0.5
|
0.1915
|
0.1950
|
0.1985
|
0.2019
|
0.2054
|
0.2088
|
0.2123
|
0.2157
|
0.2190
|
0.2224
|
|
0.6
|
0.2257
|
0.2291
|
0.2324
|
0.2357
|
0.2389
|
0.2422
|
0.2454
|
0.2486
|
0.2517
|
0.2549
|
|
0.7
|
0.2580
|
0.2611
|
0.2642
|
0.2673
|
0.2704
|
0.2734
|
0.2764
|
0.2794
|
0.2823
|
0.2852
|
|
0.8
|
0.2881
|
0.2910
|
0.2939
|
0.2967
|
0.2995
|
0.3023
|
0.3051
|
0.3078
|
0.3106
|
0.3133
|
|
0.9
|
0.3159
|
0.3186
|
0.3212
|
0.3238
|
0.3264
|
0.3289
|
0.3315
|
0.3340
|
0.3365
|
0.3389
|
|
1.0
|
0.3413
|
0.3438
|
0.3461
|
0.3485
|
0.3508
|
0.3531
|
0.3554
|
0.3577
|
0.3599
|
0.3621
|
|
1.1
|
0.3643
|
0.3665
|
0.3686
|
0.3708
|
0.3729
|
0.3749
|
0.3770
|
0.3790
|
0.3810
|
0.3830
|
|
1.2
|
0.3849
|
0.3869
|
0.3888
|
0.3907
|
0.3925
|
0.3944
|
0.3962
|
0.3980
|
0.3997
|
0.4015
|
|
1.3
|
0.4032
|
0.4049
|
0.4066
|
0.4082
|
0.4099
|
0.4115
|
0.4131
|
0.4147
|
0.4162
|
0.4177
|
|
1.4
|
0.4192
|
0.4207
|
0.4222
|
0.4236
|
0.4251
|
0.4265
|
0.4279
|
0.4292
|
0.4306
|
0.4319
|
|
1.5
|
0.4332
|
0.4345
|
0.4357
|
0.4370
|
0.4382
|
0.4394
|
0.4406
|
0.4418
|
0.4429
|
0.4441
|
|
1.6
|
0.4452
|
0.4463
|
0.4474
|
0.4484
|
0.4495
|
0.4505
|
0.4515
|
0.4525
|
0.4535
|
0.4545
|
|
1.7
|
0.4554
|
0.4564
|
0.4573
|
0.4582
|
0.4591
|
0.4599
|
0.4608
|
0.4616
|
0.4625
|
0.4633
|
|
1.8
|
0.4641
|
0.4649
|
0.4656
|
0.4664
|
0.4671
|
0.4678
|
0.4686
|
0.4693
|
0.4699
|
0.4706
|
|
1.9
|
0.4713
|
0.4719
|
0.4726
|
0.4732
|
0.4738
|
0.4744
|
0.4750
|
0.4756
|
0.4761
|
0.4767
|
|
2.0
|
0.4772
|
0.4778
|
0.4783
|
0.4788
|
0.4793
|
0.4798
|
0.4803
|
0.4808
|
0.4812
|
0.4817
|
|
2.1
|
0.4821
|
0.4826
|
0.4830
|
0.4834
|
0.4838
|
0.4842
|
0.4846
|
0.4850
|
0.4854
|
0.4857
|
|
2.2
|
0.4861
|
0.4864
|
0.4868
|
0.4871
|
0.4875
|
0.4878
|
0.4881
|
0.4884
|
0.4887
|
0.4890
|
|
2.3
|
0.4893
|
0.4896
|
0.4898
|
0.4901
|
0.4904
|
0.4906
|
0.4909
|
0.4911
|
0.4913
|
0.4916
|
|
2.4
|
0.4918
|
0.4920
|
0.4922
|
0.4925
|
0.4927
|
0.4929
|
0.4931
|
0.4932
|
0.4934
|
0.4936
|
|
2.5
|
0.4938
|
0.4940
|
0.4941
|
0.4943
|
0.4945
|
0.4946
|
0.4948
|
0.4949
|
0.4951
|
0.4952
|
|
2.6
|
0.4953
|
0.4955
|
0.4956
|
0.4957
|
0.4959
|
0.4960
|
0.4961
|
0.4962
|
0.4963
|
0.4964
|
|
2.7
|
0.4965
|
0.4966
|
0.4967
|
0.4968
|
0.4969
|
0.4970
|
0.4971
|
0.4972
|
0.4973
|
0.4974
|
|
2.8
|
0.4974
|
0.4975
|
0.4976
|
0.4977
|
0.4977
|
0.4978
|
0.4979
|
0.4979
|
0.4980
|
0.4981
|
|
2.9
|
0.4981
|
0.4982
|
0.4982
|
0.4983
|
0.4984
|
0.4984
|
0.4985
|
0.4985
|
0.4986
|
0.4986
|
|
3.0
|
0.4987
|
0.4987
|
0.4987
|
0.4988
|
0.4988
|
0.4989
|
0.4989
|
0.4989
|
0.4990
|
0.4990
|
T-table
The Shape of the Student's t distribution is determined by
the degrees of freedom. As shown in the animation above, its shape changes as
the degrees of freedom increases.
|
df\p
|
0.40
|
0.25
|
0.10
|
0.05
|
0.025
|
0.01
|
0.005
|
0.0005
|
|
1
|
0.324920
|
1.000000
|
3.077684
|
6.313752
|
12.70620
|
31.82052
|
63.65674
|
636.6192
|
|
2
|
0.288675
|
0.816497
|
1.885618
|
2.919986
|
4.30265
|
6.96456
|
9.92484
|
31.5991
|
|
3
|
0.276671
|
0.764892
|
1.637744
|
2.353363
|
3.18245
|
4.54070
|
5.84091
|
12.9240
|
|
4
|
0.270722
|
0.740697
|
1.533206
|
2.131847
|
2.77645
|
3.74695
|
4.60409
|
8.6103
|
|
5
|
0.267181
|
0.726687
|
1.475884
|
2.015048
|
2.57058
|
3.36493
|
4.03214
|
6.8688
|
|
6
|
0.264835
|
0.717558
|
1.439756
|
1.943180
|
2.44691
|
3.14267
|
3.70743
|
5.9588
|
|
7
|
0.263167
|
0.711142
|
1.414924
|
1.894579
|
2.36462
|
2.99795
|
3.49948
|
5.4079
|
|
8
|
0.261921
|
0.706387
|
1.396815
|
1.859548
|
2.30600
|
2.89646
|
3.35539
|
5.0413
|
|
9
|
0.260955
|
0.702722
|
1.383029
|
1.833113
|
2.26216
|
2.82144
|
3.24984
|
4.7809
|
|
10
|
0.260185
|
0.699812
|
1.372184
|
1.812461
|
2.22814
|
2.76377
|
3.16927
|
4.5869
|
|
11
|
0.259556
|
0.697445
|
1.363430
|
1.795885
|
2.20099
|
2.71808
|
3.10581
|
4.4370
|
|
12
|
0.259033
|
0.695483
|
1.356217
|
1.782288
|
2.17881
|
2.68100
|
3.05454
|
4.3178
|
|
13
|
0.258591
|
0.693829
|
1.350171
|
1.770933
|
2.16037
|
2.65031
|
3.01228
|
4.2208
|
|
14
|
0.258213
|
0.692417
|
1.345030
|
1.761310
|
2.14479
|
2.62449
|
2.97684
|
4.1405
|
|
15
|
0.257885
|
0.691197
|
1.340606
|
1.753050
|
2.13145
|
2.60248
|
2.94671
|
4.0728
|
|
16
|
0.257599
|
0.690132
|
1.336757
|
1.745884
|
2.11991
|
2.58349
|
2.92078
|
4.0150
|
|
17
|
0.257347
|
0.689195
|
1.333379
|
1.739607
|
2.10982
|
2.56693
|
2.89823
|
3.9651
|
|
18
|
0.257123
|
0.688364
|
1.330391
|
1.734064
|
2.10092
|
2.55238
|
2.87844
|
3.9216
|
|
19
|
0.256923
|
0.687621
|
1.327728
|
1.729133
|
2.09302
|
2.53948
|
2.86093
|
3.8834
|
|
20
|
0.256743
|
0.686954
|
1.325341
|
1.724718
|
2.08596
|
2.52798
|
2.84534
|
3.8495
|
|
21
|
0.256580
|
0.686352
|
1.323188
|
1.720743
|
2.07961
|
2.51765
|
2.83136
|
3.8193
|
|
22
|
0.256432
|
0.685805
|
1.321237
|
1.717144
|
2.07387
|
2.50832
|
2.81876
|
3.7921
|
|
23
|
0.256297
|
0.685306
|
1.319460
|
1.713872
|
2.06866
|
2.49987
|
2.80734
|
3.7676
|
|
24
|
0.256173
|
0.684850
|
1.317836
|
1.710882
|
2.06390
|
2.49216
|
2.79694
|
3.7454
|
|
25
|
0.256060
|
0.684430
|
1.316345
|
1.708141
|
2.05954
|
2.48511
|
2.78744
|
3.7251
|
|
26
|
0.255955
|
0.684043
|
1.314972
|
1.705618
|
2.05553
|
2.47863
|
2.77871
|
3.7066
|
|
27
|
0.255858
|
0.683685
|
1.313703
|
1.703288
|
2.05183
|
2.47266
|
2.77068
|
3.6896
|
|
28
|
0.255768
|
0.683353
|
1.312527
|
1.701131
|
2.04841
|
2.46714
|
2.76326
|
3.6739
|
|
29
|
0.255684
|
0.683044
|
1.311434
|
1.699127
|
2.04523
|
2.46202
|
2.75639
|
3.6594
|
|
30
|
0.255605
|
0.682756
|
1.310415
|
1.697261
|
2.04227
|
2.45726
|
2.75000
|
3.6460
|
|
inf
|
0.253347
|
0.674490
|
1.281552
|
1.644854
|
1.95996
|
2.32635
|
2.57583
|
3.2905
|
The Chi-square Table
Like the Student's t-Distribution, the Chi-square distribution's
shape is determined by its degrees of freedom. The animation above shows the
shape of the Chi-square distribution as the degrees of freedom
increase (1, 2, 5, 10, 25 and 50). As shown in the illustration below, the
values inside this table are critical values of the Chi-square distribution
with the corresponding degrees of freedom. To determine the value from a
Chi-square distribution (with a specific degree of freedom) which has a given
area above it, go to the given area column and the desired degree of freedom
row. For example, the .25 critical value for a Chi-square with 4 degrees of
freedom is 5.38527. This means that the area to the right of 5.38527 in a
Chi-square distribution with 4 degrees of freedom is .25.
|
Right tail areas for the Chi-square Distribution
|
|||||||||||||
|
df\area
|
.995
|
.990
|
.975
|
.950
|
.900
|
.750
|
.500
|
.250
|
.100
|
.050
|
.025
|
.010
|
.005
|
|
1
|
0.00004
|
0.00016
|
0.00098
|
0.00393
|
0.01579
|
0.10153
|
0.45494
|
1.32330
|
2.70554
|
3.84146
|
5.02389
|
6.63490
|
7.87944
|
|
2
|
0.01003
|
0.02010
|
0.05064
|
0.10259
|
0.21072
|
0.57536
|
1.38629
|
2.77259
|
4.60517
|
5.99146
|
7.37776
|
9.21034
|
10.59663
|
|
3
|
0.07172
|
0.11483
|
0.21580
|
0.35185
|
0.58437
|
1.21253
|
2.36597
|
4.10834
|
6.25139
|
7.81473
|
9.34840
|
11.34487
|
12.83816
|
|
4
|
0.20699
|
0.29711
|
0.48442
|
0.71072
|
1.06362
|
1.92256
|
3.35669
|
5.38527
|
7.77944
|
9.48773
|
11.14329
|
13.27670
|
14.86026
|
|
5
|
0.41174
|
0.55430
|
0.83121
|
1.14548
|
1.61031
|
2.67460
|
4.35146
|
6.62568
|
9.23636
|
11.07050
|
12.83250
|
15.08627
|
16.74960
|
|
6
|
0.67573
|
0.87209
|
1.23734
|
1.63538
|
2.20413
|
3.45460
|
5.34812
|
7.84080
|
10.64464
|
12.59159
|
14.44938
|
16.81189
|
18.54758
|
|
7
|
0.98926
|
1.23904
|
1.68987
|
2.16735
|
2.83311
|
4.25485
|
6.34581
|
9.03715
|
12.01704
|
14.06714
|
16.01276
|
18.47531
|
20.27774
|
|
8
|
1.34441
|
1.64650
|
2.17973
|
2.73264
|
3.48954
|
5.07064
|
7.34412
|
10.21885
|
13.36157
|
15.50731
|
17.53455
|
20.09024
|
21.95495
|
|
9
|
1.73493
|
2.08790
|
2.70039
|
3.32511
|
4.16816
|
5.89883
|
8.34283
|
11.38875
|
14.68366
|
16.91898
|
19.02277
|
21.66599
|
23.58935
|
|
10
|
2.15586
|
2.55821
|
3.24697
|
3.94030
|
4.86518
|
6.73720
|
9.34182
|
12.54886
|
15.98718
|
18.30704
|
20.48318
|
23.20925
|
25.18818
|
|
11
|
2.60322
|
3.05348
|
3.81575
|
4.57481
|
5.57778
|
7.58414
|
10.34100
|
13.70069
|
17.27501
|
19.67514
|
21.92005
|
24.72497
|
26.75685
|
|
12
|
3.07382
|
3.57057
|
4.40379
|
5.22603
|
6.30380
|
8.43842
|
11.34032
|
14.84540
|
18.54935
|
21.02607
|
23.33666
|
26.21697
|
28.29952
|
|
13
|
3.56503
|
4.10692
|
5.00875
|
5.89186
|
7.04150
|
9.29907
|
12.33976
|
15.98391
|
19.81193
|
22.36203
|
24.73560
|
27.68825
|
29.81947
|
|
14
|
4.07467
|
4.66043
|
5.62873
|
6.57063
|
7.78953
|
10.16531
|
13.33927
|
17.11693
|
21.06414
|
23.68479
|
26.11895
|
29.14124
|
31.31935
|
|
15
|
4.60092
|
5.22935
|
6.26214
|
7.26094
|
8.54676
|
11.03654
|
14.33886
|
18.24509
|
22.30713
|
24.99579
|
27.48839
|
30.57791
|
32.80132
|
|
16
|
5.14221
|
5.81221
|
6.90766
|
7.96165
|
9.31224
|
11.91222
|
15.33850
|
19.36886
|
23.54183
|
26.29623
|
28.84535
|
31.99993
|
34.26719
|
|
17
|
5.69722
|
6.40776
|
7.56419
|
8.67176
|
10.08519
|
12.79193
|
16.33818
|
20.48868
|
24.76904
|
27.58711
|
30.19101
|
33.40866
|
35.71847
|
|
18
|
6.26480
|
7.01491
|
8.23075
|
9.39046
|
10.86494
|
13.67529
|
17.33790
|
21.60489
|
25.98942
|
28.86930
|
31.52638
|
34.80531
|
37.15645
|
|
19
|
6.84397
|
7.63273
|
8.90652
|
10.11701
|
11.65091
|
14.56200
|
18.33765
|
22.71781
|
27.20357
|
30.14353
|
32.85233
|
36.19087
|
38.58226
|
|
20
|
7.43384
|
8.26040
|
9.59078
|
10.85081
|
12.44261
|
15.45177
|
19.33743
|
23.82769
|
28.41198
|
31.41043
|
34.16961
|
37.56623
|
39.99685
|
|
21
|
8.03365
|
8.89720
|
10.28290
|
11.59131
|
13.23960
|
16.34438
|
20.33723
|
24.93478
|
29.61509
|
32.67057
|
35.47888
|
38.93217
|
41.40106
|
|
22
|
8.64272
|
9.54249
|
10.98232
|
12.33801
|
14.04149
|
17.23962
|
21.33704
|
26.03927
|
30.81328
|
33.92444
|
36.78071
|
40.28936
|
42.79565
|
|
23
|
9.26042
|
10.19572
|
11.68855
|
13.09051
|
14.84796
|
18.13730
|
22.33688
|
27.14134
|
32.00690
|
35.17246
|
38.07563
|
41.63840
|
44.18128
|
|
24
|
9.88623
|
10.85636
|
12.40115
|
13.84843
|
15.65868
|
19.03725
|
23.33673
|
28.24115
|
33.19624
|
36.41503
|
39.36408
|
42.97982
|
45.55851
|
|
25
|
10.51965
|
11.52398
|
13.11972
|
14.61141
|
16.47341
|
19.93934
|
24.33659
|
29.33885
|
34.38159
|
37.65248
|
40.64647
|
44.31410
|
46.92789
|
|
26
|
11.16024
|
12.19815
|
13.84390
|
15.37916
|
17.29188
|
20.84343
|
25.33646
|
30.43457
|
35.56317
|
38.88514
|
41.92317
|
45.64168
|
48.28988
|
|
27
|
11.80759
|
12.87850
|
14.57338
|
16.15140
|
18.11390
|
21.74940
|
26.33634
|
31.52841
|
36.74122
|
40.11327
|
43.19451
|
46.96294
|
49.64492
|
|
28
|
12.46134
|
13.56471
|
15.30786
|
16.92788
|
18.93924
|
22.65716
|
27.33623
|
32.62049
|
37.91592
|
41.33714
|
44.46079
|
48.27824
|
50.99338
|
|
29
|
13.12115
|
14.25645
|
16.04707
|
17.70837
|
19.76774
|
23.56659
|
28.33613
|
33.71091
|
39.08747
|
42.55697
|
45.72229
|
49.58788
|
52.33562
|
|
30
|
13.78672
|
14.95346
|
16.79077
|
18.49266
|
20.59923
|
24.47761
|
29.33603
|
34.79974
|
40.25602
|
43.77297
|
46.97924
|
50.89218
|
53.67196
|
The F Distribution
The F distribution is a
right-skewed distribution used most commonly in Analysis of Variance. The F
distribution is a ratio of two Chi-square distributions, and a
specific F distribution is denoted by the degrees of freedom for the numerator
Chi-square and the degrees of freedom for the denominator Chi-square.. When
referencing the F distribution, the numerator degrees of freedom are always
given first, as switching the order of degrees of freedom changes the
distribution (e.g., F(10,12) does not equal F(12,10)). For the four F
tables below, the rows represent denominator degrees of freedom and the columns
represent numerator degrees of freedom.
By: Tito Nuevacobita Jr.
III-Gold
The tables are so crowded:)
ReplyDeleteNice comment... We will fix that thing for you...
Delete.thank you.but you haven't fix it yet..
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