1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.7. Tables for Probability Distributions
1.3.6.7.2. |
Critical Values of the Student's t Distribution |
t1-α,ν = -tα,ν.
The t table can be used for both one-sided (lower and upper) and two-sided tests using the appropriate value of α.
The significance level, α, is demonstrated in the graph below, which displays a t distribution with 10 degrees of freedom. The most commonly used significance level is α = 0.05. For a two-sided test, we compute 1 - α/2, or 1 - 0.05/2 = 0.975 when α = 0.05. If the absolute value of the test statistic is greater than the critical value (0.975), then we reject the null hypothesis. Due to the symmetry of the t distribution, we only tabulate the positive critical values in the table below.
Given a specified value for α :
- For a two-sided test, find the column corresponding to 1-α/2 and reject the null hypothesis if the absolute value of the test statistic is greater than the value of t1-α/2,ν in the table below.
- For an upper, one-sided test, find the column corresponding to 1-α and reject the null hypothesis if the test statistic is greater than the table value.
- For a lower, one-sided test, find the column corresponding to 1-α and reject the null hypothesis if the test statistic is less than the negative of the table value.
Critical values of Student's t distribution with ν degrees of freedom
Probability less than the critical value (t1-α,ν)
ν 0.90 0.95 0.975 0.99 0.995 0.999
1. 3.078 6.314 12.706 31.821 63.657 318.309
2. 1.886 2.920 4.303 6.965 9.925 22.327
3. 1.638 2.353 3.182 4.541 5.841 10.215
4. 1.533 2.132 2.776 3.747 4.604 7.173
5. 1.476 2.015 2.571 3.365 4.032 5.893
6. 1.440 1.943 2.447 3.143 3.707 5.208
7. 1.415 1.895 2.365 2.998 3.499 4.785
8. 1.397 1.860 2.306 2.896 3.355 4.501
9. 1.383 1.833 2.262 2.821 3.250 4.297
10. 1.372 1.812 2.228 2.764 3.169 4.144
11. 1.363 1.796 2.201 2.718 3.106 4.025
12. 1.356 1.782 2.179 2.681 3.055 3.930
13. 1.350 1.771 2.160 2.650 3.012 3.852
14. 1.345 1.761 2.145 2.624 2.977 3.787
15. 1.341 1.753 2.131 2.602 2.947 3.733
16. 1.337 1.746 2.120 2.583 2.921 3.686
17. 1.333 1.740 2.110 2.567 2.898 3.646
18. 1.330 1.734 2.101 2.552 2.878 3.610
19. 1.328 1.729 2.093 2.539 2.861 3.579
20. 1.325 1.725 2.086 2.528 2.845 3.552
21. 1.323 1.721 2.080 2.518 2.831 3.527
22. 1.321 1.717 2.074 2.508 2.819 3.505
23. 1.319 1.714 2.069 2.500 2.807 3.485
24. 1.318 1.711 2.064 2.492 2.797 3.467
25. 1.316 1.708 2.060 2.485 2.787 3.450
26. 1.315 1.706 2.056 2.479 2.779 3.435
27. 1.314 1.703 2.052 2.473 2.771 3.421
28. 1.313 1.701 2.048 2.467 2.763 3.408
29. 1.311 1.699 2.045 2.462 2.756 3.396
30. 1.310 1.697 2.042 2.457 2.750 3.385
31. 1.309 1.696 2.040 2.453 2.744 3.375
32. 1.309 1.694 2.037 2.449 2.738 3.365
33. 1.308 1.692 2.035 2.445 2.733 3.356
34. 1.307 1.691 2.032 2.441 2.728 3.348
35. 1.306 1.690 2.030 2.438 2.724 3.340
36. 1.306 1.688 2.028 2.434 2.719 3.333
37. 1.305 1.687 2.026 2.431 2.715 3.326
38. 1.304 1.686 2.024 2.429 2.712 3.319
39. 1.304 1.685 2.023 2.426 2.708 3.313
40. 1.303 1.684 2.021 2.423 2.704 3.307
41. 1.303 1.683 2.020 2.421 2.701 3.301
42. 1.302 1.682 2.018 2.418 2.698 3.296
43. 1.302 1.681 2.017 2.416 2.695 3.291
44. 1.301 1.680 2.015 2.414 2.692 3.286
45. 1.301 1.679 2.014 2.412 2.690 3.281
46. 1.300 1.679 2.013 2.410 2.687 3.277
47. 1.300 1.678 2.012 2.408 2.685 3.273
48. 1.299 1.677 2.011 2.407 2.682 3.269
49. 1.299 1.677 2.010 2.405 2.680 3.265
50. 1.299 1.676 2.009 2.403 2.678 3.261
51. 1.298 1.675 2.008 2.402 2.676 3.258
52. 1.298 1.675 2.007 2.400 2.674 3.255
53. 1.298 1.674 2.006 2.399 2.672 3.251
54. 1.297 1.674 2.005 2.397 2.670 3.248
55. 1.297 1.673 2.004 2.396 2.668 3.245
56. 1.297 1.673 2.003 2.395 2.667 3.242
57. 1.297 1.672 2.002 2.394 2.665 3.239
58. 1.296 1.672 2.002 2.392 2.663 3.237
59. 1.296 1.671 2.001 2.391 2.662 3.234
60. 1.296 1.671 2.000 2.390 2.660 3.232
61. 1.296 1.670 2.000 2.389 2.659 3.229
62. 1.295 1.670 1.999 2.388 2.657 3.227
63. 1.295 1.669 1.998 2.387 2.656 3.225
64. 1.295 1.669 1.998 2.386 2.655 3.223
65. 1.295 1.669 1.997 2.385 2.654 3.220
66. 1.295 1.668 1.997 2.384 2.652 3.218
67. 1.294 1.668 1.996 2.383 2.651 3.216
68. 1.294 1.668 1.995 2.382 2.650 3.214
69. 1.294 1.667 1.995 2.382 2.649 3.213
70. 1.294 1.667 1.994 2.381 2.648 3.211
71. 1.294 1.667 1.994 2.380 2.647 3.209
72. 1.293 1.666 1.993 2.379 2.646 3.207
73. 1.293 1.666 1.993 2.379 2.645 3.206
74. 1.293 1.666 1.993 2.378 2.644 3.204
75. 1.293 1.665 1.992 2.377 2.643 3.202
76. 1.293 1.665 1.992 2.376 2.642 3.201
77. 1.293 1.665 1.991 2.376 2.641 3.199
78. 1.292 1.665 1.991 2.375 2.640 3.198
79. 1.292 1.664 1.990 2.374 2.640 3.197
80. 1.292 1.664 1.990 2.374 2.639 3.195
81. 1.292 1.664 1.990 2.373 2.638 3.194
82. 1.292 1.664 1.989 2.373 2.637 3.193
83. 1.292 1.663 1.989 2.372 2.636 3.191
84. 1.292 1.663 1.989 2.372 2.636 3.190
85. 1.292 1.663 1.988 2.371 2.635 3.189
86. 1.291 1.663 1.988 2.370 2.634 3.188
87. 1.291 1.663 1.988 2.370 2.634 3.187
88. 1.291 1.662 1.987 2.369 2.633 3.185
89. 1.291 1.662 1.987 2.369 2.632 3.184
90. 1.291 1.662 1.987 2.368 2.632 3.183
91. 1.291 1.662 1.986 2.368 2.631 3.182
92. 1.291 1.662 1.986 2.368 2.630 3.181
93. 1.291 1.661 1.986 2.367 2.630 3.180
94. 1.291 1.661 1.986 2.367 2.629 3.179
95. 1.291 1.661 1.985 2.366 2.629 3.178
96. 1.290 1.661 1.985 2.366 2.628 3.177
97. 1.290 1.661 1.985 2.365 2.627 3.176
98. 1.290 1.661 1.984 2.365 2.627 3.175
99. 1.290 1.660 1.984 2.365 2.626 3.175
100. 1.290 1.660 1.984 2.364 2.626 3.174
Normal Values
100. 1.282 1.645 1.960 2.326 2.576 3.090
1.282 1.645 1.960 2.326 2.576 3.090
