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Appendix F — Critical values of Student’s \(t\) distribution with \(\nu\) degrees of freedom

The following table displays the critical values \(|c|\), where \(c = t_{1-\alpha, \nu}\) is a quantile of the Student \(t\) distribution with \(\nu\) degrees of freedom and a significance level \(\alpha\): P\((X \leq c) = 1 - \alpha\).

For a symmetric (i.e. two-sided) interval we use the column corresponding to \(1 - \alpha/2\) and the row which is equal to \(\nu = N - K\) (where \(K\) is the number of parameters of the model; in a pooled two-sample \(t\) test, \(\nu = N_1 + N_2 - 2\); Welch’s \(t\) test uses the Welch-Satterthwaite formula for \(\nu\)). For a one-sided test or interval, use the column corresponding to \(1 - \alpha\) directly. For instance, if \(N\) is very large (i.e. \(N - K > 100\), which for the sample mean with \(K = 1\) means \(N > 101\)), the 0.975 column at \(\nu = 100\) gives 1.984, while 1.960 is the limiting value at \(\nu = \infty\) (the Normal Distribution).

Table F.1: Probability less than the critical value \(\left( t_{1-\alpha, \nu} \right)\) – (source: NIST/SEMATECH (n.d.); URL: web page )
\(\nu\) 0.90 0.95 0.975 0.99 0.995 0.999
1 3.078 6.314 12.706 31.821 63.657 318.313
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 4.782
8 1.397 1.860 2.306 2.896 3.355 4.499
9 1.383 1.833 2.262 2.821 3.250 4.296
10 1.372 1.812 2.228 2.764 3.169 4.143
11 1.363 1.796 2.201 2.718 3.106 4.024
12 1.356 1.782 2.179 2.681 3.055 3.929
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
\(\infty\) 1.282 1.645 1.960 2.326 2.576 3.090