Appendix B: Table B

Table B: Critical Values for the Distribution of t (Student's t)

How to use this table: The top two rows of the critical values table lists the most common levels of significance for one-tailed and two-tailed tests, respectively. The column of figures on the far left indicates the degrees of freedom for the test.

For the One-Sample t Test: df = n - 1
where n represents the sample size.

For the Paired Student's t Test: df = n_D - 1
where n_D represents the number of differences between pairs of data.

For the Two-Sample t Test: df = n₁ + n₂ - 2
where n₁ represents the number of elements in sample set one and n₂ represents the number of elements in sample set two.

Once the degrees of freedom have been determined and the level of significance has been selected, simply locate the critical value from the table on page 610. If the calculated value of t is equal to or greater than the critical value obtained from the t table, reject the null hypothesis and support the alternative hypothesis. If the calculated value of t is less than the critical value, retain the null hypothesis and reject the alternative hypothesis.

Example: Suppose a researcher has collected data from two independent samples with n₁ = 13 and n₂ = 10. A student's t of 2.375 is calculated. For a two-tailed test with α = .05, df = 21, the test statistic exceeds the critical value of 2.080; therefore, the null hypothesis of no significant difference between the means is rejected and the alternative hypothesis is supported.

Note

When the combined degrees of freedom for a two-sample student's t test exceed 30, the distribution of t and the distribution of z are approximately the same. Therefore, with df > 30, the distribution of z (Table A) can be used for student's t critical values, as well as for finding exact p values.

Critical Values for the Distribution of t (Student's t)

df	Levels of significance for a one-tailed test
	.20	.10	.05	.025	.01	.005	.0005
	Levels of significance for a two-tailed test
	.40	.20	.10	.05	.02	.01	.001
1	1.376	3.078	6.314	12.707	31.821	63.657	636.619
2	1.061	1.886	2.920	4.303	6.965	9.925	31.598
3	0.978	1.638	2.353	3.182	4.541	5.841	12.941
4	0.941	1.533	2.132	2.776	3.747	4.604	8.610
5	0.920	1.476	2.015	2.571	3.365	4.032	6.859
6	0.906	1.440	1.943	2.447	3.143	3.707	5.959
7	0.896	1.415	1.895	2.365	2.998	3.499	5.404
8	0.889	1.397	1.860	2.306	2.896	3.355	5.041
9	0.883	1.383	1.833	2.262	2.821	3.250	4.781
10	0.879	1.372	1.812	2.228	2.764	3.169	4.587
11	0.876	1.363	1.796	2.201	2.718	3.106	4.437
12	0.873	1.356	1.782	2.179	2.681	3.055	4.318
13	0.870	1.350	1.771	2.160	2.650	3.012	4.221
14	0.868	1.345	1.761	2.145	2.624	2.977	4.140
15	0.866	1.341	1.753	2.131	2.602	2.947	4.073
16	0.865	1.337	1.746	2.120	2.583	2.921	4.015
17	0.863	1.333	1.740	2.110	2.567	2.898	3.965
18	0.862	1.330	1.734	2.101	2.552	2.878	3.922
19	0.861	1.328	1.729	2.093	2.539	2.861	3.883
20	0.860	1.325	1.725	2.086	2.528	2.845	3.850
21	0.859	1.323	1.721	2.080	2.518	2.831	3.819
22	0.858	1.321	1.717	2.074	2.508	2.819	3.792
23	0.858	1.319	1.714	20.69	2.500	2.807	3.767
24	0.857	1.318	1.711	20.64	2.492	2.797	3.745
25	0.856	1.316	1.708	20.60	2.485	2.787	3.725
26	0.856	1.315	1.706	20.56	2.479	2.779	3.707
27	0.855	1.314	1.703	2.052	2.473	2.771	3.690
28	0.855	1.313	1.701	2.048	2.467	2.763	3.674
29	0.854	1.311	1.699	2.045	2.462	2.756	3.659
30	0.854	1.310	1.697	2.042	2.457	2.750	3.646
t = z	0.842	1.282	1.645	1.960	2.326	2.576	3.291
Critical values computed by John Timko. Used by permission.