T Table: Critical Values of Student’s t Distribution

Degrees of freedomUsually n − 1 for one sample. Type ∞ for the z value. Significance level (α) Test

t* = 2.228

Two-tailed α = 0.05 (0.025 in each tail), 10 degrees of freedom. Reject the null hypothesis if |t| > 2.228.

Critical values of t. Column headings are the area in one tail; the second row gives the matching two-tailed α.
df	0.1	0.05	0.025	0.01	0.005	0.001	0.0005
Two tails	0.2	0.1	0.05	0.02	0.01	0.002	0.001
1	3.078	6.314	12.706	31.821	63.657	318.309	636.619
2	1.886	2.920	4.303	6.965	9.925	22.327	31.599
3	1.638	2.353	3.182	4.541	5.841	10.215	12.924
4	1.533	2.132	2.776	3.747	4.604	7.173	8.610
5	1.476	2.015	2.571	3.365	4.032	5.893	6.869
6	1.440	1.943	2.447	3.143	3.707	5.208	5.959
7	1.415	1.895	2.365	2.998	3.499	4.785	5.408
8	1.397	1.860	2.306	2.896	3.355	4.501	5.041
9	1.383	1.833	2.262	2.821	3.250	4.297	4.781
10	1.372	1.812	2.228	2.764	3.169	4.144	4.587
11	1.363	1.796	2.201	2.718	3.106	4.025	4.437
12	1.356	1.782	2.179	2.681	3.055	3.930	4.318
13	1.350	1.771	2.160	2.650	3.012	3.852	4.221
14	1.345	1.761	2.145	2.624	2.977	3.787	4.140
15	1.341	1.753	2.131	2.602	2.947	3.733	4.073
16	1.337	1.746	2.120	2.583	2.921	3.686	4.015
17	1.333	1.740	2.110	2.567	2.898	3.646	3.965
18	1.330	1.734	2.101	2.552	2.878	3.610	3.922
19	1.328	1.729	2.093	2.539	2.861	3.579	3.883
20	1.325	1.725	2.086	2.528	2.845	3.552	3.850
21	1.323	1.721	2.080	2.518	2.831	3.527	3.819
22	1.321	1.717	2.074	2.508	2.819	3.505	3.792
23	1.319	1.714	2.069	2.500	2.807	3.485	3.768
24	1.318	1.711	2.064	2.492	2.797	3.467	3.745
25	1.316	1.708	2.060	2.485	2.787	3.450	3.725
26	1.315	1.706	2.056	2.479	2.779	3.435	3.707
27	1.314	1.703	2.052	2.473	2.771	3.421	3.690
28	1.313	1.701	2.048	2.467	2.763	3.408	3.674
29	1.311	1.699	2.045	2.462	2.756	3.396	3.659
30	1.310	1.697	2.042	2.457	2.750	3.385	3.646
40	1.303	1.684	2.021	2.423	2.704	3.307	3.551
50	1.299	1.676	2.009	2.403	2.678	3.261	3.496
60	1.296	1.671	2.000	2.390	2.660	3.232	3.460
80	1.292	1.664	1.990	2.374	2.639	3.195	3.416
100	1.290	1.660	1.984	2.364	2.626	3.174	3.390
120	1.289	1.658	1.980	2.358	2.617	3.160	3.373
1000	1.282	1.646	1.962	2.330	2.581	3.098	3.300
∞ (z)	1.282	1.645	1.960	2.326	2.576	3.090	3.291

Every value is calculated in your browser from the standard normal distribution (Marsaglia’s series for Φ) and rounded to four decimals; the tables were checked cell by cell against an independent calculation. These tables give the area to the left of z. Some textbooks, including the NIST handbook, print the area from 0 to z instead, which is the left area minus 0.5 for positive z.

How to read the t table

Find the row for your degrees of freedom and the column for the area in one tail. For a two-tailed test at α = 0.05, use the 0.025 column (the second header row shows the two-tailed α). With 10 degrees of freedom the critical value is 2.228. The last row, ∞, matches the z table: 1.960 for 0.025.

Common questions

What if my degrees of freedom are not in the table?

Use the calculator above, which works for any whole number of degrees of freedom. With a printed table, a common choice is the next smaller df, which gives a slightly larger, more conservative critical value.

When do I use t instead of z?

Use t when the population standard deviation is unknown and estimated from the sample, which is most real data. As df grows, t values approach z values.

Sources and further reading

NIST/SEMATECH e-Handbook: Critical Values of the Student’s t Distribution

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