PlanetMath: table of critical values of t distributions

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table of critical values of t distributions

(Data Structure)

Below is a table of the two-tailed critical values of the t distribution corresponding to various degrees of freedom (first column) and p-values (first row in red).

	0.2	0.1	0.05	0.02	0.01	0.005	0.002	0.001
1	3.078	6.314	12.706	31.821	63.657	127.321	318.309	636.619
2	1.886	2.920	4.303	6.965	9.925	14.089	22.327	31.599
3	1.638	2.353	3.182	4.541	5.841	7.453	10.215	12.924
4	1.533	2.132	2.776	3.747	4.604	5.598	7.173	8.610
5	1.476	2.015	2.571	3.365	4.032	4.773	5.893	6.869
6	1.440	1.943	2.447	3.143	3.707	4.317	5.208	5.959
7	1.415	1.895	2.365	2.998	3.499	4.029	4.785	5.408
8	1.397	1.860	2.306	2.896	3.355	3.833	4.501	5.041
9	1.383	1.833	2.262	2.821	3.250	3.690	4.297	4.781
10	1.372	1.812	2.228	2.764	3.169	3.581	4.144	4.587
11	1.363	1.796	2.201	2.718	3.106	3.497	4.025	4.437
12	1.356	1.782	2.179	2.681	3.055	3.428	3.930	4.318
13	1.350	1.771	2.160	2.650	3.012	3.372	3.852	4.221
14	1.345	1.761	2.145	2.624	2.977	3.326	3.787	4.140
15	1.341	1.753	2.131	2.602	2.947	3.286	3.733	4.073
16	1.337	1.746	2.120	2.583	2.921	3.252	3.686	4.015
17	1.333	1.740	2.110	2.567	2.898	3.222	3.646	3.965
18	1.330	1.734	2.101	2.552	2.878	3.197	3.610	3.922
19	1.328	1.729	2.093	2.539	2.861	3.174	3.579	3.883
20	1.325	1.725	2.086	2.528	2.845	3.153	3.552	3.850
21	1.323	1.721	2.080	2.518	2.831	3.135	3.527	3.819
22	1.321	1.717	2.074	2.508	2.819	3.119	3.505	3.792
23	1.319	1.714	2.069	2.500	2.807	3.104	3.485	3.768
24	1.318	1.711	2.064	2.492	2.797	3.091	3.467	3.745
25	1.316	1.708	2.060	2.485	2.787	3.078	3.450	3.725
26	1.315	1.706	2.056	2.479	2.779	3.067	3.435	3.707
27	1.314	1.703	2.052	2.473	2.771	3.057	3.421	3.690
28	1.313	1.701	2.048	2.467	2.763	3.047	3.408	3.674
29	1.311	1.699	2.045	2.462	2.756	3.038	3.396	3.659
30	1.310	1.697	2.042	2.457	2.750	3.030	3.385	3.646
40	1.303	1.684	2.021	2.423	2.704	2.971	3.307	3.551
50	1.299	1.676	2.009	2.403	2.678	2.937	3.261	3.496
60	1.296	1.671	2.000	2.390	2.660	2.915	3.232	3.460
70	1.294	1.667	1.994	2.381	2.648	2.899	3.211	3.435
80	1.292	1.664	1.990	2.374	2.639	2.887	3.195	3.416
90	1.291	1.662	1.987	2.368	2.632	2.878	3.183	3.402
100	1.290	1.660	1.984	2.364	2.626	2.871	3.174	3.390

These values can be related by the equation:

$\displaystyle p=P(\mid\! X\!\mid\ \ge c)$ where $\displaystyle X\sim t(d).$

Graphically, this looks like

$\includegraphics{t_dist}$

where the curve is the probability density function of the t distribution and

is the darkened area. It is evident from the graph where the name two-tailed comes from.

The one-tailed critical value for the t distribution (with degrees of freedom ) is defined to be the value such that

$\displaystyle p=P(X\ge c)$ where $\displaystyle X\sim t(d),$

for any given p-value

. One can use the same table to find the one-tailed critical value

using the following two rules:

if the p-value $p\le0.5$ then is the same as the critical value for the two-tailed case with p-value .
if then corresponds to the negative of the one-tailed critical value with p-value , which then, is the same as the negative of the critical value for the two-tailed case with p-value .