T Table (Student’s t-Distribution Critical Values)

The T Table provides critical values for the Student’s t-distribution. Use it to find the cutoff points for t-tests when sample sizes are small.

Students, teachers, and researchers use this table for hypothesis testing, confidence intervals, and comparing means. Look up values by degrees of freedom and significance level (alpha) for one-tailed or two-tailed tests.

This page includes a standard t table with common alpha levels for quick reference.

What Is a T Table?

A T Table lists critical values for the Student’s t-distribution. It helps you find cutoff points for t-tests when you work with small samples.

The t-distribution looks like the normal distribution but has thicker tails. This accounts for extra uncertainty when sample size is small.

Real-world example
You test if a new medicine lowers blood pressure using 15 patients. You calculate a t statistic from the sample mean. The t table tells you if the result is significant or likely due to chance.

Use the t table instead of a z table when your sample is under 30 or population standard deviation is unknown.

How to Use a T Table

A T Table shows critical values for the Student’s t-distribution. Use it to find cutoff points for your t statistic in hypothesis testing.

Step 1: Calculate degrees of freedom (df). For one sample: df = n − 1. For two independent samples: df = n₁ + n₂ − 2 (or use the smaller df for conservative test).

Step 2: Decide if your test is one-tailed or two-tailed. One-tailed: look at the alpha column (example: 0.05). Two-tailed: use alpha/2 (example: 0.025 for 0.05 two-tailed).

Step 3: Find the row for your df in the left column.

Step 4: Move to the column for your chosen alpha or confidence level.

Step 5: Read the critical value. For two-tailed tests, the critical values are ± the table number.

Step 6: Compare your calculated t statistic to the table value. If |t| > critical value, reject the null hypothesis.

The table gives positive values only. Use symmetry for negative t scores.

T Table Critical Values

Critical Values for Student’s t-Distribution
Upper Tail Probability: Pr(T > t)

df	0.2	0.1	0.05	0.04	0.03	0.025	0.02	0.01	0.005	0.0005
1	1.376	3.078	6.314	7.916	10.579	12.706	15.895	31.821	63.657	636.619
2	1.061	1.886	2.920	3.320	3.896	4.303	4.849	6.965	9.925	31.599
3	0.978	1.638	2.353	2.605	2.951	3.182	3.482	4.541	5.841	12.924
4	0.941	1.533	2.132	2.333	2.601	2.776	2.999	3.747	4.604	8.610
5	0.920	1.476	2.015	2.191	2.422	2.571	2.757	3.365	4.032	6.869
6	0.906	1.440	1.943	2.104	2.313	2.447	2.612	3.143	3.707	5.959
7	0.896	1.415	1.895	2.046	2.241	2.365	2.517	2.998	3.499	5.408
8	0.889	1.397	1.860	2.004	2.189	2.306	2.449	2.896	3.355	5.041
9	0.883	1.383	1.833	1.973	2.150	2.262	2.398	2.821	3.250	4.781
10	0.879	1.372	1.812	1.948	2.120	2.228	2.359	2.764	3.169	4.587
11	0.876	1.363	1.796	1.928	2.096	2.201	2.328	2.718	3.106	4.437
12	0.873	1.356	1.782	1.912	2.076	2.179	2.303	2.681	3.055	4.318
13	0.870	1.350	1.771	1.899	2.060	2.160	2.282	2.650	3.012	4.221
14	0.868	1.345	1.761	1.887	2.046	2.145	2.264	2.624	2.977	4.140
15	0.866	1.341	1.753	1.878	2.034	2.131	2.249	2.602	2.947	4.073
16	0.865	1.337	1.746	1.869	2.024	2.120	2.235	2.583	2.921	4.015
17	0.863	1.333	1.740	1.862	2.015	2.110	2.224	2.567	2.898	3.965
18	0.862	1.330	1.734	1.855	2.007	2.101	2.214	2.552	2.878	3.922
19	0.861	1.328	1.729	1.850	2.000	2.093	2.205	2.539	2.861	3.883
20	0.860	1.325	1.725	1.844	1.994	2.086	2.197	2.528	2.845	3.850
21	0.859	1.323	1.721	1.840	1.988	2.080	2.189	2.518	2.831	3.819
22	0.858	1.321	1.717	1.835	1.983	2.074	2.183	2.508	2.819	3.792
23	0.858	1.319	1.714	1.832	1.978	2.069	2.177	2.500	2.807	3.768
24	0.857	1.318	1.711	1.828	1.974	2.064	2.172	2.492	2.797	3.745
25	0.856	1.316	1.708	1.825	1.970	2.060	2.167	2.485	2.787	3.725
26	0.856	1.315	1.706	1.822	1.967	2.056	2.162	2.479	2.779	3.707
27	0.855	1.314	1.703	1.819	1.963	2.052	2.158	2.473	2.771	3.690
28	0.855	1.313	1.701	1.817	1.960	2.048	2.154	2.467	2.763	3.674
29	0.854	1.311	1.699	1.814	1.957	2.045	2.150	2.462	2.756	3.659
30	0.854	1.310	1.697	1.812	1.955	2.042	2.147	2.457	2.750	3.646
35	0.852	1.306	1.690	1.803	1.944	2.030	2.133	2.438	2.724	3.591
40	0.851	1.303	1.684	1.796	1.936	2.021	2.123	2.423	2.704	3.551
50	0.849	1.299	1.676	1.787	1.924	2.009	2.109	2.403	2.678	3.496
60	0.848	1.296	1.671	1.781	1.917	2.000	2.099	2.390	2.660	3.460
70	0.847	1.294	1.667	1.776	1.912	1.994	2.093	2.381	2.648	3.435
80	0.846	1.292	1.664	1.773	1.908	1.990	2.088	2.374	2.639	3.416
90	0.846	1.291	1.662	1.770	1.905	1.987	2.084	2.368	2.632	3.402
100	0.845	1.290	1.660	1.768	1.902	1.984	2.081	2.364	2.626	3.390
∞	0.842	1.282	1.645	1.751	1.881	1.960	2.054	2.326	2.576	3.291
Confidence Level	60%	80%	90%	92%	94%	95%	96%	98%	99%	99.9%

Note: t(∞)α/2 = Zα/2 in our notation.
For two-sided confidence intervals or two-tailed tests, use the column that matches your desired confidence level (which corresponds to α/2 upper tail probability).

T Table Examples

These examples show how to use the t table to find critical values and make decisions in t-tests.

Example 1: One-sample t-test (two-tailed)

You have a sample of 19 students. You test if their average score differs from 75. Degrees of freedom: df = 19 − 1 = 18. Significance level: alpha = 0.05 (two-tailed).

Look in the t table row df=18, column for two-tailed 0.05 (or alpha/2 = 0.025). Critical value ≈ ±2.101.

If your calculated |t| > 2.101, reject the null hypothesis.

Example 2: Independent two-sample t-test (one-tailed)

You compare test scores between two groups (n=12 and n=15). df = 12 + 15 − 2 = 25 (or use conservative df=11). Alpha = 0.05 (one-tailed, expecting higher scores in group 1).

Row df=25, column for one-tailed 0.05. Critical value ≈ 1.708.

If your calculated t > 1.708, reject the null (group 1 higher).

Example 3: Confidence interval for mean

Sample size n=10, want 95% confidence interval (two-tailed). df = 10 − 1 = 9. Alpha = 0.05 two-tailed → critical t ≈ 2.262.

Use ±2.262 in the margin of error formula.

These steps help you apply the t table in real statistics problems.

Common Mistakes When Using a T Table

Many students and beginners make these errors when reading a t table. Avoid them to get correct results.

• Using the wrong degrees of freedom. Always calculate df = n − 1 for one sample or df = n₁ + n₂ − 2 for two samples.
• Mixing one-tailed and two-tailed tests. For two-tailed, use alpha/2 in the column. For one-tailed, use full alpha.
• Using a z table instead for small samples. The t table is needed when n < 30 or population standard deviation is unknown.
• Forgetting the negative sign for two-tailed tests. Critical values are ± the table number.
• Reading the wrong row or column. Double-check df row and alpha column before noting the value.
• Assuming the table gives p-values. The t table only gives critical values — compare your t statistic to them.

Check these points each time you use the t table.

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