Statistics Reference

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

The t-table lists critical values of Student's t-distribution: the cutoffs a t-statistic must exceed to be significant at a given level. Rows are degrees of freedom (df), columns are significance levels (α), with headers for both one-tailed and two-tailed tests. The distribution is symmetric, so only positive values are printed; for a lower tail, use the negative of the table value.

How to Read the T-Table

You need three things: the degrees of freedom, the significance level, and whether your test is one-tailed or two-tailed. The critical value sits where the df row meets the α column.

Example: 95% confidence interval from a sample of 15

Degrees of freedom: df = n − 1 = 15 − 1 = 14, so use the row labeled 14.
A 95% confidence interval leaves 5% split across both tails, so read the two-tail α = 0.05 column (the same column as one-tail α = 0.025).
The cell gives 2.145. The interval is the sample mean ± 2.145 standard errors.

The same lookup works for hypothesis tests: a two-tailed t-test with df = 10 at α = 0.05 rejects the null hypothesis when |t| exceeds 2.228, while a one-tailed test with df = 1 at α = 0.05 uses 6.314.

T-Distribution Critical Values

Each cell is the positive critical value t* for the given degrees of freedom and significance level. The final row, computed at df = 1000, stands in for infinite degrees of freedom and sits within 0.005 of the z critical values in every column.

Critical values of Student's t-distribution by degrees of freedom and significance level
One-tail α	0.10	0.05	0.025	0.01	0.005
Two-tail α	0.20	0.10	0.05	0.02	0.01
df	critical values t*
1	3.078	6.314	12.706	31.821	63.657
2	1.886	2.920	4.303	6.965	9.925
3	1.638	2.353	3.182	4.541	5.841
4	1.533	2.132	2.776	3.747	4.604
5	1.476	2.015	2.571	3.365	4.032
6	1.440	1.943	2.447	3.143	3.707
7	1.415	1.895	2.365	2.998	3.499
8	1.397	1.860	2.306	2.896	3.355
9	1.383	1.833	2.262	2.821	3.250
10	1.372	1.812	2.228	2.764	3.169
11	1.363	1.796	2.201	2.718	3.106
12	1.356	1.782	2.179	2.681	3.055
13	1.350	1.771	2.160	2.650	3.012
14	1.345	1.761	2.145	2.624	2.977
15	1.341	1.753	2.131	2.602	2.947
16	1.337	1.746	2.120	2.583	2.921
17	1.333	1.740	2.110	2.567	2.898
18	1.330	1.734	2.101	2.552	2.878
19	1.328	1.729	2.093	2.539	2.861
20	1.325	1.725	2.086	2.528	2.845
21	1.323	1.721	2.080	2.518	2.831
22	1.321	1.717	2.074	2.508	2.819
23	1.319	1.714	2.069	2.500	2.807
24	1.318	1.711	2.064	2.492	2.797
25	1.316	1.708	2.060	2.485	2.787
26	1.315	1.706	2.056	2.479	2.779
27	1.314	1.703	2.052	2.473	2.771
28	1.313	1.701	2.048	2.467	2.763
29	1.311	1.699	2.045	2.462	2.756
30	1.310	1.697	2.042	2.457	2.750
40	1.303	1.684	2.021	2.423	2.704
60	1.296	1.671	2.000	2.390	2.660
80	1.292	1.664	1.990	2.374	2.639
100	1.290	1.660	1.984	2.364	2.626
120	1.289	1.658	1.980	2.358	2.617
∞ (z)	1.282	1.646	1.962	2.330	2.581

One-Tailed vs Two-Tailed

The two header rows describe the same columns from two points of view. A one-tailed test places all of α in a single tail; a two-tailed test splits α across both tails, so each tail holds α/2.

One-tailed test

Directional hypothesis, such as “the new process takes longer than the old one.” At α = 0.05 with df = 10, reject when t exceeds 1.812 (or falls below −1.812 for a lower-tail test).

Two-tailed test

Non-directional hypothesis, such as “the two processes differ.” At α = 0.05 with df = 10, each tail holds 0.025, so reject when |t| exceeds 2.228.

Confidence intervals always use the two-tail reading: a 95% interval corresponds to two-tail α = 0.05, a 99% interval to two-tail α = 0.01.

Degrees of Freedom for Common Tests

One-sample t-test: df = n − 1. A sample of 20 observations has df = 19.
Two-sample t-test (pooled variance): df = n₁ + n₂ − 2. Groups of 12 and 14 give df = 24. The Welch version uses a smaller, non-integer df computed from the two variances.
Paired t-test: df = n − 1, where n is the number of pairs, because the analysis runs on the within-pair differences.
Regression slope: df = n − 2 in simple linear regression, one degree lost to each estimated coefficient.

If your exact df is missing from the table, round down to the nearest listed row for a conservative critical value, or compute the exact value with the critical value calculator.

Why the Last Row Matches z

As degrees of freedom grow, the sample standard deviation becomes a precise estimate of the population value, and the t-distribution converges to the standard normal distribution. You can watch the two-tail 0.05 column shrink toward the z value of 1.960: from 12.706 at df = 1 to 2.228 at df = 10, 2.042 at df = 30, 1.980 at df = 120, and 1.962 in the final row. That row is computed at df = 1000 as a stand-in for infinity, which is why it reads 1.962 rather than exactly 1.960; the remaining gap is under 0.005 in every column.

This convergence is why the common shortcut “use z beyond n = 30” exists. It is a reasonable approximation, but the t values are the exact ones whenever the standard deviation is estimated, and the z-table values are simply the limit the t-table approaches.

Where These Values Are Used

Critical t-values appear in two places: hypothesis tests, where the test statistic is compared against the critical value, and confidence intervals, where the critical value scales the standard error into a margin of error. To run the full procedures rather than look up cutoffs, use the t-test calculator or the confidence interval calculator, both of which report the exact critical values and p-values for your data.

Frequently Asked Questions

Why do degrees of freedom matter in the t-table?

Degrees of freedom control the shape of the t-distribution. With few degrees of freedom, the sample standard deviation is an unreliable estimate of the population value, so the distribution has heavy tails and large critical values (6.314 for a one-tail test with df = 1 at alpha = 0.05). As degrees of freedom increase, the estimate stabilizes, the tails thin out, and critical values shrink toward the z values in the bottom row.

How do I choose between the one-tail and two-tail header rows?

Use the one-tail row when your hypothesis has a direction, such as testing whether a mean is greater than a target value. Use the two-tail row when you test for any difference in either direction, or when you build a confidence interval. Both headers point at the same columns: a one-tail alpha of 0.025 and a two-tail alpha of 0.05 give the same critical value, because the two-tail test splits its alpha across both tails.

When should I use the t-table instead of the z-table?

Use the t-table whenever the population standard deviation is unknown and you estimate it from the sample, which covers most real t-tests and confidence intervals for means. Use the z-table when the population standard deviation is genuinely known or the sample is very large. The practical difference fades as samples grow: by df = 120 the two-tail 0.05 critical value is 1.980, already close to the z value of 1.960.

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

The standard conservative rule is to round down to the nearest listed row, so for df = 47 use the df = 40 row (1.684 rather than the exact 1.678 for a one-tail alpha of 0.05). Linear interpolation between the neighboring rows gives a closer approximation when the gap matters. For an exact value at any degrees of freedom, use the critical value calculator on this site instead of interpolating by hand.

Why are t critical values always larger than z critical values?

Estimating the population standard deviation from a sample adds uncertainty on top of the sampling variation in the mean. The t-distribution accounts for that extra uncertainty with heavier tails, so you must go further from zero to enclose the same probability. The penalty is largest for tiny samples (12.706 versus 1.960 for two-tail 0.05 at df = 1) and becomes negligible past a few hundred degrees of freedom.

Can I still use the t-table for large samples?

Yes. The t-distribution is always the exact reference when the standard deviation is estimated, no matter the sample size, so using it for large samples is never wrong. The table simply stops distinguishing t from z at some point: beyond df = 120 the values change so little that the infinity row, which sits within 0.005 of the z critical values, is accurate for practical purposes.