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Distribution

t-Distribution Calculator — Critical t Table

Free Student's t-distribution calculator. Compute one- and two-tailed p-values from a t-statistic and degrees of freedom, plus critical t values for any α level.

📊 t = 2.262, df = 9 → two-tailed p = .0500

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Student's t-distribution, developed by William Sealy Gosset in 1908, is the correct reference distribution whenever a sample's standard deviation is estimated from the data itself rather than known in advance — which describes almost every real t-test, confidence interval, and regression coefficient test. It looks like the normal distribution but has heavier tails, reflecting the extra uncertainty introduced by estimating σ. Its shape depends on a single parameter, the degrees of freedom (df), typically n − 1 for a one-sample test. As df grows, the t-distribution converges to the standard normal — by df ≈ 100 the two are nearly indistinguishable.

t-statistic

t = \frac{\overline{x} - μ_{0}}{s / \sqrt{n}}, df = n - 1

Critical t Values (two-tailed, α = .05 unless noted)

df	t.90 (α=.10)	t.95 (α=.05)	t.975 (α=.025)	t.99 (α=.01)
1	3.078	6.314	12.706	31.821
2	1.886	2.920	4.303	6.965
3	1.638	2.353	3.182	4.541
4	1.533	2.132	2.776	3.747
5	1.476	2.015	2.571	3.365
6	1.440	1.943	2.447	3.143
7	1.415	1.895	2.365	2.998
8	1.397	1.860	2.306	2.896
9	1.383	1.833	2.262	2.821
10	1.372	1.812	2.228	2.764
12	1.356	1.782	2.179	2.681
15	1.341	1.753	2.131	2.602
20	1.325	1.725	2.086	2.528
24	1.318	1.711	2.064	2.492
30	1.310	1.697	2.042	2.457
40	1.303	1.684	2.021	2.423
60	1.296	1.671	2.000	2.390
120	1.289	1.658	1.980	2.358

Generated directly from src/core/distributions.js (tInv) — never hand-typed.

Frequently Asked Questions

Why does the t-distribution have 'heavier tails' than normal?

Because the sample standard deviation s is itself a random quantity estimated from a finite sample, it sometimes comes out too small by chance — which would make t look artificially extreme if the reference distribution didn't account for it. The t-distribution's heavier tails build in exactly this extra source of variability, so critical values are appropriately larger (more conservative) than the normal distribution's, especially at small df.

What degrees of freedom (df) should I use?

It depends on the test: a one-sample t-test uses df = n − 1; an independent-samples pooled-variance t-test uses df = n₁ + n₂ − 2; Welch's t-test (unequal variances) uses the non-integer Welch–Satterthwaite df; a paired t-test uses df = (number of pairs) − 1. MindStat's t-test panels compute and report the correct df automatically for each design.

At what df does t become essentially the same as z?

By df = 30 the critical two-tailed t.975 value (≈2.042) is already close to z=1.96; by df ≈ 100 the difference is under 0.01 and by df = ∞ they are mathematically identical. This is why some older textbooks simplify and use the normal table once n exceeds about 30 — modern software makes that shortcut unnecessary.

Why is the critical t for df=1 so large?

At df = 1 the t-distribution is the Cauchy distribution, which has such heavy tails that it has no defined mean or variance — extreme values are far more common than under the normal distribution, so a large t is needed to reach the same tail probability. This is a mathematical reminder that a 2-observation sample carries very little information about the population standard deviation.

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t-Distribution Calculator — Critical t Table

Critical t Values (two-tailed, α = .05 unless noted)

Frequently Asked Questions

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