About the T Distribution Calculator
This t distribution calculator works with Student's t distribution in both directions. Give it a t score and the degrees of freedom and it returns the cumulative probability to the left, the area to the right, the two-tailed p-value and the probability between two t values. Give it a cumulative probability and it returns the matching percentile (the inverse t, like Excel's T.INV), together with one- and two-tailed critical values for your chosen α.
It replaces the printed t table for students, researchers and analysts running t-tests, building confidence intervals for a mean with an unknown standard deviation, or checking regression coefficients. Unlike a table, it handles any degrees of freedom — including fractional values from Welch's test — and gives exact areas rather than bracketing a p-value between columns.
Probabilities are computed from the regularized incomplete beta function, the standard exact method, so results match Excel (T.DIST, T.DIST.RT, T.DIST.2T, T.INV) and statistics software to many decimals. The density curve compares the t distribution with the standard normal so you can see its heavier tails at low degrees of freedom.
With the default inputs, the selected probability is 0.062077. Change any value above to recalculate instantly.
How to use the t distribution calculator
- 1Enter the degrees of freedom (n − 1 for a one-sample test, n₁ + n₂ − 2 for a pooled two-sample test).
- 2Enter your t value, and choose whether you want the left, right, two-tailed or between probability.
- 3For a between probability, enter the second t value.
- 4To find a percentile, enter a cumulative probability (for example 97.5% for the upper 2.5% point).
- 5Set α to get one- and two-tailed critical values for your test.
Formula and method
Student's t distribution describes a standardized sample mean when the population standard deviation is estimated from the sample. It is symmetric around 0 like the normal curve but has heavier tails; as the degrees of freedom ν grow, it approaches the standard normal. The mean is 0 (for ν > 1) and the variance is ν ÷ (ν − 2) for ν > 2.
Tail areas are computed exactly with the regularized incomplete beta function I: the upper tail beyond |t| equals ½ I_{ν/(ν+t²)}(ν/2, ½), and symmetry gives the rest. Percentiles (inverse t) are found by numerically solving P(T ≤ t) = p to 12+ significant digits. Two-tailed critical values use p = 1 − α/2 and one-tailed values use p = 1 − α.
- ν
- Degrees of freedom
- t
- Value of the t statistic
- Γ
- Gamma function
- I_x(a, b)
- Regularized incomplete beta function
- α
- Significance level
Worked examples
Two-tailed p-value for t = 2.1 with 10 df
With 10 degrees of freedom, the area above t = 2.1 is about 0.0310, so the two-tailed p-value is 0.0621 — not significant at 5%, because the two-tailed critical value is 2.228. The 97.5th percentile is the same 2.228, and the variance of t with 10 df is 10 ÷ 8 = 1.25.
Left-tail probability for t = −1.5 with 24 df
P(T ≤ −1.5) with 24 df is about 0.0733. The 95th percentile of t(24) is 1.711, and at α = 0.01 the critical values are ±2.797 (two-tailed) and 2.492 (one-tailed), matching a printed t table.
Probability between −1 and 2 with 5 df
P(T ≤ 2) is 0.9490 and P(T ≤ −1) is 0.1816 for t with 5 df, so the probability of landing between −1 and 2 is 0.9490 − 0.1816 ≈ 0.767. With so few degrees of freedom the tails are heavy: 5.1% of the distribution lies above 2, versus 2.3% for a normal curve.
Frequently asked questions
What is the t distribution used for?+
It is used whenever a mean is standardized with a standard deviation estimated from the sample: one-sample, paired and two-sample t-tests, confidence intervals for a mean, and tests of regression coefficients. With large samples it becomes almost identical to the normal distribution.
How do I find the p-value from a t score?+
Enter the t statistic and degrees of freedom and choose two-tailed for a ≠ alternative, right tail for a > alternative or left tail for a < alternative. The selected probability is your p-value.
How do I find a t critical value?+
Enter α and the degrees of freedom. The two-tailed critical value is the 1 − α/2 percentile (2.228 for α = 0.05 and 10 df); the one-tailed value is the 1 − α percentile (1.812 for the same inputs).
What degrees of freedom should I use?+
Use n − 1 for a one-sample or paired t-test, n₁ + n₂ − 2 for a pooled two-sample test, the Welch–Satterthwaite value for unequal variances, and n − k − 1 for a regression with k predictors. Fractional degrees of freedom are allowed.
When does the t distribution become normal?+
As degrees of freedom increase the t distribution approaches the standard normal. By about 30 df the 97.5th percentile is 2.042 versus 1.960 for z, and by 120 df it is 1.980, which is why old tables stop there and use z beyond it.
How does this compare to Excel?+
Left tail matches T.DIST(t, df, TRUE), right tail matches T.DIST.RT, the two-tailed value matches T.DIST.2T(|t|, df), and the percentile matches T.INV(p, df). Two-tailed critical values equal T.INV.2T(α, df).