T Distribution Calculator
Calculate probabilities and critical values for Student's t-distribution using degrees of freedom and a t-value or probability.
Your t probability or threshold.
Enter your values, or try an example.
Your results and calculation will appear here.
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How to use this calculator
- Choose a probability mode or find a t-value from a probability.
- Enter degrees of freedom. In a basic one-sample t procedure df is often sample size minus one, but that is not a universal rule.
- For probabilities, enter a t-value or interval. For inverse modes, choose decimal or percent and enter the cumulative or tail probability requested.
- Calculate, then compare the shaded region with the result and open the calculation steps.
Worked example: right tail, df 10
This example uses sample values, not your current inputs.
Right tail · df = 10 · t = 2
Probability = 0.03669402; Percentage = 3.6694%
P(T ≥ 2) ≈ 0.03669402 (3.6694%). Areas represent probability; density height at a single point does not.
Example calculation
- Degrees of freedom: df = 10. This is the central Student's t distribution.
- P(T ≥ 2) = 1 − F(10, 2).
- F(10, 2) ≈ 0.96330598; right-tail probability ≈ 0.03669402. Small tails are evaluated directly, not by subtracting rounded values.
- P(T ≥ 2) ≈ 0.03669402; percentage ≈ 3.6694%.
- These are continuous probabilities: including or excluding one boundary does not change the area. Displayed intermediate values are rounded; the answer uses full precision.
Formula & assumptions
f(t) = Γ((ν + 1)/2) / [√(νπ) Γ(ν/2)] × (1 + t²/ν)⁻⁽ν⁺¹⁾/²; F(t) = P(T ≤ t); right tail = F(−t); two tails = 2F(−|t|).
- This calculator uses the central Student's t distribution, symmetric around zero; it does not perform a complete t-test.
- Degrees of freedom may be fractional. Supported df: 0.1 to 10,000. Enter df directly, not sample size.
- Probabilities are areas under a continuous density curve; a single point has zero probability. Including an endpoint does not change an interval area.
- Inverse inputs must be strictly between 0 and 1 (or 0% and 100%). Two-tailed α is the total area in both tails, not the confidence level.
- Numerical evaluation uses regularized incomplete beta tails and a bounded inverse solver. Inputs outside the supported finite range fail explicitly.
Sources
- NIST: t distribution ↗
- NIST: Student's t critical values ↗
- R documentation: Student t distribution ↗
- SciPy reference: Student t ↗
Reviewed: October 2026
Common questions
What are degrees of freedom?
Degrees of freedom describe the parameter controlling the t distribution’s shape. They come from the statistical procedure being used; they are not always the sample size minus one. Fractional values are valid.
How is Student’s t different from the standard normal distribution?
Student’s t has heavier tails, especially with small degrees of freedom. As degrees of freedom increase, its shape approaches the standard normal distribution.
What does a two-tailed probability mean?
It is the combined area at least as far from zero as the entered t-value in either direction. A negative t-value gives the same two-tailed probability as its positive counterpart.
How do I find a critical value for 95% two-tailed confidence?
Choose Two-tailed critical t, select Percent format and enter 5 for the total tail probability. Each tail has area 2.5%, so the positive threshold has cumulative probability 97.5%.
Does this calculator perform a t-test?
No. It calculates central t-distribution areas and thresholds. A complete test also needs an appropriate statistic, sampling assumptions and a defined hypothesis.
Reference tables
Read Student's t critical values in the T Distribution Table. Browse all statistical tables.
Use these tools for mathematical information, not as a substitute for checking assumptions. Read our methodology or disclaimer.