Inverse T-Distribution Calculator
The t value for a given tail probability and degrees of freedom — the number behind every confidence interval built from a small sample.
The t distribution has heavier tails than the normal, and how much heavier depends on the degrees of freedom.
How the inverse t-distribution calculator works
The t distribution has heavier tails than the normal, and how much heavier depends on the degrees of freedom. This inverts its CDF by bisection, so any α and any df work.
At 1 df the 95% two-tailed value is 12.71; at 10 df it is 2.23; at 1,000 it is 1.962, essentially the normal's 1.960. The extra width is what a small sample costs.
Formula: t such that P(|T| > t) = α
Worked examples
| Inputs | t value | Note |
|---|---|---|
| 5% two-tailed, 20 df | 2.085963 | t = 2.086 |
| A tiny sample | 12.706205 | t = 12.71 — enormous |
| A large sample | 1.960201 | t = 1.960, indistinguishable from z |
FAQFrequently asked questions
What is the t distribution for?
Inference when the standard deviation is estimated from the sample rather than known. Its heavier tails pay for that extra uncertainty.
How many degrees of freedom do I have?
n − 1 for a one-sample test, n₁ + n₂ − 2 for a pooled two-sample test, n − k − 1 for a regression with k predictors.
When can I just use 1.96?
Above about 30 df the difference is under 4%, and above 100 it is under 1%. Below 10 it matters a great deal.
Why is t so large at 1 df?
With one degree of freedom it becomes the Cauchy distribution, whose tails are so heavy it has no mean at all.
Is this the same as a t-table?
Yes, but continuous — a printed table gives a handful of α and df values; this inverts the distribution for any pair.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.