StatReason

t Distribution Calculator

Given degrees of freedom, find either the cumulative probability at a t value, or the critical t value at a target probability.

What this answers

This calculator gives direct access to Student's t distribution, the distribution behind every t-test and t-based confidence interval on this site. In cumulative-probability mode, it answers "what fraction of the distribution falls below (or above, or beyond) this t value?" In critical-value mode, it answers the reverse: "what t value corresponds to this target probability?"

How it is calculated

Both directions use the same underlying relationship between the t distribution and the regularized incomplete beta function that every t-test on this site relies on internally. Critical-value mode solves for the t value by bisection on the cumulative distribution function, since no closed-form inverse exists for the t distribution.

Worked example

For 10 degrees of freedom and a two-sided 95% confidence level: the critical value is approximately 2.228, matching the widely published reference value for t(10, .975) used throughout introductory statistics texts. As degrees of freedom grow large, this value approaches the standard normal critical value of about 1.96.

Assumption audit

Calculated from your data: the exact cumulative probability or critical value for your stated degrees of freedom, mode, and tail.
Evidence to review: whether your degrees of freedom actually match your study design (typically sample size minus 1 or minus 2, depending on the test); an incorrect df silently shifts every downstream result.
You must verify: that a t-based method is actually appropriate for your data in the first place; this calculator is a raw distribution lookup, not a test that checks its own assumptions.

Common mistakes

A common mistake is confusing a one-sided critical value with a two-sided one; a two-sided 95% confidence level uses the .975 one-sided quantile, not .95. Another is using this calculator as a substitute for running an actual t-test on real data; it looks up distribution values only and does not compute a sample statistic from your data. A third is forgetting that as degrees of freedom grow, the t distribution converges to the normal distribution, so very large df values should produce results very close to the Normal Distribution Calculator's equivalent output.

Limitations

This is a raw distribution lookup tool; it does not compute a t statistic from your own data or check whether a t-based method's underlying assumptions (roughly normal data, independence) actually hold for your situation.