What are degrees of freedom?

Degrees of freedom (df) represent the number of values in a calculation that are free to vary. For a one-sample t-test with n observations, df = n - 1. For a two-sample t-test, df is typically approximated by the Welch–Satterthwaite equation.

When should I use a one-tail vs. two-tail test?

Use a two-tailed test when you have no prior expectation of the direction of the effect (e.g., "is A different from B?"). Use a one-tailed test only when theory or prior knowledge justifies a directional hypothesis (e.g., "is A greater than B?").

How accurate is the p-value approximation?

This calculator uses a continued-fraction approximation of the regularized incomplete beta function, accurate to roughly 6–7 significant figures for typical statistical work. Results match standard t-tables and software (R, Python scipy) to at least 4 decimal places.

What is a critical t-value?

The critical t-value is the threshold t-statistic at which a result becomes statistically significant for a given significance level (α) and degrees of freedom. If |t| exceeds the critical value, you reject the null hypothesis.

Can I use this for large samples?

Yes. As degrees of freedom increase, the t-distribution approaches the standard normal distribution. For df > 120 the results are virtually identical to z-test p-values.

Student T-Distribution Calculator

Name: Student T-Distribution Calculator
Availability: InStock
Author: Nham Vu

Enter a t-statistic and degrees of freedom to get one-tail and two-tail p-values instantly.

Inputs

t-statistic

Degrees of freedom

Critical Value Finder

Significance level (α)

Tail

Enter a t-statistic and degrees of freedom, then click Calculate P-Values.

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Summary

Enter a t-statistic and degrees of freedom to get one-tail and two-tail p-values instantly.

How it works

Enter the t-statistic from your hypothesis test.
Enter the degrees of freedom (typically n - 1 for one-sample tests).
The calculator converts |t| to an F-ratio and evaluates the regularized incomplete beta function I(df/(df+t²), df/2, 1/2).
The one-tail p-value is half the two-tail value; pick the tail based on your alternative hypothesis direction.
Results update instantly; you can also enter a desired p-value to find the corresponding critical t-value.

Use cases

Check whether a sample mean differs significantly from a known population mean.
Compare p-values from two independent or paired sample t-tests.
Verify t-test output from Excel, R, or Python.
Find critical t-values for custom significance levels (e.g., α = 0.01, 0.05, 0.10).
Use in classroom statistics exercises to build intuition for the t-distribution.
Assess regression coefficients: is a predictor statistically significant?
Quick sanity-check during data analysis before finalizing a report.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu