Find the critical t-value for a given confidence level and degrees of freedom, for one-tailed or two-tailed hypothesis tests.
The critical t-value (or t*) is the cutoff value on Student's t-distribution used to build confidence intervals and run hypothesis tests when the population standard deviation is unknown and the sample size is small. This calculator looks up the correct critical value for your confidence level, degrees of freedom, and test type.
A critical t-value marks the boundary of the "rejection region" in a hypothesis test, or the edge of a confidence interval. If your calculated t-statistic is more extreme than the critical value, you reject the null hypothesis. The t-distribution is used instead of the normal (z) distribution whenever the sample size is small and the population standard deviation is unknown, since the t-distribution has heavier tails to account for that extra uncertainty.
df = n − 1α = 1 − confidence levelA two-tailed test checks whether a value is significantly different from expectation in either direction (higher or lower), and is used for most confidence intervals. A one-tailed test checks for a difference in only one specific direction and is used when the hypothesis specifically predicts that direction.
When you know the true population standard deviation, or your sample size is very large, the z-distribution is appropriate. But with small samples and an estimated standard deviation (the common real-world case), the t-distribution's wider tails better account for the added uncertainty, giving more honest, slightly wider confidence intervals. This calculator pairs well with the standard deviation calculator when estimating sample variability.
Khan Academy's introduction to t-statistics covers this topic with additional worked examples.
For a one-sample t-test or confidence interval, degrees of freedom equal your sample size minus 1 (df = n − 1). Other test types, like two-sample t-tests, use different formulas for df.
Use a two-tailed test when you're checking for any significant difference in either direction (most confidence intervals use two-tailed values). Use a one-tailed test only when your hypothesis specifically predicts a difference in one particular direction.
Both mark critical cutoffs on a distribution, but the t-distribution has heavier tails to account for the extra uncertainty of estimating standard deviation from a small sample. As degrees of freedom grow large, the t-distribution converges to the standard normal (z) distribution.
With fewer degrees of freedom (smaller samples), there's more uncertainty in the standard deviation estimate, so the t-distribution has heavier tails and a larger critical value. As df increases, the critical value shrinks toward the corresponding z-value.