One-Sample T-Test Calculator
Calculate the t-statistic, standard error and degrees of freedom for a one-sample t-test, comparing a sample mean to a hypothesized value when the population standard deviation is unknown.
About the One-Sample T-Test Calculator
The one-sample t-test checks whether a sample's mean differs significantly from a hypothesized value, for situations where the population standard deviation isn't known and must be estimated from the sample itself — the most common real-world case.
Formula: t = (x̄ − μ₀) ÷ (s ÷ √n), with degrees of freedom df = n − 1
This differs from the one-sample z-test, which requires knowing the true population standard deviation (σ) in advance — something rarely available outside of controlled or theoretical settings. Because the t-test instead uses the sample's own standard deviation (s), it accounts for the extra uncertainty that comes with estimating spread from limited data, especially for small samples. Compare the resulting t-statistic against a critical value from the critical value calculator, using the reported degrees of freedom, to decide whether to reject the hypothesis that the true mean equals the hypothesized value. As the sample size grows, the t-distribution converges toward the normal distribution, so the t-test and z-test produce increasingly similar results for large n.