One-Sample t-Test Calculator
Calculate the t-statistic for a one-sample t-test from the mean, standard deviation and sample size. Compare your sample mean against a reference value.
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One-sample t-test
The one-sample t-statistic checks the sample mean against a hypothesized value in the situation where you don't know the population standard deviation: t = (x̄ − μ₀) / (s / √n), with s standing in for the sample standard deviation. Student's t-distribution carries heavier tails than the normal and is governed by the degrees of freedom df = n − 1; as n → ∞ it settles into the standard normal. You then compare |t| to the critical value t_crit pulled from a t-table, and that critical value drifts toward 1.96 as df grows. Take x̄ = 102, μ₀ = 100, s = 8, n = 30: that works out to t ≈ 1.37 with df = 29, so you don't reject H₀ at 5%.
Applications
Think of lab experiments running on small samples (n between 5 and 30), method validation in analytical chemistry, or checking a sample mean against a theoretical or specified target. It also shows up in clinical trials with modest sample sizes and in acceptance testing whenever the historical σ can't be trusted.
FAQ
Why use t instead of z? Because s only estimates σ, and that estimate is shaky when n is small. The heavier tails of t are what absorb that extra uncertainty.
What is df = n − 1? Those are the degrees of freedom. You lose one because s gets computed from the sample mean itself. The more df you have, the tighter and more normal-like the distribution becomes.
Does t-test require normality? With small n you really want it, ideally. Once n > 30, the central limit theorem keeps the test robust against mild departures from normality.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.