A t-test answers one question: is the difference you observed — between a sample and a target, between two groups, or between paired before/after measurements — bigger than random sampling noise would produce? This calculator runs all three common variants from summary statistics you already have (means, standard deviations, sample sizes), so you don't need raw data or statistical software.
Which t-test do I need?
Use the one-sample test when you're comparing a single group's average against a known or hypothesized value (a spec, a historical average, a claimed population mean). Use the two-sample test when you have two separate, independent groups and want to know if their means differ — this calculator uses Welch's version, which does not assume the two groups have equal variance and is the safer default in practice. Use the paired test when the same subjects were measured twice (before/after, left/right, pre/post) — pairing removes subject-to-subject variability so the test is more sensitive to a real effect.
Reading the t-statistic, degrees of freedom, and p-value
The t-statistic measures how many standard errors separate the observed difference from zero (or from µ₀ for the one-sample test) — the larger its absolute value, the less compatible the data are with the null hypothesis of no difference. Degrees of freedom (df) reflect how much independent information went into estimating the variability; for the one-sample and paired tests df = n − 1, while Welch's two-sample test uses the Welch–Satterthwaite formula, which usually gives a non-integer df between the two group sizes. The p-value converts (t, df) into a probability: it's the chance of seeing a t-statistic this extreme (or more) if the null hypothesis were actually true. Compare it to your significance threshold α (conventionally 0.05) — p < α means reject the null hypothesis, p ≥ α means you don't have enough evidence to do so.
One-tailed vs. two-tailed, and what the assumptions mean
Two-tailed is the standard, conservative default — it tests whether the difference is significantly different from zero in either direction. A one-tailed test only makes sense if you committed, before collecting data, to a specific predicted direction; it halves the p-value for the same t-statistic, which makes it easier to find significance and easier to misuse after the fact. All variants here assume the underlying data are approximately normally distributed (or the sample is large enough for the Central Limit Theorem to apply) and that observations within each group are independent. With small, heavily skewed samples, treat a borderline p-value with caution and consider a nonparametric alternative.