T-Test Calculator

The one-sample t-test addresses the question: given a sample drawn from a population, is there sufficient evidence that the true population mean differs from some pre-specified value μ₀? The procedure rests on the assumption that the underlying data are approximately normally distributed — a condition that, by the Central Limit Theorem, holds reasonably well for sample sizes of 30 or more even when the population distribution is not perfectly normal. When the population standard deviation σ is unknown (the common case), the t-distribution is used instead of the standard normal distribution because it accounts for the extra uncertainty introduced by estimating variability from the sample itself.

The formula is t = (&x̄; − μ₀) / (s / √n). The numerator measures the raw difference between what you observed (the sample mean &x̄;) and what you expected under the null hypothesis (μ₀). The denominator — the standard error of the mean — scales this difference by how much sampling variability you would expect given your sample size n and sample standard deviation s. A larger sample size shrinks the standard error, making the same raw difference yield a larger t-statistic and therefore stronger evidence against the null hypothesis. The degrees of freedom df = n − 1 determine the shape of the reference t-distribution; as df increases the t-distribution approaches the standard normal.

In practice the t-statistic is compared to a critical value from the t-distribution at a chosen significance level α (most commonly 0.05 for a two-tailed test). If |t| exceeds the critical value, or equivalently if the p-value is less than α, the null hypothesis is rejected. The t-test is used across virtually every quantitative discipline: a pharmaceutical company testing whether a new drug produces a different mean biomarker level than the known baseline, a manufacturer checking whether produced components meet a specification, an educator evaluating whether a new curriculum changes average test scores, or a financial analyst testing whether a portfolio's mean return differs from a benchmark.

The one-sample t-test makes several assumptions that should be checked before drawing conclusions. First, observations must be independent — if the data contain repeated measurements, time-series autocorrelation, or cluster effects, the standard error calculation is biased. Second, the test assumes the data (or the sampling distribution of the mean) are approximately normally distributed; for very small samples (n < 15) from heavily skewed or heavy-tailed distributions, consider a non-parametric alternative such as the Wilcoxon signed-rank test. Third, the sample standard deviation s must be a reasonable estimate of population variability — outliers can inflate s dramatically, distorting the t-statistic. Fourth, this calculator covers only the one-sample case; testing the difference between two group means requires an independent-samples or paired t-test. Finally, statistical significance does not imply practical significance — a very large sample can yield a highly significant t-statistic for a trivially small effect. Always report effect size (e.g. Cohen's d) alongside the t-statistic.