One-Sample t-Test Calculator

A one-sample t test evaluates whether a sample mean differs from a hypothesized population mean. It is used when the population standard deviation is unknown and must be estimated from the sample. Examples include testing whether average fill weight differs from a label claim or whether mean response time differs from a target.

The test compares the observed difference with the standard error of the mean. A large difference relative to sampling uncertainty produces a larger t statistic. A small difference or noisy sample produces weaker evidence.

The t distribution accounts for the extra uncertainty introduced by estimating standard deviation from the sample. It has heavier tails than the normal distribution, especially with small sample sizes. As sample size grows, it approaches the standard normal distribution.

Degrees of freedom for the one-sample t test are n - 1. Fewer degrees of freedom mean wider uncertainty and a more cautious test. This is why small samples need stronger evidence to reach the same significance threshold.

The test assumes independent observations and a population distribution that is roughly normal, or a sample size large enough for the mean to be approximately normal. Strong outliers, dependence, and skew can distort results.

The t test is fairly robust in many practical settings, but it is not immune to bad data. A plot of the sample, a check for outliers, and an understanding of how observations were collected are part of responsible use.

A statistically significant result means the observed mean would be unlikely under the null hypothesis, given the assumptions. It does not automatically mean the difference is large enough to matter.

Confidence intervals help bridge that gap. They show a range of plausible mean differences. If every value in the interval is practically important, the result is stronger for decision-making. If the interval includes trivial differences, the practical conclusion may be cautious even when the p-value is small.