One-Sample T-Test Calculator

Enter at least two numeric observations and the null-hypothesis mean to receive the t statistic, degrees of freedom, and two-sided p-value. The calculation uses the sample standard deviation, so it is appropriate when the population standard deviation is unknown. The result is deterministic and is computed directly from the values you supply, without network calls or random simulation.

What the one-sample t test measures

The one-sample t test compares the mean of one quantitative sample with a fixed reference value. That reference is the population mean proposed by the null hypothesis, often written as mu zero. The calculator first finds the sample mean and measures how far it lies from the hypothesized mean. It then scales that difference by the estimated standard error, producing the t statistic. A positive statistic means the sample mean is above the reference; a negative statistic means it is below. Its magnitude indicates how many estimated standard errors separate the two means. This calculation is useful when the population standard deviation is unknown and must be estimated from the sample itself. Because that estimate adds uncertainty, the test uses Student's t distribution rather than the normal distribution. The returned degrees of freedom equal the number of observations minus one and determine the exact shape of that reference distribution. The reported p-value is two-sided, so departures in either direction count as evidence against equality.

How to enter data and read the result

Provide the observations in sample order as a numeric array and enter the hypothesized mean in the separate field. At least two values are required, and every value must be finite. The order does not change the test because the calculation depends on the mean and variance rather than sequence. The t statistic preserves direction: for example, a negative result indicates that the observed mean is lower than the null value. Degrees of freedom are always one less than the sample size. The p-value is the probability, assuming the null hypothesis and its model are correct, of obtaining a t statistic at least as extreme in absolute value as the one calculated. A smaller p-value indicates greater incompatibility with the hypothesized mean, but it is not the probability that the null hypothesis is true. Compare it with a significance level selected before examining the data. Statistical significance also does not measure practical importance, so consider the raw difference, domain context, and an effect size or confidence interval when making a decision.

Assumptions, limitations, and responsible interpretation

A valid interpretation depends on how the observations were obtained. Values should be independent, measured on a meaningful quantitative scale, and drawn through a process that supports inference to the population of interest. For a small sample, the underlying population should be approximately normal and should not contain severe outliers, because either condition can distort the mean and standard deviation. With larger samples, the procedure is often more tolerant of moderate non-normality, although dependence, selection bias, or clustered measurements remain important problems. The calculator rejects a constant sample because its estimated standard deviation and standard error are zero, leaving the ordinary t statistic undefined. It also rejects fewer than two observations, nonnumeric entries, and non-finite values. A low p-value can motivate rejection of the stated mean under a prespecified decision rule, but it does not establish causation or guarantee replication. Report the sample size, t statistic, degrees of freedom, p-value, direction of the difference, and the substantive context rather than presenting the p-value alone.