One-Sample t-Test Calculator

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t-Statistic

2.5

one-sample t-test

Standard error (s/√n)	0.8
Degrees of freedom (n − 1)	24

What is a one-sample t-test?

A one-sample t-test checks whether the mean of a single sample differs significantly from a known or hypothesized population mean ($\mu_0$). It is used when the population standard deviation is unknown and is estimated from the sample. This calculator returns the t-statistic, the standard error, and the degrees of freedom so you can complete the test.

How to use this calculator

Enter four values: the sample mean ($\bar{x}$), the hypothesized population mean ($\mu_0$) you are testing against, the sample standard deviation ($s$), and the sample size ($n$). The calculator instantly returns the t-statistic. Compare its absolute value against a critical t-value (from a t-table at your chosen significance level and degrees of freedom), or convert it to a p-value, to decide whether to reject the null hypothesis.

The formula explained

The statistic is $$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$$ The numerator ($\bar{x} - \mu_0$) is the observed difference between your sample mean and the hypothesized value. The denominator $s/\sqrt{n}$ is the standard error of the mean — how much sample means typically vary. Dividing the difference by the standard error expresses it in standard-error units. Degrees of freedom are $df = n - 1$.

t-distribution bell curve with sample mean offset from hypothesized mean, showing the difference divided by standard error — The t-statistic measures how far the sample mean sits from the hypothesized mean in standard-error units.

Worked example

Suppose $\bar{x} = 52$, $\mu_0 = 50$, $s = 4$, and $n = 25$. The standard error is $$\frac{4}{\sqrt{25}} = \frac{4}{5} = 0.8$$ Then $$t = \frac{52 - 50}{0.8} = \frac{2}{0.8} = 2.5$$ with $df = 24$. At a two-tailed $\alpha = 0.05$ the critical value is about 2.064, so 2.5 exceeds it and the result is statistically significant.

Two-tailed t-distribution with shaded rejection regions in both tails and the computed t-statistic marked — Comparing the computed t-statistic against the critical values in the tails of the t-distribution.

FAQ

When should I use a t-test instead of a z-test? Use a t-test when the population standard deviation is unknown (estimated from the sample), especially with small samples.

What does a negative t mean? It simply means the sample mean is below $\mu_0$. The sign indicates direction; the magnitude indicates strength.

How do I get a p-value? Use the t-statistic and df with a t-distribution table or software to find the area in the tail(s).

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