Last updated: 2026-07-01

What This Calculator Does

This calculator runs a one-sample t-test from summary statistics. Given a sample mean, its standard deviation, and the sample size, it compares the sample against a hypothesized population mean $\mu_0$ and reports the t statistic, the degrees of freedom, and a p-value for the two-tailed or one-tailed alternative you choose.

What a one-sample t-test answers

Suppose a process is supposed to produce a certain average — a target fill weight of 50 grams, a benchmark test score, a specified reaction time. You collect a sample and its mean lands a little away from that target. Some gap is expected from random variation alone. The one-sample t-test asks a precise question: if the true population mean really were $\mu_0$, how surprising would a sample mean this far away be? The answer is the p-value.

The t formula

The t statistic measures the distance between the sample mean $\bar{x}$ and the hypothesized mean $\mu_0$, expressed in standard errors:

where $s$ is the sample standard deviation and $n$ is the sample size. The denominator $s/\sqrt{n}$ is the standard error of the mean — it shrinks as the sample grows, so the same gap between $\bar{x}$ and $\mu_0$ becomes more convincing with more data.

Degrees of freedom

The reference distribution is Student's t with

degrees of freedom. Estimating the standard deviation from the data itself costs one degree of freedom. Fewer degrees of freedom give the t-distribution heavier tails, which makes the test appropriately more cautious with small samples. As $n$ grows, the t-distribution approaches the standard normal.

The p-value and how to read it

Once $t$ and $df$ are known, the p-value depends on the alternative hypothesis. Writing $F_t$ for the cumulative distribution function of the t-distribution:

• Two-tailed ($\mu \ne \mu_0$): $\;p = 2\left[1 - F_t(|t|,\, df)\right]$
• Right-tailed ($\mu > \mu_0$): $\;p = 1 - F_t(t,\, df)$
• Left-tailed ($\mu < \mu_0$): $\;p = F_t(t,\, df)$

Compare the p-value to a significance level $\alpha$ chosen before looking at the data, conventionally 0.05:

• p < α — the result is statistically significant; the data give evidence against $\mu = \mu_0$.
• p ≥ α — the data do not provide enough evidence to reject $\mu_0$ (which is not the same as proving the mean equals $\mu_0$).

The p-value is not the probability that the null hypothesis is true, and a non-significant result is not proof of "no difference." It only measures how compatible the data are with the hypothesized mean.

Worked example

A sample of $n = 30$ observations has mean $\bar{x} = 52$ and standard deviation $s = 5$. We test against a hypothesized mean of $\mu_0 = 50$ using a two-tailed alternative.

With $p \approx 0.037$, below 0.05, the sample mean differs significantly from 50 at the 5% level. Had we specified a right-tailed alternative in advance, the p-value would be half as large, $\approx 0.0183$.

A significant p-value tells you a difference is detectable, not how large it is relative to the spread of the data. To attach a range of plausible values to the true mean instead of a single yes/no verdict, pair the test with a Confidence Interval Calculator.

Assumptions

The one-sample t-test rests on a few conditions:

• Independence — the observations are drawn independently from the population.
• Approximate normality — the underlying population is roughly normal, or the sample is large enough that the central limit theorem makes the sampling distribution of the mean approximately normal. With moderate sample sizes and no extreme skew, the test is fairly robust.
• A standard deviation estimated from the sample — this is exactly why the t-distribution, rather than the normal, is the correct reference.

When you are comparing two independent groups rather than one sample against a fixed value, use the Two-Sample t-Test Calculator (Welch) instead. When your test statistic is already a standardized z, the P-Value from Z-Score Calculator converts it to a p-value using the normal distribution.