One-sample t-test calculator with worked example

When to use a one-sample t-test

Use a one-sample t-test when you have one quantitative sample and want to know whether its population mean, written $\mu$ (the true average for the whole group), differs from a specific claimed value $\mu_0$ (the number you are testing against). You know the sample but not the population's true spread, so you estimate it from the data. That estimate is what makes this a t-test rather than a z-test.

The sample mean is written $\bar{x}$ (the average of your data), and the sample standard deviation is $s$ (how spread out your data is). You also need the sample size $n$. If you actually knew the population standard deviation $\sigma$ (the true spread of the whole group), you would use a z-test, but that is rare with real data.

Set up your hypotheses

The null hypothesis says the population mean equals the claimed value:

The alternative hypothesis takes one of three forms, depending on your question:

• $H_a: \mu \neq \mu_0$ for a two-sided test (the mean is different)
• $H_a: \mu > \mu_0$ for a one-sided test (the mean is larger)
• $H_a: \mu < \mu_0$ for a one-sided test (the mean is smaller)

Pick the alternative before you look at the data. A two-sided test is the safe default when you have no reason to expect a particular direction. For more on setting these up, see null vs alternative hypothesis.

The one-sample t-test formula

The College Board writes every standardized test statistic as (statistic minus parameter) divided by the standard error of the statistic. For a mean that gives:

The denominator $s / \sqrt{n}$ is the standard error (SE), the estimated standard deviation of the sample mean. The degrees of freedom, written $df$ (the value that sets the shape of the t-distribution), equal

A larger $|t|$ means your sample mean sits farther from $\mu_0$ in standard-error units, which produces a smaller p-value.

Conditions to check first

Three conditions support a one-sample t-test:

• Random: the data come from a random sample or a randomized experiment.
• Independent: observations are independent. When you sample without replacement, the population should be at least 10 times the sample size.
• Normal: the sampling distribution of $\bar{x}$ is approximately normal. This holds if the population is roughly normal, if $n \geq 30$ by the central limit theorem, or if a plot of a smaller sample shows no strong skew or outliers.

State these conditions in any free-response answer before you compute the statistic.

How to use this one sample t test calculator

Enter four numbers into the one sample t test calculator: the sample mean $\bar{x}$, the sample standard deviation $s$, the sample size $n$, and the claimed mean $\mu_0$. Then choose your alternative hypothesis: two-sided, greater than, or less than.

The tool returns the standard error, the t statistic, the degrees of freedom, and the p-value. If you have raw data instead of summary numbers, find $\bar{x}$ and $s$ first with the standard deviation calculator, then bring those values here.

Read the result and write a conclusion

Compare the p-value to your significance level $\alpha$ (alpha, the threshold you set in advance, often 0.05). If the p-value is less than or equal to $\alpha$, reject $H_0$. If it is greater than $\alpha$, fail to reject $H_0$.

Here is a conclusion template you can reuse at $\alpha = 0.05$: "Because the p-value of [value] is [less than / greater than] 0.05, we [reject / fail to reject] $H_0$. We [do / do not] have convincing evidence that the population mean [differs from / is greater than / is less than] [$\mu_0$ stated in context]."

If you want the interpretation in more depth, read what does a p-value mean.

Paired data: run the test on the differences

A paired design gives two measurements on each subject, such as a before value and an after value. Do not run two separate tests. Instead, subtract to get one difference for each subject, then run a one-sample t-test on that single column of differences.

The hypotheses use the mean difference $\mu_d$ (the true average of the paired differences), usually with $H_0: \mu_d = 0$. The AP course treats this as the same procedure, a test for a population mean difference, so you enter the mean, standard deviation, and count of the differences into the same calculator.