How do I interpret the t-statistic?

A t-statistic near 0 means your sample mean is close to the hypothesised value mu0. As |t| grows, the sample mean is further from mu0 relative to sampling variability. Compare t to the critical value from a t-table with n-1 degrees of freedom at your chosen significance level.

What are the degrees of freedom for this test?

For a one-sample t-test, df = n - 1. You use this to look up the critical t-value or compute the p-value. Larger samples have more df and the t-distribution approaches the standard normal.

When should I use a one-sample t-test versus a z-test?

Use a t-test when the population standard deviation is unknown and you estimate it from the sample (the usual case). Use a z-test only when the population standard deviation is known exactly and n is large.

One-Sample t-Statistic Calculator

Calculate the one-sample t-statistic from sample mean, hypothesized population mean, standard deviation, and sample size. Use this to test whether your sample mean differs significantly from a known or assumed value.

By AbraCalc Editorial Team · Reviewed by AbraCalc Methodology Review · Last reviewed: 2026-06-25

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How to use this tool

Enter sample mean, hypothesized mean (mu0), standard deviation and sample size (n) in the fields above.
Results update instantly as you type — or click Calculate.
Read your t-statistic and the full breakdown beneath it.

Formula

t = (x_bar - mu0) / (SD / sqrt(n))

How it works

Subtract the hypothesized mean from the sample mean, then divide by the standard error (SD divided by the square root of n). Degrees of freedom = n - 1.

Worked example

IQ intervention study

Sample mean = 105, mu0 = 100, SD = 15, n = 36
SE = 15 / sqrt(36) = 2.5
t = (105 - 100) / 2.5 = 2.0
With df = 35, p < 0.05 (two-tailed critical t ~2.03); borderline significant.

Common mistakes to avoid

Using the population standard deviation sigma instead of the sample standard deviation SD in the denominator — the one-sample t-test requires SD estimated from the sample, not a known population parameter.
Forgetting to divide SD by sqrt(n) to form the standard error — computing (x_bar - mu0) / SD (without sqrt(n)) ignores sample size and gives an incorrect result.
Confusing the t-statistic with the p-value — a large |t| means more evidence against the null hypothesis, but you still need to compare it to a t-distribution with n-1 degrees of freedom to get a p-value.

Key terms

Frequently asked questions

How do I interpret the t-statistic?: A t-statistic near 0 means your sample mean is close to the hypothesised value mu0. As |t| grows, the sample mean is further from mu0 relative to sampling variability. Compare t to the critical value from a t-table with n-1 degrees of freedom at your chosen significance level.
What are the degrees of freedom for this test?: For a one-sample t-test, df = n - 1. You use this to look up the critical t-value or compute the p-value. Larger samples have more df and the t-distribution approaches the standard normal.
When should I use a one-sample t-test versus a z-test?: Use a t-test when the population standard deviation is unknown and you estimate it from the sample (the usual case). Use a z-test only when the population standard deviation is known exactly and n is large.