T-Score Calculator (t Statistic and p-Value)
Calculate the t-score of a sample mean, its degrees of freedom and p-value, the critical t value for your confidence level and a confidence interval.
A t-score (t statistic) measures how far a sample mean is from a hypothesized population mean, in units of standard error. It is the heart of the one-sample t-test, used when the population standard deviation is unknown and has to be estimated from the sample. Enter your sample summary to get the t-score, degrees of freedom, p-value, critical value, a reject or keep decision and a confidence interval.
How to use this tool
- Enter the sample mean x̄ and the population mean μ₀ you are testing against.
- Enter the sample standard deviation s and the sample size n.
- Choose a confidence level (95% means α = 0.05) and the test direction.
- Compare the t-score with the critical value, or the p-value with α.
Formula
t = (x̄ − μ₀) / (s / √n), with df = n − 1
s / √n is the standard error of the mean. The p-value is the area under Student’s t distribution with n − 1 degrees of freedom beyond your t-score (both tails for a two-tailed test). The confidence interval is x̄ ± t* × s / √n, where t* is the two-sided critical value.
Example
A sample of 25 bulbs has a mean life of 105 hours with s = 15, and the maker claims 100 hours. The standard error is 15 / √25 = 3, so t = (105 − 100) / 3 = 1.6667 with 24 degrees of freedom. The two-tailed critical value at 95% is 2.0639, so the result is not significant (p ≈ 0.1086), and the 95% confidence interval is 98.8083 to 111.1917 hours.
Frequently asked questions
What is the difference between a t-score and a z-score?
A z-score uses a known population standard deviation and the normal distribution. A t-score uses the sample standard deviation, so it follows the wider t distribution, which approaches the normal as n grows.
What are degrees of freedom here?
For a one-sample t-test, df = n − 1, because one degree of freedom is used up estimating the mean.
Is this the T-score used in psychology tests?
No. That standardized T-score rescales a z-score to mean 50 and SD 10: T = 50 + 10z. Compute z with the z-score calculator and apply that formula.
Should I use a one- or two-tailed test?
Use two-tailed unless you decided in advance that only one direction matters. A one-tailed test has a smaller critical value but ignores effects in the other direction.
Last updated: 2026-10-30