June 7th, 2026
The t-statistic is used in a t-test when deciding if you should support to reject a null hypothesis.
It measures how far a result is from a reference value after adjusting for variability.
It is used when there is a small sample size, or if the population standard deviation is unknown.
There isn’t an exact formula for the t-statistic but they all follow a core structure.
\[t = \frac{estimate-hypothesized\ value}{standard\ error}\] Common t-test
\[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\]
| Symbol | Meaning |
|---|---|
| \(\bar{X}\) | Sample mean |
| \(\mu_0\) | Hypothesized population mean |
| \(s\) | Sample standard deviation |
| \(n\) | Sample size |
\[t = \frac{\bar{x_1} - \bar{x_2}}{\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}}\]
| Symbol | Meaning |
|---|---|
| \(\bar{x_1}\), \(\bar{x_2}\) | Sample means |
| \(s^2_1\), \(s^2_2\) | Sample variances |
| \(n_1\), \(n_2\) | Sample sizes |
\[t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}}\]
| Symbol | Meaning |
|---|---|
| \(\bar{d}\) | Average of the difference between 2 observations |
| \(\mu_d\) | Hypothesized mean difference |
| \(s_d\) | Sample standard deviation of the differences |
| \(n\) | Number of pairs |
Probability density function
\[f(t \mid \nu) = \frac{\Gamma\!\left(\tfrac{\nu+1}{2}\right)} {\sqrt{\nu\pi}\;\Gamma\!\left(\tfrac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-(\nu+1)/2}\]
Decision rule
We reject \(H_0: \mu = \mu_0\) at significance level \(\alpha\) when
\[|t_{\text{obs}}| > t_{\alpha/2,\; n-1}\]
where \(t_{\alpha/2,\; n-1}\) is the upper \(\alpha/2\) critical value of the t-distribution with \(n - 1\) degrees of freedom.
The two-tailed p-value is
\[p = 2\,P\!\left(T_{n-1} \geq |t_{\text{obs}}|\right)\]
Two-sided \(100(1-\alpha)\%\) confidence interval for \(\mu\)
\[\bar{X} \;\pm\; t_{\alpha/2,\; n-1} \cdot \frac{s}{\sqrt{n}}\]
The half-width \(h = t_{\alpha/2,\; n-1} \cdot s/\sqrt{n}\) shrinks as:
A wider CI does not mean the interval is “bad” — it reflects uncertainty in small samples.
