Paired t-Test

1 min read Updated Fri Apr 24 2026 03:19:45 GMT+0000 (Coordinated Universal Time)

Used when two related measurements are taken on the same sample population. Evaluates whether the mean difference between the paired observations is significantly different.

Let each pair have values $(X_1, X_2)$ . Suppose $d_i = X_{1i} - X_{2i}$ .

\bar{d} = \frac{1}{n} \sum_{i=1}^{n} d_i \quad \text{and} \quad s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n - 1}}

Hypotheses Setup

H_0 : \mu_d = d_0 \,\, \text{(no mean difference)}

H_1 : \mu_d < d_0 \text{ or } \mu_d > d_0 \text{ or } \mu_d \ne d_0

Test Statistic

Under $H_0$ with $\nu = n - 1$ :

t = \frac{\bar{d} - d_0}{s_d / \sqrt{n}} \sim t_\nu