Paired t-Test

1 min read Last updated Sat Jun 27 2026 08:46:52 GMT+0000 (Coordinated Universal Time)

Used when 2 related measurements are taken on the same subjects. Evaluates whether the mean difference between paired observations is significantly different from zero.

Use when: observations are naturally paired (before/after, matched subjects, repeated measures). Use an independent 2-sample $t$ -test instead when samples are unrelated.

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

H_0 : \mu_d = d_0 \quad \text{(no mean difference, typically } d_0 = 0\text{)}

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

Decision Rule

Reject $H_0$ if:

Left-tail: $t < -t_{\nu, \alpha}$
Right-tail: $t > t_{\nu, \alpha}$
Two-tail: $|t| > t_{\nu, \alpha/2}$

Examples:

Blood pressure measured before and after a drug treatment on the same patients
Exam scores for the same students before and after a tutoring program

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