Paired Samples t-Test#

The paired samples t-test (also: dependent samples t-test) tests whether the means of two related measurements differ significantly. Typical use cases include before-and-after measurements on the same subjects.

When to Use#

Use the paired t-test when you want to:

Compare two related measurements (e.g., before vs. after)
The dependent variable is metric (continuous)
The differences between paired values are approximately normally distributed
Each observation in one group can be matched to exactly one observation in the other group

Assumptions#

Paired observations (each measurement has a corresponding pair)
Metric scale of the dependent variable
Normal distribution of the differences (apply Shapiro-Wilk test to the differences)
Independence of pairs (the pairs themselves are independent of each other)

Note: The normality assumption refers to the differences of the paired values, not the raw values themselves. If this assumption is violated, the Wilcoxon signed-rank test is the appropriate alternative.

Formula#

The test statistic is based on the differences $d_i = X_{1i} - X_{2i}$ :

t = \frac{\bar{d}}{s_d / \sqrt{n}}

where:

$\bar{d}$ is the mean of the differences
$s_d$ is the standard deviation of the differences
$n$ is the number of pairs

The mean and standard deviation of the differences are calculated as:

\bar{d} = \frac{1}{n} \sum_{i=1}^{n} d_i

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

The test statistic follows a t-distribution with $df = n - 1$ degrees of freedom.

Example#

Practical Example: Training Effect on Endurance

A sports physician wants to investigate whether an 8-week endurance training program improves maximum oxygen uptake (VO₂max). They measure VO₂max in 25 participants before and after the training.

Measurement 1 (Before): VO₂max before the training program
Measurement 2 (After): VO₂max after 8 weeks of training

Since each person is measured twice, the data are paired. The paired t-test checks whether the mean difference in VO₂max values differs significantly from zero.

Effect Size#

Cohen's $d_z$ as a measure of effect size for paired designs:

d_z = \frac{\bar{d}}{s_d}

Effect Size	Cohen's d_z
Small	0.2
Medium	0.5
Large	0.8

Tip: Cohen's $d_z$ refers to the standardized mean difference. Alternatively, Cohen's $d_{av}$ can be calculated, which uses the average standard deviation of the two measurement points.