Paired-Samples t-Test

paired-t-test

dependent-samples

cohen-dz

pre-post

Comparing two dependent measurements per unit on a continuous outcome

Published

April 17, 2026

Research question

Use the paired t-test when each unit contributes two related measurements and you want to know if their means differ. Biomedical examples: (1) does a twelve-week supervised exercise programme reduce systolic blood pressure (pre vs. post within the same patient)?; (2) do matched tumour-normal tissue pairs differ in the expression of a candidate biomarker measured by qPCR?

Assumptions

Assumption	How to verify in R
Pairs are naturally linked (same unit, matched case-control, left-right eye)	design
The differences are approximately normal	`shapiro_test(d_i)` on the paired differences
No extreme outliers among the differences	boxplot of differences

Normality applies to the differences, not to the two raw distributions.

Hypotheses

Let \(D_i = X_i - Y_i\) be the paired difference. Then

\[H_0: \mu_D = 0 \qquad H_1: \mu_D \ne 0\]

R code

library(tidyverse); library(rstatix); library(effectsize); library(ggstatsplot)
set.seed(42)

# 40 patients; pre- and post-exercise SBP in mmHg
ex <- tibble(
  id  = 1:40,
  pre  = round(rnorm(40, 142, 12)),
  post = NA_real_
) |>
  mutate(post = round(pre - rnorm(40, 6, 5)))

long <- ex |> pivot_longer(c(pre, post), names_to = "time", values_to = "sbp") |>
  mutate(time = factor(time, levels = c("pre", "post")))

# Differences + assumption check
diffs <- ex$post - ex$pre
shapiro_test(diffs)

# Paired t-test
long |> t_test(sbp ~ time, paired = TRUE, detailed = TRUE)

# Effect size: Cohen's d_z (mean of differences / SD of differences)
effectsize::cohens_d(ex$post, ex$pre, paired = TRUE)

ggwithinstats(data = long, x = time, y = sbp, type = "parametric",
              xlab = "Time", ylab = "Systolic BP (mmHg)",
              title = "Pre vs. post exercise programme")

Interpreting the output

The mean difference (post minus pre) is about \(-6\) mmHg with a 95 % CI roughly \([-7.6, -4.3]\). The paired \(t\) is approximately \(-7.6\) on \(39\) df, \(p < .001\). The effect size \(d_z \approx 1.20\) indicates a very large within-subject change.

Effect size

Cohen’s \(d_z\) = mean(differences) / SD(differences). Conventional thresholds: small 0.20, medium 0.50, large 0.80.

Reporting (APA 7)

The twelve-week programme reduced systolic blood pressure by 6.0 mmHg on average (paired t(39) = -7.59, p < .001, d_z = 1.20, 95 % CI for the mean difference [-7.6, -4.3] mmHg).

Common pitfalls

Running an independent t-test on paired data inflates the standard error and wastes power.
Reporting two-sample Cohen’s \(d\) instead of \(d_z\) for paired designs gives a misleadingly small effect.
Pairs with missing values in one measurement must be dropped or imputed; t.test() removes them silently.

Parametric vs. non-parametric alternative

If the paired differences are non-normal, use the Wilcoxon signed-rank test. If only the direction of each difference is meaningful, use the sign test.