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Paired two samples t-test

Tests whether the mean of the within-pair differences is zero. The right test for before/after or two-methods-on-same-subjects designs.

What is Paired two samples t-test?

The paired t-test takes the row-wise difference dᵢ = bᵢ − aᵢ and runs a one-sample t-test against μ₀ = 0. By collapsing each pair into a single difference, it removes between-subject variability — typically much larger than the within-subject treatment effect — and so has dramatically more power than a two-sample t-test on the same data.

The non-parametric counterpart is the Wilcoxon signed-rank test, which is rank-based on the absolute differences. Use signed-rank when the differences are heavily skewed or the sample is very small.

A common mistake: applying a two-sample t-test to paired data. This treats the two columns as independent and discards the pairing structure entirely. Always pair when the design pairs.

When should I use Paired two samples t-test?

Before / after measurements on the same subject.
Two methods (e.g. two devices, two raters) on the same samples.
Crossover trial with two conditions in the same patients (after washout).

What data does it need?

Two numeric columns of equal length, row-paired.

What does it report?

t, df = n − 1, two-sided p-value, mean difference + 95% CI on the difference.

What does it assume?

Independent pairs.
The within-pair differences are approximately normal (or n ≥ 30).

Formula

dᵢ = bᵢ − aᵢ;  t = d̄ / (s_d / √n);  df = n − 1

How do I interpret the result?

Check the distribution of the differences (not the raw columns) for normality. A skewed dᵢ is the case where Wilcoxon signed-rank is a better choice.

Pair the test with a Bland-Altman plot when the design is method-comparison — the t-test tells you about a constant bias but the BA plot reveals proportional bias or heteroscedasticity.