When to use it
Use this for a paired design — the same subjects measured twice (before/after a treatment, left vs. right, matched pairs) — or a one-sample test comparing a mean to a fixed reference value. If your two groups contain different people, use the two-means sample size calculator instead.
The inputs explained
- Mean change (Δ) — the smallest average before/after difference worth detecting.
- SD of the differences (σd) — the spread of each subject's change, not the raw-score SD. Because before and after are correlated, σd is usually smaller, which is the whole efficiency advantage of pairing. Estimate it from a pilot.
- Significance and power — 0.05 and 80% are standard.
The effect size here is dz = Δ / σd.
Worked example
A training program is expected to raise scores by 5 points, with the SD of individual changes around 10 (dz = 0.5). At α = 0.05 (two-sided) and 80% power, you need about 34 pairs. Compare that with 64 per group (128 total) for the unpaired version of the same effect — pairing is dramatically cheaper when the measurements are correlated.
The formula
A paired t-test is a one-sample test on the differences, so
n = (z1−α/2 + z1−β)² / dz², with dz = Δ / σd
This normal approximation slightly underestimates small samples; the exact t-based value is a little larger, so round up. See the methodology page.
Common mistakes
- Using the raw-score SD instead of the SD of the differences — this inflates the required sample and throws away the benefit of pairing.
- Treating paired data as two independent groups — wrong test and a much larger sample.
- Ignoring dropout — in before/after designs, attrition between measurements is common; recruit extra.