Paired T-Test Calculator
Test whether a before-and-after change is real — the paired t-test, which works on the differences and is far more powerful than comparing two groups.
When each observation is measured twice, the pairing removes between-subject variation entirely.
How the paired t-test calculator works
When each observation is measured twice, the pairing removes between-subject variation entirely. Compute each difference, then run a one-sample t-test on those differences against zero: t = d̄ ÷ (s_d / √n).
That is why paired designs are so much more sensitive. The same data analysed as two independent groups often fails to reach significance, because it is fighting variation the pairing had already eliminated.
Formula: t = d̄ / (s_d / √n), df = n − 1
Worked examples
| Inputs | t statistic | Note |
|---|---|---|
| Eight before-and-after pairs | 8.282512 | a clear drop, strongly significant |
| No change | 0 | the changes cancel |
| A small consistent shift | 5 | small but consistent — highly significant |
FAQFrequently asked questions
When is a paired test right?
When the same subjects, items or matched pairs are measured twice. Before-and-after, left-and-right, treatment-and-control on matched pairs.
Why is it more powerful?
Because it removes all the variation between subjects. Only the change matters, so a small consistent effect stands out clearly.
What if I analysed it as two independent groups?
You would throw away the pairing and usually fail to detect a real effect. It is one of the most common analysis errors.
What does Cohen's dz mean?
The mean change divided by the standard deviation of the changes — the effect size for a paired design. It is not comparable to the two-group d.
What if the differences are badly skewed?
Use the Wilcoxon signed-rank test, or the sign test, which make no distributional assumption.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.