The interval a paired t-test leaves behind
A paired t-test asks whether the mean difference is zero; the confidence interval answers the more useful question: how large is the mean difference, plausibly? The construction is the one-sample interval applied to the differences: mean of the differences, plus and minus t* times the standard error of the differences. Pairing does the heavy lifting before the interval is even built: by subtracting each unit's baseline, the differences usually vary far less than either measurement does, so the interval comes out narrow where an unpaired interval would sprawl.
Scores before and after a course, eight students: differences 6, 7, 5, 13, 14, 5, 6, 15. Mean difference 8.875, SD of differences 4.324, standard error 1.529, t* on 7 degrees of freedom 2.365. The 95% interval: 8.875 ± 3.615 = 5.260 to 12.490. Every value in it is a gain, so at this confidence level the average improvement is positive; the interval also says gains as small as 5.3 or as large as 12.5 points remain compatible with the data.
Interval first, test second
The test and the interval are the same machinery read differently: the 95% interval excludes exactly those null values a two-tailed test at 0.05 would not reject. But the interval shows magnitude and precision, while the p-value compresses both into one tail probability. Report the interval when the size of the change matters (it nearly always does), and note the level: the same data at 99% give a wider interval, the price of the stronger guarantee. The calculator switches levels so you can watch that trade directly.
Building the interval from the two groups' separate SDs as if the samples were independent. In paired data the correlation between measurements is the whole asset; ignoring it throws the asset away and can double the interval's width.
Frequently asked questions
How is this interval calculated?
From the differences: mean difference plus and minus t* (df = n - 1) times SD(differences)/sqrt(n). It is the one-sample t interval applied to the difference scores.
Does the interval include zero in the example?
No: 5.260 to 12.490 lies entirely above zero, matching a paired t-test that rejects zero at the 0.05 level (t = 5.805, p = 0.0007).
Paired interval or independent interval: which is narrower?
Paired, whenever the pairing induces positive correlation, which real before-after data almost always has. The differences strip out person-to-person level variation.