The simplest test in statistics
The sign test asks one brutally simple question of paired data: ignoring size completely, do the improvements outnumber the declines more than a fair coin would allow? Each pair is a coin flip under the null, so the count of positives follows an exact binomial distribution, and the p-value this calculator reports is exact, computed from that distribution, not approximated by any curve.
The revision week again: 8 improvements, 0 declines, no ties. Eight heads in eight flips: exact two-tailed p = 0.0078. Significant, and note the ladder of the three paired tests on identical data: sign test 0.0078, Wilcoxon 0.014 (approximation aside), paired t 0.0007. Each step up uses more information (direction, then rank, then full magnitude) and, when its assumptions hold, sees more.
Where the sign test earns its place
- When only direction is trustworthy: "better or worse", with no honest scale.
- When difference distributions are so wild that ranks mislead too.
- As a five-second sanity check before the heavier machinery.
Expecting it to find what the paired t-test found. Discarding magnitudes is a real cost: with modest, consistent gains, the sign test can shrug where the t-test is certain.
Frequently asked questions
How are ties handled?
Tied pairs (no change) are dropped and the test runs on the remaining pairs, reported openly on the results card.
Is the p-value really exact?
Yes, it is summed directly from the binomial distribution, so it is trustworthy at any sample size, including very small ones.