Sign Test Calculator — Exact Paired Analysis

The Sign Test. Paired changes can carry trustworthy directions even when their magnitudes do not. The sign test keeps only plus or minus and asks whether those directions balance. Keep every pair intact and subtract sample two from sample one. Positive means sample one is larger; negative means sample two is larger. An exactly zero difference has no direction. This page drops every tie before inference, while preserving the original pair count for reporting. Count positives and negatives after ties leave. Their sum is informative n, the effective sample size used by the sign test. Under the directional null, positive and negative are equally likely among informative pairs. The positive count follows a binomial distribution with probability one half. This calculator evaluates those finite binomial tail probabilities directly at every n. It uses no normal approximation and no continuity correction. For a two-sided test, the page finds the smaller inclusive tail, doubles it, and caps the result at one. That is its exact convention.

Pairing is the design. Row i in sample one and row i in sample two must describe the same person or deliberately matched unit. Different pairs must remain independent. Households, clinics, or one person contributing several pairs require an analysis that represents that dependence. Only direction needs a meaningful order. The test does not require normality, equal spacing, or a symmetric distribution of paired differences. That robustness has a price: a tiny improvement and a huge improvement both contribute one positive sign. Their magnitudes disappear. The operational null is equal positive and negative probabilities among non-ties. Calling it a zero-median test needs extra continuity and point-mass care. Tail direction follows sample one minus sample two. Right-tailed favors positives, left-tailed favors negatives, and two-sided allows either imbalance. Exact calculation does not make small samples powerful. Discrete p-values move in coarse steps, so the attained error rate may sit below nominal alpha.

Now compare nine matched measurements under two conditions. Keep sample one minus sample two, a two-sided alternative, and alpha point zero five fixed. The nine paired differences contain seven positives, two negatives, and no ties. All nine pairs are informative for the sign count. Under the null, imagine nine independent fair signs. Every positive-count pattern lies on a binomial reference with five hundred twelve equally likely sign sequences. The smaller side is two. Counts from zero through two contain forty-six of those five hundred twelve sequences, giving tail probability zero point zero eight nine eight four four. Double that smaller inclusive tail. The exact two-sided p-value is zero point one seven nine six eight eight, above alpha point zero five. Fail to reject equal directional probabilities. This does not prove no change, perfect balance, or a zero median in the population. Seven ninths, or zero point seven seven eight, were positive among non-ties. Wilcoxon rejected on these same pairs because magnitudes added information; that is not a contradiction.

With five informative pairs, even five positives and no negatives give two-sided p zero point zero six two five. Alpha point zero five is unreachable. Many tied pairs shrink informative n and can make the nonzero subset selective. Always report original pairs, ties dropped, and directions retained. Unequal sample lengths fail validation. If every paired difference is zero, no direction remains, so the calculator stops instead of inventing evidence. Swap the sample order and seven positives become two. The positive share and one-sided tails reverse, while the two-sided p-value stays unchanged. Lock pair order, subtraction direction, tail, and alpha before results. Choosing a favorable tail or test afterward invalidates the evidence. Report pair design, sign orientation, positive, negative, and tied counts, informative n, exact convention, p, alpha, and guarded conclusion. This page provides no change magnitude or confidence interval. Reduce valid paired changes to honest directions, count exact binomial evidence, and run the Sign Test calculator free at Distri Scope dot com.