StatReason

Sign Test

Test whether paired data (or a single sample against a reference value) shows a consistent direction of change, using only whether each difference is positive or negative, the lowest-assumption test on this site.

Enter differences directly. To compare a single sample against a reference median instead of matched pairs, subtract the reference from each value yourself before entering it here.

What this answers

This test answers "ignoring how large each change was, is there a consistent direction to the changes at all?" It is the simplest paired test on this site, trading the power of magnitude-aware methods for minimal assumptions: it does not require symmetric differences, an interval scale, or even numeric data, only a way to say which of two directions each observation falls on.

How it is calculated

Every difference of exactly zero is removed first (a documented convention, since a tie at the reference value carries no directional information), leaving n usable differences. The test then asks: if positive and negative directions were equally likely by chance (a fair coin flip for each pair), how surprising is it to see this many, or more, of the less common sign? That question has an exact answer from the binomial distribution with p equal to .5, so this test needs no normal approximation at any sample size.

Worked example

For four positive differences and zero negative differences, n is 4 and the observed minority count is 0. Under a fair coin-flip null, getting all 4 in the same direction (or the equally extreme all-4-negative outcome) has an exact two-sided probability of 2 times 1 over 16, exactly .125, not small enough to reject a fair-coin null at the conventional .05 threshold despite the visually perfect pattern, a useful reminder of how little power this test has with very few pairs.

Assumption audit

Calculated from your data: the count of positive and negative differences, how many exact zeros were dropped, and the exact binomial p-value from the remaining count.
Evidence to review: how many pairs were dropped as zeros; if a large share of your data ties at the reference value, the remaining usable sample may be too small to detect a real effect.
You must verify: that each difference represents an independent, genuinely matched comparison, and that discarding the magnitude of each change (keeping only its direction) still answers the question you actually care about.

What this result does not mean

A significant sign test means the direction of change is consistent more often than chance would predict, not that the typical change is large; this test discards magnitude entirely by design. A non-significant result with a visually consistent pattern, as in the worked example, often just reflects low statistical power with a small sample, not evidence of no effect.

Limitations

Because it uses direction only, the sign test is markedly less powerful than the Wilcoxon Signed-Rank Test whenever magnitude information is available and meaningful; use this test when you specifically cannot or do not want to assume the sizes of the differences are comparable to one another.