Sign Test (paired, direction-only)

Verified

Nonparametric

Independently verified. Every statistic this test reports has been re-derived against an independent reference — never the library the pipeline itself calls — the rendered output was read back in a browser, and the result is locked with a committed regression suite.

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Tests whether the median paired difference is zero by counting the proportion of positive vs negative paired differences and comparing to the binomial(n_nonzero, 0.5) null.

Discards the magnitude of differences entirely — only their sign is used. The most assumption-light paired test available: requires only that paired observations are independent across pairs and the null is symmetric in direction. Apt when the difference distribution is wildly non-normal AND the Wilcoxon signed-rank's symmetry assumption is also untenable.

Worked example

Do more people prefer version B over version A?

24 people tried both versions. The sign test counts only who preferred which (direction), ignoring by how much — useful when magnitudes aren't meaningful.

Result

18 of 24 preferred version B and 6 preferred A; the sign test gives p = .023 — a significant majority for B.

How you'd report it (APA)

A sign test showed a significant preference for version B (18 of 24 favouring B), p = .023.

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

  • Paired ordinal or severely skewed continuous DV
    30 patients rate symptom severity on a 5-point Likert scale before and after a single dose of an experimental analgesic.
  • Matched-pair directional outcome (preference / improvement)
    60 wine-tasters compare two vintages and report which they prefer.

When NOT to — use instead

Hypotheses

H₀: P(v1 > v2 | v1 ≠ v2) = 0.5 — among non-tied paired observations, v1 is as likely to exceed v2 as the other way around. This is the operative parameter the sign test actually evaluates; a median-shift interpretation only follows as a corollary under continuity and a symmetric paired-difference distribution, which is exactly the assumption the sign test is designed not to require.
Hₐ: P(v1 > v2 | v1 ≠ v2) ≠ 0.5 — non-zero paired differences favour one direction more often than would be expected under the null.

Parameter tested: P(v1 > v2) — probability that measurement 1 exceeds measurement 2 on a randomly chosen pair

Assumptions (and what to do if they fail)

Effective sample size (non-zero paired differences)medium

Check: Count pairs with v1 ≠ v2.

If violated: Too few non-zero pairs leave the sign test with almost no power.

Tie burden on the paired differenceshigh

Check: tie_ratio ≤ 10% → negligible; 10–25% → declare the power loss; 25–50% → the test is running on a remnant and inference is weak; > 50% → the paired-difference distribution is zero-inflated and the sign

If violated: Many tied pairs erode the effective sample size and may hide a ceiling/floor effect.

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