Wilcoxon Signed-Rank

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Nonparametric

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Nonparametric test for paired/matched observations.

Under the strict Wilcoxon null the paired-difference distribution is symmetric about zero; under approximate symmetry this reduces to the familiar claim that the median of the paired differences is zero, and under asymmetry it becomes a stochastic-dominance test of positive vs negative signed-rank mass. The canonical alternative to the paired t-test when the differences are not approximately normally distributed, when the DV is ordinal, or when the sample is small and outliers are a concern.

Worked example

Does a 4-week mindfulness app reduce self-reported stress?

30 users rated stress (0–10) before and after. The paired differences were skewed, so a Wilcoxon signed-rank test is used instead of a paired t-test.

Result

Stress fell significantly (Mdn 6 → 5), W = 48, z = −2.55, p = .008, r = .54.

How you'd report it (APA)

A Wilcoxon signed-rank test showed stress decreased significantly after the programme (Mdn 6 → 5), z = −2.55, p = .008, r = .54.

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

When to use it

  • Pre/post with an ordinal outcome
    A physiotherapy clinic rates 35 patients' pain on a 0–10 NRS scale before and after a 4-week exercise programme.
  • Matched-pairs design with skewed/non-normal DV
    A health-economics study matches 30 patients on a new drug one-to-one with 30 on the standard of care by age, sex, and baseline severity; cost of care (right-skewed) is the DV.
  • Small paired sample where normality is doubtful
    A pilot study tests a new wearable glucose sensor on 15 diabetic volunteers, comparing morning and evening readings.

When NOT to — use instead

  • Two independent groups
    Wilcoxon signed-rank requires PAIRED data. → Mann-Whitney U
  • Three or more related groups
    Wilcoxon handles exactly two related measurements.
  • Binary paired outcomes
    When each subject's 'difference' is gain/no-change/loss on a binary outcome, signed-rank magnitudes are meaningless.
  • Symmetric difference distribution holds (paired-t valid)
    Paired-t is more powerful when its assumptions hold. → Paired Samples t-test

Hypotheses

H₀: The paired-difference distribution is symmetric about zero (under approximate symmetry this is equivalent to the median of the paired differences being zero; under asymmetry it reduces to a stochastic-dominance claim that positive signed-rank mass equals negative signed-rank mass)
Hₐ: The paired-difference distribution is centered away from zero (under approximate symmetry, the median of the paired differences is non-zero)

Parameter tested: location of the paired-difference distribution (D = X₂ − X₁) — interpretable as the median under approximate symmetry

Assumptions (and what to do if they fail)

Differences are approximately symmetric around the medianmedium

Check: Create histogram of differences (Time2 - Time1).

If violated: If differences are skewed, Wilcoxon still valid but tests stochastic dominance (not specifically median).

Zero differences are excluded from analysislow

Check: Count number of zero differences (Time2 - Time1 = 0).

If violated: Many zero differences reduce effective sample size and may indicate measurement insensitivity

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