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Non-parametric

Wilcoxon Signed-Rank Test — Paired Non-parametric

Non-parametric alternative to the paired t-test. Before/after coping scores for 9 employees after a mindfulness program, using the exact signed-rank distribution (n<50, no ties).

📊 W = 2, n = 9, p = .012 (exact)
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🔢 30 · Wilcoxon Signed-Rank Test — Paired Non-parametric

Before vs. After an 8-Week Mindfulness Program — Coping Scores
Non-parametric
Research Question: Did coping scores (0–100 scale) change for the same 9 employees after an 8-week mindfulness program? The scores are skewed, so a paired t-test's normality assumption is doubtful — the Wilcoxon signed-rank test is the non-parametric alternative for paired data.
Employee Before After Diff (After−Before)
16066+6
25862+4
36574+9
47270−2
55562+7
66366+3
75964+5
86876+8
96162+1
Sum of ranks for positive and negative differences
W + = ∑ rank ( | d i | )   for   d i > 0 , W − = ∑ rank ( | d i | )   for   d i < 0 , W = min ( W + , W − )
  1. 1
    Rank the absolute differences |d| from smallest to largest (no ties here):
    | d | :   1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9   →   ranks   1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9   (each unique)
  2. 2
    Sum the ranks separately for positive and negative differences:
    W + = 6 + 4 + 9 + 7 + 3 + 5 + 8 + 1 = 43 , W − = 2 , W = min ( 43 , 2 ) = 2
    Only employee #4 got worse (diff = −2, rank 2); every other employee improved.
  3. 3
    Exact test (n=9 ≤ 49, no ties, no zero differences): MindStat and R's wilcox.test() both use the exact distribution of W under H₀ whenever n < 50 with no ties — the normal approximation is only a fallback for larger or tied samples.
    p exact ( W = 2 ,   n = 9 ) = .0117 ( two-tailed )
  4. 4
    Decision rule: p = .012 < .05 → reject H₀ → coping scores changed significantly after the program.
2
W
9
n
.012
p (exact)
🔴 Significant — W = 2, n = 9, p = .012 (exact, two-tailed). Coping scores increased after the program.
APA-7
A Wilcoxon signed-rank test indicated that coping scores were significantly higher after the mindfulness program than before, W = 2, p = .012 (exact, two-tailed).

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