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Non-parametric

Kruskal-Wallis H Test — Three Groups Non-parametric ANOVA

Non-parametric alternative to one-way ANOVA. Rank all observations, compute H statistic, compare to chi-square critical value.

📊 H(2) ≥ χ²(2,.05) = 5.991
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🔢 4 · Non-parametric Tests

4a · Mann-Whitney U Test
Non-parametric
Research Question: Do satisfaction scores differ between two departments (n₁=5, n₂=5)? Data is ordinal/skewed.
Dept A68249
Dept B751031
U Statistic
U 1 = n 1 n 2 + n 1 ( n 1 + 1 ) 2 − R 1 , U = min ( U 1 , U 2 )
  1. 1
    Rank all 10 values combined:
    Value12345678910
    Rank12345678910
    GroupBABABABAAB
  2. 2
    R 1 = 2 + 4 + 6 + 8 + 9 = 29 ( sum of ranks for Dept A ) U 1 = 5 × 5 + 5 × 6 2 − 29 = 25 + 15 − 29 = 11 U 2 = n 1 n 2 − U 1 = 25 − 11 = 14 , U = min ( 11 , 14 ) = 11
  3. 3
    Normal approximation (n>10 each; shown for demonstration):
    z = U − n 1 n 2 / 2 n 1 n 2 ( n 1 + n 2 + 1 ) / 12 = 11 − 12.5 25 × 11 / 12 = − 1.5 4.787 = − 0.313
  4. 4
    Effect size r:
    r = | z | N = 0.313 10 = 0.099 ( small )
11
U
.754
p (exact)
0.099
r
⚪ Fail to Reject H₀ — No significant difference between departments, U=11, p=.754.
APA-7
A Mann-Whitney U test revealed no significant difference in satisfaction scores between Department A (Mdn=6) and Department B (Mdn=6), U=11, z=−0.31, p=.754, r=.10.
4b · Kruskal-Wallis H Test
Non-parametric
Research Question: Do pain levels differ across three clinics (non-normal data)?
Clinic AClinic BClinic C
3,5,4,2,67,9,8,10,65,4,6,3,5
Kruskal-Wallis H
H = 12 N ( N + 1 ) ∑ j = 1 k R j 2 n j − 3 ( N + 1 )
  1. 1
    Rank all 15 values, sum ranks per group:
    R A = 2 + 6.5 + 4 + 1 + 9 = 22.5 , R B = 11 + 14 + 13 + 15 + 9 = 62 , R C = 6.5 + 4 + 9 + 2 + 6.5 = 28
    (ties at value 6 receive average rank (9+10)/2=9.5; value 5: (5+6+7)/3=6)
  2. 2
    H = 12 15 × 16 ( 22.5 2 5 + 62 2 5 + 28 2 5 ) − 3 ( 16 ) = 12 240 ( 101.25 + 768.8 + 156.8 ) − 48 = 0.05 ( 1026.85 ) − 48 = 51.34 − 48 = 3.34
  3. 3
    Compare to χ² distribution with df=k−1=2: χ²(2,0.05)=5.99. Since H=3.34 < 5.99 → fail to reject H₀.
3.34
H
2
df
.188
p
⚪ Fail to Reject H₀ — Pain levels do not differ significantly across clinics, H(2)=3.34, p=.188.
APA-7
A Kruskal-Wallis test indicated no significant difference in pain levels across the three clinics, H(2) = 3.34, p = .188, η² = .10.

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