Module 3: Non-Parametric Methods
Essential Exam Notes
Core Concepts
1. Key Characteristics
• Advantages:
– No normality assumption required
– Robust to outliers
– Works with ordinal data
• Disadvantages:
– Less powerful than parametric tests when assumptions are met
– Limited for complex relationships
Frequently Tested Tests
1. Sign Test (2023 Q10, 2022 Q17)
• Purpose: Test median for single sample or paired data
• Steps:
1. Calculate differences from hypothesized median (θ0 )
2. Count positive signs (S + ) and negative signs (S − )
3. Test statistic: S = min(S + , S − )
4. Compare to binomial distribution Bin(n, 0.5)
• Example (2022 Q17): Rainfall median testing
2. Wilcoxon Signed-Rank Test
• Improvement over: Sign test (uses magnitude)
• Steps:
1. Rank absolute differences
2. Sum ranks for positive (W + ) and negative (W − ) differences
3. Test statistic: W = min(W + , W − )
1
3. Mann-Whitney U Test (2023 Q17)
• Purpose: Compare two independent samples
• Steps:
1. Combine and rank all observations
2. Calculate rank sums R1 and R2
3. Test statistics:
n1 (n1 + 1)
U1 = n1 n2 + − R1
2
U2 = n1 n2 − U1
4. Use min(U1 , U2 )
4. Kruskal-Wallis Test (2024 Q9, 2022 Q10)
• Purpose: Non-parametric ANOVA for ≥ 2 groups
• Test statistic:
k
X R2
12 i
H= − 3(N + 1)
N (N + 1) i=1 ni
where Ri = sum of ranks in group i
• Adjustment for ties:
H
Hadj = P
(t3j −tj )
1− N 3 −N
PYQ Problem Framework
1. Hypothesis Setup
• Always state:
– H0 : No difference/distribution as specified
– H1 : Difference exists (one/two-tailed)
2. Test Selection Guide
Scenario Appropriate Test
Single sample median Sign test
Paired samples Wilcoxon signed-rank
Two independent samples Mann-Whitney U
≥ 3 independent samples Kruskal-Wallis
Randomness Runs test
2
Critical Values Reference
• Small samples (n ≤ 20): Use exact tables
• Large samples: Normal approximation:
n1 n2
U−
– Mann-Whitney: Z = q 2
n1 n2 (n1 +n2 +1)
12
– Kruskal-Wallis: χ2 approximation (df = k-1)
Common Exam Mistakes
• Using parametric critical values
• Incorrect handling of ties
• Wrong test selection (e.g., Mann-Whitney for paired data)
• Forgetting continuity correction in normal approximations
