Non-Parametric Tests

What Are Non-Parametric Tests?

Non-parametric tests (distribution-free tests) don’t assume the data follows a specific distribution like normal.

Parametric Test	Non-Parametric Alternative
One-sample t-test	Wilcoxon signed-rank
Independent t-test	Mann-Whitney U
Paired t-test	Wilcoxon signed-rank (paired)
One-way ANOVA	Kruskal-Wallis
Pearson correlation	Spearman correlation

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When to Use Non-Parametric Tests

Advantages

• No distribution assumptions
• Work with ordinal data
• Robust to outliers
• Valid for small samples

Disadvantages

• Less statistical power than parametric tests
• Harder to generalize
• Results may be harder to interpret

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1. Mann-Whitney U Test

Alternative to the independent two-sample t-test. Tests if two groups come from the same population.

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2. Wilcoxon Signed-Rank Test

Alternative to paired t-test. Tests if the median difference is zero.

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3. Kruskal-Wallis Test

Alternative to one-way ANOVA. Compares three or more independent groups.

If H is significant, use post-hoc tests (like Dunn’s test) to find which groups differ.

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4. Spearman Rank Correlation

Non-parametric alternative to Pearson correlation. Measures monotonic relationship.

Use when:

• Data is ordinal
• Relationship is monotonic but not linear
• Outliers present

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Choosing the Right Test

Research Question	Parametric	Non-Parametric
One sample vs value	t-test	Wilcoxon signed-rank
Two independent groups	t-test	Mann-Whitney U
Two paired measurements	Paired t	Wilcoxon signed-rank
3+ independent groups	ANOVA	Kruskal-Wallis
Linear relationship	Pearson r	Spearman rₛ

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Handling Ties

When values are tied (equal), assign the average rank to all tied values.

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Summary

In this lesson, you learned:

• Non-parametric tests don’t require distributional assumptions
• Mann-Whitney U: Compares two independent groups (t-test alternative)
• Wilcoxon signed-rank: Compares paired observations (paired t alternative)
• Kruskal-Wallis: Compares 3+ groups (ANOVA alternative)
• Spearman correlation: Non-parametric correlation
• Based on ranks rather than actual values
• Use when assumptions fail or data is ordinal

Practice Problems

1. When would you choose Mann-Whitney U over an independent t-test?

2. Group A scores: 12, 15, 18 Group B scores: 22, 25, 28 Calculate the sum of ranks for each group.

3. Five subjects have before/after differences: -3, +5, -2, -4, -1 Calculate W+ and W- for the Wilcoxon signed-rank test.

4. A Kruskal-Wallis test yields H = 8.5 with 3 groups (df=2). Critical χ² = 5.99 at α = 0.05. What’s the conclusion?

Click to see answers

1. Use Mann-Whitney when:

• Sample sizes are small (< 30)
• Data is not normally distributed
• Data is ordinal (rankings, scales)
• Outliers are present
• Variances are very unequal

2. Combined and ranked: 12(1), 15(2), 18(3), 22(4), 25(5), 28(6)

• Group A sum: 1 + 2 + 3 = 6
• Group B sum: 4 + 5 + 6 = 15

3. Absolute differences and ranks:

• |-3| = 3 → rank 3 (negative)
• |+5| = 5 → rank 5 (positive)
• |-2| = 2 → rank 2 (negative)
• |-4| = 4 → rank 4 (negative)
• |-1| = 1 → rank 1 (negative)

W+ = 5 (only the +5) W- = 1 + 2 + 3 + 4 = 10

4. H = 8.5 > 5.99 (critical value) Reject H₀. There is significant evidence that at least one group differs from the others.

Next Steps

Continue with advanced statistics:

• Effect Size and Power - Beyond p-values
• Bayesian Statistics Introduction - Alternative framework
• T-Test Calculator - Practice calculations