Non-Parametric Tests

Non-parametric tests are alternatives to t-tests and ANOVA when:

• The data is not normally distributed
• The data is ordinal (ranks, ratings)
• There are outliers that violate assumptions
• Sample sizes are small
• You want a “distribution-free” version of a test

Mann–Whitney U Test

Independent two-group comparison. Alternative to the independent t-test.

Used when:

• Two groups are independent
• Data is not normal
• Data is ordinal or skewed
• Sample sizes are small

Example

from scipy import stats​group1 = [12, 14, 15, 16, 18]group2 = [20, 22, 19, 23, 21]​stat, p = stats.mannwhitneyu(group1, group2)print(stat, p)

Interpretation

• p < 0.05 → groups differ significantly
• Direction: compare medians or inspect raw data

With “two-sided” alternative

stat, p = stats.mannwhitneyu(group1, group2, alternative='two-sided')

Wilcoxon Signed-Rank Test

Paired / matched samples. Alternative to the paired t-test.

Used when:

• Same subjects measured twice
• Non-normal data
• Median difference instead of mean difference

Example

before = [10, 12, 9, 14, 11]after  = [13, 15, 10, 17, 12]​stat, p = stats.wilcoxon(before, after)print(stat, p)

Interpretation

• p < 0.05 → median difference is significant
• Use for before/after experiments with skewed or ordinal data

Note: requires paired data and no zero-differences for the default behaviour.

Kruskal–Wallis Test

Three or more independent groups. Alternative to one-way ANOVA.

Used when:

• 3+ independent groups
• Data is not normal
• Data is ordinal
• Variances differ

Example

group_a = [5, 6, 7, 5, 6]group_b = [8, 9, 7, 8, 9]group_c = [4, 5, 3, 4, 5]​stat, p = stats.kruskal(group_a, group_b, group_c)print(stat, p)

Interpretation

• p < 0.05 → at least one group differs
• Like ANOVA, it does not tell which groups differ

Post-hoc Testing

SciPy does not provide post-hoc pairwise non-parametric tests, but you can use:

• Dunn’s Test (via scikit-posthocs)
• Pairwise Mann–Whitney with Bonferroni correction

Example (Dunn test):

pip install scikit-posthocs​

import scikit_posthocs as spimport pandas as pd​data = pd.DataFrame({    'value': group_a + group_b + group_c,    'group': ['A']*5 + ['B']*5 + ['C']*5})​sp.posthoc_dunn(data, val_col='value', group_col='group', p_adjust='bonferroni')

Friedman Test

Three or more repeated-measures groups. Alternative to repeated-measures ANOVA.

Used when:

• Same subjects measured under 3+ conditions
• Non-normal repeated measurements

Example

condition1 = [5, 6, 7, 8]condition2 = [6, 7, 8, 9]condition3 = [7, 8, 9, 10]​stat, p = stats.friedmanchisquare(condition1, condition2, condition3)print(stat, p)

Interpretation

• p < 0.05 → at least one repeated condition differs

Useful for experiments involving multiple tests on the same participants.

Sign Test (Not in SciPy)

The Sign Test is a simple non-parametric test for paired data.

SciPy does not include it, but you can write your own:

def sign_test(x, y):    import numpy as np    diff = np.array(y) - np.array(x)    pos = np.sum(diff > 0)    neg = np.sum(diff < 0)    # Two-sided binomial test    return stats.binomtest(pos, pos+neg, p=0.5)​result = sign_test(before, after)print(result.statistic, result.pvalue)

Use only when Wilcoxon assumptions fail (e.g., many zeros).

Choosing Non-Parametric Tests