Non-Parametric Tests

SciPy - Statistical Testing

2 min read

Published Nov 17 2025

PythonSciPyStatistics

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 sp

import 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

Goal	Parametric Equivalent	Non-Parametric Version
Compare 2 independent groups	Independent t-test	Mann–Whitney U
Compare 2 related groups	Paired t-test	Wilcoxon
Compare 3+ independent groups	ANOVA	Kruskal–Wallis
Compare 3+ repeated measures	Repeated-measures ANOVA	Friedman
Before/After with weak assumptions	Paired t-test	Sign Test