# How to do a Wilcoxon signed-rank test (in Python, using SciPy)

## Task

Assume we a sample of data, $x_1, x_2, x_3, \ldots x_k$ and either the sample size is small or the population is not normally distributed. But we still want to perform tests that compare the sample median to a hypothesized value (equal, greater, or less). One method is the Wilcoxon Signed-Rank Test.

Related tasks:

- How to do a Kruskal-Wallis test
- How to do a Wilcoxon rank-sum test
- How to do a Wilcoxon signed-rank test for matched pairs

## Solution

We’re going to use fake data for illustrative purposes, but you can replace our fake data with your real data. Say our sample, $x_1, x_2, x_3, \ldots x_k$, has median $m$.

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import numpy as np
# Replace the next line with your data
sample = np.array([19, 4, 23, 16, 1, 8, 30, 25, 13])

We choose a value, $0 \le \alpha \le 1$, as the Type I Error Rate. We’ll let $\alpha$ be 0.05. In the examples below, we will be comparing the median $m$ to a hypothesized value of $a=10$, but you can use any value for $a$.

### Two-tailed test

To test the null hypothesis $H_0: m=a$, we use a two-tailed test:

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from scipy import stats
from scipy.stats import wilcoxon
a = 10 # or your chosen value for comparison
wilcoxon(sample - a)

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WilcoxonResult(statistic=10.0, pvalue=0.1640625)

Our p-value, 0.1640625, is greater than $\alpha=0.05$, so we do not have sufficient evidence to reject the null hypothesis. We may continue to assume the population median is equal to 10.

### Right-tailed test

To test the null hypothesis $H_0: m\ge a$, we use a right-tailed test:

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wilcoxon(sample - a, alternative = 'less')

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WilcoxonResult(statistic=35.0, pvalue=0.935546875)

Our p-value, 0.935546875, is greater than $\alpha=0.05$, so we do not have sufficient evidence to reject the null hypothesis. We may continue to assume the population median is less than (or equal to) 10.

### Left-tailed test

To test the null hypothesis $H_0: m\le a$, we use a left-tailed test:

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wilcoxon(sample - a, alternative = 'greater')

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WilcoxonResult(statistic=35.0, pvalue=0.08203125)

Our p-value, 0.08203125, is greater than $\alpha$, so we do not have sufficient evidence to reject the null hypothesis. We may continue to assume the population median is greater than (or equal to) 10.

Content last modified on 24 July 2023.

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Contributed by Elizabeth Czarniak (CZARNIA_ELIZ@bentley.edu)