Wilcoxon signed-rank test in SciPy - Step-by-Step Execution

Concept Flow - Wilcoxon signed-rank test

Start with paired data

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Calculate differences

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Remove zero differences

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Rank absolute differences

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Assign signs to ranks

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Sum positive and negative ranks

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Calculate test statistic W

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Compare W to distribution

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Output p-value and decision

The test compares paired samples by ranking their differences and checking if median difference is zero.

Execution Sample

SciPy

from scipy.stats import wilcoxon
x = [20, 22, 19, 24, 30]
y = [21, 21, 20, 23, 29]
stat, p = wilcoxon(x, y)
print(stat, p)

Runs Wilcoxon signed-rank test on two paired samples and prints test statistic and p-value.

Execution Table

Step	Action	Calculation/Condition	Result
1	Calculate differences	diff = x - y	[ -1, 1, -1, 1, 1 ]
2	Remove zeros	No zeros to remove	[ -1, 1, -1, 1, 1 ]
3	Absolute differences	abs_diff = [1, 1, 1, 1, 1]	[1, 1, 1, 1, 1]
4	Rank absolute differences	All equal, ranks = [3.0,3.0,3.0,3.0,3.0]	[3.0, 3.0, 3.0, 3.0, 3.0]
5	Assign signs to ranks	Signed ranks = [-3.0, +3.0, -3.0, +3.0, +3.0]	[-3.0, 3.0, -3.0, 3.0, 3.0]
6	Sum positive ranks	Sum_pos = 3.0 + 3.0 + 3.0	9.0
7	Sum negative ranks	Sum_neg = -3.0 + -3.0	-6.0
8	Test statistic W	W = min(Sum_pos, abs(Sum_neg))	6.0
9	Calculate p-value	Using Wilcoxon distribution	p = 0.6875
10	Decision	p > 0.05	Fail to reject null hypothesis

💡 Test ends after p-value calculation and decision based on significance level

Variable Tracker

Variable	Start	After Step 1	After Step 3	After Step 4	After Step 5	After Step 6	After Step 7	After Step 8	Final
diff	N/A	[-1, 1, -1, 1, 1]	[-1, 1, -1, 1, 1]	[-1, 1, -1, 1, 1]	[-1, 1, -1, 1, 1]	[-1, 1, -1, 1, 1]	[-1, 1, -1, 1, 1]	[-1, 1, -1, 1, 1]	[-1, 1, -1, 1, 1]
abs_diff	N/A	N/A	[1, 1, 1, 1, 1]	[1, 1, 1, 1, 1]	[1, 1, 1, 1, 1]	[1, 1, 1, 1, 1]	[1, 1, 1, 1, 1]	[1, 1, 1, 1, 1]	[1, 1, 1, 1, 1]
ranks	N/A	N/A	N/A	[3.0, 3.0, 3.0, 3.0, 3.0]	[3.0, 3.0, 3.0, 3.0, 3.0]	[3.0, 3.0, 3.0, 3.0, 3.0]	[3.0, 3.0, 3.0, 3.0, 3.0]	[3.0, 3.0, 3.0, 3.0, 3.0]	[3.0, 3.0, 3.0, 3.0, 3.0]
signed_ranks	N/A	N/A	N/A	N/A	[-3.0, 3.0, -3.0, 3.0, 3.0]	[-3.0, 3.0, -3.0, 3.0, 3.0]	[-3.0, 3.0, -3.0, 3.0, 3.0]	[-3.0, 3.0, -3.0, 3.0, 3.0]	[-3.0, 3.0, -3.0, 3.0, 3.0]
sum_pos	N/A	N/A	N/A	N/A	N/A	9.0	9.0	9.0	9.0
sum_neg	N/A	N/A	N/A	N/A	N/A	N/A	-6.0	-6.0	-6.0
W	N/A	N/A	N/A	N/A	N/A	N/A	N/A	6.0	6.0
p-value	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	0.6875

Key Moments - 3 Insights

Why do we rank the absolute differences instead of the raw differences?

Ranking absolute differences (see Step 4 in execution_table) allows us to measure the size of differences ignoring direction, which is needed before assigning signs back.

What happens if some differences are zero?

Zero differences are removed before ranking (Step 2). They do not affect the test statistic because they show no change.

Why do we take the minimum of sum of positive and negative ranks as test statistic W?

The test statistic W (Step 8) is the smaller sum to measure how much ranks lean to one side, indicating if median difference is zero or not.

Visual Quiz - 3 Questions

Look at the execution_table at Step 5. What is the signed rank for the third difference?

A -3.0

B 1.5

C 0

D 3.0

💡 Hint

Check the 'Assign signs to ranks' row in execution_table, third value in signed ranks array.

At which step does the test remove zero differences?

A Step 1

B Step 4

C Step 2

D Step 6

💡 Hint

Look for 'Remove zeros' action in execution_table.

If one difference was zero, how would the number of ranks change?

A Ranks would increase by one

B Ranks would decrease by one

C Ranks would stay the same

D Ranks would become zero

💡 Hint

Zero differences are removed before ranking (Step 2), so ranks count decreases.

Concept Snapshot

Wilcoxon signed-rank test compares paired samples.
Calculate differences, remove zeros.
Rank absolute differences.
Sum ranks by sign.
Test statistic W = smaller sum.
Use p-value to decide if median difference is zero.

Full Transcript

The Wilcoxon signed-rank test compares two related samples to check if their median difference is zero. First, it calculates the difference between paired values. Then it removes any zero differences because they don't affect the test. Next, it ranks the absolute values of these differences. After ranking, it assigns the original signs back to the ranks. The sums of positive and negative signed ranks are calculated. The test statistic W is the smaller of these sums. Finally, the test calculates a p-value from W to decide if the difference is statistically significant. If the p-value is greater than 0.05, we do not reject the null hypothesis, meaning no significant difference is found.