Mann-Whitney U test in SciPy - Step-by-Step Execution

Concept Flow - Mann-Whitney U test

Start with two independent samples

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Rank all values together

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Calculate U statistic for each sample

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Compare U to distribution or calculate p-value

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Decide if samples differ significantly

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End

The test ranks combined data, calculates U statistics, then finds the p-value to check if two groups differ.

Execution Sample

SciPy

from scipy.stats import mannwhitneyu

sample1 = [5, 7, 8, 9]
sample2 = [1, 2, 3, 4]

stat, p = mannwhitneyu(sample1, sample2, alternative='two-sided')
print(stat, p)

This code runs the Mann-Whitney U test on two small samples and prints the U statistic and p-value.

Execution Table

Step	Action	Details	Result
1	Combine samples	Samples: [5,7,8,9] and [1,2,3,4]	Combined: [1,2,3,4,5,7,8,9]
2	Rank combined data	Assign ranks to combined values	Ranks: [1,2,3,4,5,6,7,8]
3	Sum ranks for sample1	Ranks for sample1 values (5,7,8,9)	Sum ranks = 5+6+7+8 = 26
4	Sum ranks for sample2	Ranks for sample2 values (1,2,3,4)	Sum ranks = 1+2+3+4 = 10
5	Calculate U for sample1	U1 = n1*n2 + n1(n1+1)/2 - R1	U1 = 4*4 + 4*5/2 - 26 = 6
6	Calculate U for sample2	U2 = n1*n2 - U1	U2 = 16 - 6 = 10
7	Select smaller U	U = min(U1, U2)	U = 6
8	Calculate p-value	Using U and sample sizes	p ≈ 0.0286
9	Interpret result	p < 0.05 means significant difference	Reject null hypothesis
10	End	Test complete	Output: U=6, p=0.0286

💡 Test ends after calculating p-value and interpreting significance.

Variable Tracker

Variable	Start	After Step 3	After Step 5	After Step 7	Final
sample1	[5,7,8,9]	[5,7,8,9]	[5,7,8,9]	[5,7,8,9]	[5,7,8,9]
sample2	[1,2,3,4]	[1,2,3,4]	[1,2,3,4]	[1,2,3,4]	[1,2,3,4]
combined	[]	[1,2,3,4,5,7,8,9]	[1,2,3,4,5,7,8,9]	[1,2,3,4,5,7,8,9]	[1,2,3,4,5,7,8,9]
ranks	[]	[1,2,3,4,5,6,7,8]	[1,2,3,4,5,6,7,8]	[1,2,3,4,5,6,7,8]	[1,2,3,4,5,6,7,8]
R1 (sum ranks sample1)	0	26	26	26	26
U1	0	0	6	6	6
U2	0	0	0	10	10
U	0	0	0	6	6
p-value	None	None	None	None	0.0286

Key Moments - 3 Insights

Why do we rank all values together instead of comparing raw values?

Ranking all values together (see Step 2 in execution_table) allows the test to compare relative positions without assuming normal distribution.

Why do we pick the smaller U value for the test?

The Mann-Whitney U test uses the smaller U (Step 7) because it reflects the difference between groups more conservatively.

What does a small p-value mean in this test?

A small p-value (Step 8) means the two samples likely come from different groups, so we reject the idea they are the same.

Visual Quiz - 3 Questions

Look at the execution_table at Step 3, what is the sum of ranks for sample1?

A 10

B 6

C 26

D 16

💡 Hint

Check the 'Result' column in Step 3 of execution_table.

At which step does the test decide which U value to use?

A Step 7

B Step 6

C Step 5

D Step 8

💡 Hint

Look for 'Select smaller U' in the Action column.

If sample1 had all values smaller than sample2, how would the U statistic change?

A U would be unchanged

B U would be zero

C U would be maximum possible

D U would be negative

💡 Hint

Recall U measures how many times values in one sample exceed the other; all smaller means minimum U.

Concept Snapshot

Mann-Whitney U test compares two independent samples.
It ranks combined data and calculates U statistics.
Uses smaller U to find p-value.
No assumption of normal distribution.
Small p-value means groups differ significantly.

Full Transcript

The Mann-Whitney U test compares two independent groups by ranking all their values together. It sums ranks for each group and calculates U statistics. The smaller U is used to find a p-value, which tells us if the groups differ significantly. This test does not assume normal distribution, making it useful for non-normal data. A small p-value means we reject the idea that the groups are the same.