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Chapter 80 · Solutions

Nonparametric — Worked Solutions ✅

Five challenges, each verified in code.

⚙️ Setup

import numpy as np, pandas as pd
from scipy import stats
BASE="https://raw.githubusercontent.com/johnfisher-ai/Statistics-Data-Science-AI-Visual-Book/main/data/"
rng=np.random.default_rng(800)

CHALLENGE 1

Mann-Whitney U

Compare lognormal(1.0,0.6,n=50) and lognormal(0.6,0.6,n=50) with the rank-sum test.

a=rng.lognormal(1.0,0.6,50); b=rng.lognormal(0.6,0.6,50)
u,p=stats.mannwhitneyu(a,b,alternative="two-sided"); print(f"U={u:.1f}, p={p:.4f}")

U=1772.0, p=0.0003

CHALLENGE 2

Outlier robustness

Add one outlier of 500 to a clean N(50,8,40) sample; compare how the t-test p and Mann-Whitney p vs N(52,8,40) change.

clean=rng.normal(50,8,40); other=rng.normal(52,8,40); dirty=clean.copy(); dirty[0]=500
print(f"t-test (dirty) p={stats.ttest_ind(dirty,other).pvalue:.3f}")
print(f"Mann-Whitney (dirty) p={stats.mannwhitneyu(dirty,other).pvalue:.3f}  (ranks ignore the outlier\u2019s size)")

t-test (dirty) p=0.522
Mann-Whitney (dirty) p=0.071  (ranks ignore the outlier’s size)

CHALLENGE 3

Kruskal-Wallis

Compare three skewed groups lognormal at mu=1.0,1.2,1.5 (n=40 each).

g=[rng.lognormal(m,0.5,40) for m in (1.0,1.2,1.5)]; H,p=stats.kruskal(*g)
print(f"H={H:.2f}, p={p:.4f}")

H=24.73, p=0.0000

CHALLENGE 4

Wilcoxon signed-rank

Paired: before=lognormal(1.2,0.4,30), after=before*uniform(0.7,0.95). Test the paired drop.

before=rng.lognormal(1.2,0.4,30); after=before*rng.uniform(0.7,0.95,30)
w,p=stats.wilcoxon(after,before); print(f"W={w:.1f}, p={p:.4f}")

W=0.0, p=0.0000

CHALLENGE 5

Real data: ticket times

Load nonparametric-tests--ticket_times.xlsx; run Mann-Whitney (Alpha vs Bravo) and Kruskal-Wallis (by priority).

try:    d = pd.read_excel("../../data/nonparametric-tests--ticket_times.xlsx", sheet_name="Tickets")
except FileNotFoundError: d = pd.read_excel(BASE+"nonparametric-tests--ticket_times.xlsx", sheet_name="Tickets")
A=d[d.team=="Alpha"].resolution_hours; B=d[d.team=="Bravo"].resolution_hours
print(f"Mann-Whitney p={stats.mannwhitneyu(A,B).pvalue:.4f}")
g=[x.resolution_hours.values for _,x in d.groupby("priority")]
print(f"Kruskal-Wallis p={stats.kruskal(*g).pvalue:.2e}")

Mann-Whitney p=0.0469
Kruskal-Wallis p=3.76e-07

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Statistics, Data Science and AI: A Visual Handbook · © 2026 John Fisher

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