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Chapter 40 · Probability Distributions

The Normal Distribution in Depth 🔔

The bell curve earns its own chapter. This notebook verifies the 68-95-99.7 rule by simulation, standardizes data into z-scores, computes probabilities and percentiles, and standardizes features the way every machine-learning pipeline does.

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

⚙️ Setup

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
rng = np.random.default_rng(39)
plt.rcParams.update({"figure.dpi":110,"font.size":11,"axes.spines.top":False,"axes.spines.right":False})
TEAL="#0d9488"; PINK="#db2777"; AMBER="#d97706"
print("ready")

ready

DEMO 1 · THE BELL CURVE

Two parameters, one shape

Every normal curve is the same bell set by its mean mu and standard deviation sigma. The PDF peaks at the mean and is perfectly symmetric.

xs=np.linspace(-6,12,500)
fig,ax=plt.subplots(figsize=(7,3.4))
for mu,sd,c,lab in [(0,1,TEAL,"N(0,1)"),(3,2,AMBER,"N(3,2)"),(6,0.7,PINK,"N(6,0.7)")]:
    ax.plot(xs, stats.norm.pdf(xs,mu,sd), color=c, lw=2.4, label=lab)
    ax.axvline(mu, color=c, ls=":", lw=1)
ax.set_xlabel("x"); ax.set_ylabel("density"); ax.set_title("Same bell, relocated by mu and stretched by sigma"); ax.legend()
plt.tight_layout(); plt.show()

The peak always sits at mu, and the curve is symmetric, so mean = median = mode. A smaller sigma concentrates the area into a tall narrow spike; a larger sigma flattens it. These two numbers fully determine the entire distribution.

DEMO 2 · THE EMPIRICAL RULE

68 - 95 - 99.7

For any normal distribution, fixed fractions of the data fall within 1, 2, and 3 standard deviations of the mean. This is the famous 68-95-99.7 rule.

for k in (1,2,3):
    frac = stats.norm.cdf(k) - stats.norm.cdf(-k)
    print(f"within +/- {k} sigma: {frac*100:.2f}% of the data")

# verify by simulation
data = rng.normal(0,1,size=1_000_000)
for k in (1,2,3):
    print(f"simulated within +/- {k} sigma: {np.mean(np.abs(data)<k)*100:.2f}%")

within +/- 1 sigma: 68.27% of the data
within +/- 2 sigma: 95.45% of the data
within +/- 3 sigma: 99.73% of the data
simulated within +/- 1 sigma: 68.28%
simulated within +/- 2 sigma: 95.51%
simulated within +/- 3 sigma: 99.74%

xs=np.linspace(-4,4,400); pdf=stats.norm.pdf(xs)
fig,ax=plt.subplots(figsize=(7,3.4))
ax.plot(xs,pdf,color="#1a2138",lw=1.5)
for k,c,a in [(3,"#a7f3d0",0.5),(2,"#5eead4",0.6),(1,TEAL,0.7)]:
    m=(xs>=-k)&(xs<=k); ax.fill_between(xs[m],pdf[m],color=c,alpha=a)
ax.set_title("68% within 1 sigma, 95% within 2, 99.7% within 3"); ax.set_xlabel("standard deviations from the mean")
ax.set_xticks([-3,-2,-1,0,1,2,3]); plt.tight_layout(); plt.show()

The percentages, 68.27%, 95.45%, and 99.73%, hold for every normal, whatever its mu and sigma. This is why "within two standard deviations" is shorthand for "almost all" and why a value beyond 3 sigma is genuinely surprising (under 0.3% of cases).

DEMO 3 · STANDARDIZING TO z-SCORES

One scale to compare them all

A z-score, z = (x - mu) / sigma, says how many standard deviations a value sits from its mean. It turns ANY normal into the standard normal N(0,1), making different scales comparable.

sat = rng.normal(1050, 200, size=8)        # SAT-like scores
z = (sat - sat.mean()) / sat.std()
for s,zi in zip(sat, z):
    print(f"score {s:7.1f}  ->  z = {zi:+.2f}")
print(f"\nstandardized data: mean = {z.mean():.2f}, std = {z.std():.2f}")

score   759.3  ->  z = -0.97
score  1310.6  ->  z = +2.29
score   848.6  ->  z = -0.44
score   762.7  ->  z = -0.95
score   982.2  ->  z = +0.35
score   930.8  ->  z = +0.05
score   983.7  ->  z = +0.36
score   807.6  ->  z = -0.68

standardized data: mean = -0.00, std = 1.00

After standardizing, the data has mean 0 and standard deviation 1 by construction, the standard normal scale. A z of +2 means "two standard deviations above average", whether the raw value was a test score, a height, or a temperature. Standardization is what lets you compare apples to oranges.

DEMO 4 · PROBABILITIES & PERCENTILES

Working in z

Once standardized, probabilities come straight from the standard normal CDF (the z-table). The value z = 1.96 marks the central 95%, the number behind nearly every confidence interval.

print(f"P(Z < 1)        = {stats.norm.cdf(1):.4f}")
print(f"P(Z > 2)        = {1-stats.norm.cdf(2):.4f}")
print(f"central 95% is z in +/- {stats.norm.ppf(0.975):.4f}")

# applied to N(100,15): what IQ is the 99th percentile?
print(f"99th percentile of N(100,15) = {stats.norm.ppf(0.99,100,15):.1f}")

P(Z < 1)        = 0.8413
P(Z > 2)        = 0.0228
central 95% is z in +/- 1.9600
99th percentile of N(100,15) = 134.9

The single number z = 1.96 captures the central 95% of any normal, which is why it appears in confidence intervals and significance tests throughout the rest of the book. Convert a value to z, look up the area, and you have its probability or percentile.

DEMO 5 · STANDARDIZATION IN MACHINE LEARNING

Why models z-score their features

Most ML algorithms work far better when features share a scale. Standardizing each feature to mean 0, std 1 (a z-score) is the most common preprocessing step in all of machine learning.

# two features on wildly different scales
age    = rng.normal(40, 12, size=1000)        # ~ tens
income = rng.normal(60000, 20000, size=1000)  # ~ tens of thousands
X = np.column_stack([age, income])
print(f"raw means {X.mean(0).round(1)},  raw stds {X.std(0).round(1)}")

Xz = (X - X.mean(0)) / X.std(0)               # standardize each column
print(f"scaled means {Xz.mean(0).round(3)},  scaled stds {Xz.std(0).round(3)}")

# z-scores also flag anomalies: |z| > 3 is a rare outlier
outliers = np.abs(Xz) > 3
print(f"points beyond 3 sigma (anomalies): {outliers.any(axis=1).sum()}")

raw means [4.00000e+01 5.96914e+04],  raw stds [1.22000e+01 1.99733e+04]
scaled means [-0.  0.],  scaled stds [1. 1.]
points beyond 3 sigma (anomalies): 7

Before scaling, income dwarfs age simply because its numbers are bigger, and distance-based or gradient-based models would be dominated by it. Standardizing puts every feature on the z-scale (mean 0, std 1) so each contributes fairly. The same z-scores double as an anomaly detector: any point beyond 3 sigma is, by the empirical rule, a rare outlier worth a look.

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✅ What you built

You confirmed the 68-95-99.7 rule, standardized data into z-scores, pulled probabilities and percentiles from the standard normal, and scaled features the way every ML pipeline does. The z-score, how many standard deviations from the mean, is one of the most useful numbers in statistics.

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

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