←Back to the bookChapter 1 · What Is Statistics?

Part I · Foundations  |  Chapter 1

What Is Statistics? 🐍 Notebook

Six hands-on demos that turn the chapter's ideas into runnable code, descriptive statistics, the normal curve, sampling & inference, the law of large numbers, and how statistics powers machine learning.

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

🎯 What you'll build in this notebook

#	Demo	Concept it illustrates
1	Exam-score summary	Descriptive statistics, mean, median, mode, spread
2	The bell curve	Normal distribution & the 68–95–99.7 rule
3	Population → Sample	Inferential statistics & the Central Limit Theorem
4	Rolling the dice	Law of Large Numbers, why sampling works
5	Experience → Salary	Statistics → Machine Learning (least-squares = an ML model)
6	A biased sample	Sampling bias, why a random sample matters

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⚙️ Setup, imports & a book-matched plotting style

We define a small color palette and Matplotlib defaults so every chart in this handbook shares the same clean, modern look as the infographics.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.ticker as mticker
from matplotlib.patches import FancyBboxPatch

# Reproducible randomness
rng = np.random.default_rng(42)

# ---- Book palette (matches the HTML chapters) ----
NAVY="#0a1230"; INK="#1a2138"; INK_SOFT="#4a5578"
CYAN="#0891b2"; PURPLE="#7c3aed"; AMBER="#d97706"
GREEN="#059669"; PINK="#db2777"; BLUE="#2563eb"
GRID="#e6e9f2"

# ---- Global Matplotlib style ----
plt.rcParams.update({
    "figure.facecolor":"white", "axes.facecolor":"white",
    "figure.dpi":110, "savefig.dpi":110,
    "font.family":"DejaVu Sans", "font.size":11,
    "axes.edgecolor":GRID, "axes.linewidth":1.2,
    "axes.grid":True, "grid.color":GRID, "grid.linewidth":1,
    "axes.spines.top":False, "axes.spines.right":False,
    "axes.titlesize":15, "axes.titleweight":"bold", "axes.titlecolor":INK,
    "axes.labelcolor":INK_SOFT, "axes.labelsize":11.5,
    "xtick.color":INK_SOFT, "ytick.color":INK_SOFT,
    "legend.frameon":False,
})

def titlecard(ax, title, subtitle=None):
    """Add an infographic-style left-aligned title above an axis."""
    ax.set_title(title, loc="left", pad=18)
    if subtitle:
        ax.text(0, 1.02, subtitle, transform=ax.transAxes,
                fontsize=10.5, color=INK_SOFT, va="bottom")

print("✅ Environment ready — palette loaded, RNG seeded.")

✅ Environment ready — palette loaded, RNG seeded.

DEMO 1 · DESCRIPTIVE STATISTICS

📊 Describing a class of exam scores

Descriptive statistics answers "what does the data look like?", its center (mean, median, mode) and its spread (range, standard deviation).

# Simulate exam scores for 200 students (integers, 40–100)
scores = np.clip(rng.normal(75, 11, 200).round(), 40, 100).astype(int)
data = pd.Series(scores, name="score")

mean   = data.mean()
median = data.median()
mode   = data.mode().iloc[0]
std    = data.std()

summary = pd.DataFrame({
    "Statistic":["Count","Mean","Median","Mode","Std Dev","Min","Max","Range"],
    "Value":[len(data), round(mean,1), median, mode, round(std,1),
             data.min(), data.max(), data.max()-data.min()]
})
summary

	Statistic	Value
0	Count	200.0
1	Mean	74.6
2	Median	74.0
3	Mode	71.0
4	Std Dev	9.5
5	Min	52.0
6	Max	100.0
7	Range	48.0

fig, ax = plt.subplots(figsize=(9,4.8))

n, bins, patches = ax.hist(data, bins=range(40,103,3), color=CYAN, alpha=0.78,
                           edgecolor="white", linewidth=1.4)

# Mark the three measures of central tendency
for value, label, color in [(mean,"Mean","%s"%PURPLE),
                            (median,"Median","%s"%AMBER),
                            (mode,"Mode","%s"%GREEN)]:
    ax.axvline(value, color=color, linewidth=2.6, linestyle="--")
    ax.text(value, ax.get_ylim()[1]*0.96, f"  {label}: {value:.0f}",
            color=color, fontweight="bold", fontsize=10.5, ha="left", va="top")

titlecard(ax, "Distribution of Exam Scores",
          f"200 students · the shape, center, and spread at a glance")
ax.set_xlabel("Score"); ax.set_ylabel("Number of students")
ax.set_axisbelow(True)
plt.tight_layout(); plt.show()

Reading the chart: the bars show how many students scored in each range (the shape). The three dashed lines show the center three different ways. When mean ≈ median ≈ mode, the distribution is roughly symmetric, a hint of the normal distribution we explore next.

DEMO 2 · THE NORMAL DISTRIBUTION

🔔 The bell curve & the 68–95–99.7 rule

The most important shape in all of statistics. The empirical rule tells us how data clusters around the mean, the backbone of confidence intervals, z-scores, and countless ML methods.

def normal_pdf(x, mu=0, sigma=1):
    """Probability density of the normal distribution (no scipy needed)."""
    return np.exp(-0.5*((x-mu)/sigma)**2) / (sigma*np.sqrt(2*np.pi))

x = np.linspace(-4, 4, 500)
y = normal_pdf(x)

fig, ax = plt.subplots(figsize=(9,4.8))
ax.plot(x, y, color=NAVY, linewidth=2.5)

# Shade ±1, ±2, ±3 sigma with the empirical-rule percentages
shades = [(1, "#7c3aed", 0.45, "68.2%"),
          (2, "#a855f7", 0.28, "95.4%"),
          (3, "#c4b5fd", 0.20, "99.7%")]
for s, color, alpha, label in reversed(shades):
    mask = (x >= -s) & (x <= s)
    ax.fill_between(x[mask], y[mask], color=color, alpha=alpha)

# Annotate the bands
ax.annotate("68%", xy=(0,0.18), ha="center", fontsize=12, fontweight="bold", color="white")
ax.annotate("95%", xy=(1.5,0.04), ha="center", fontsize=11, fontweight="bold", color=PURPLE)
ax.annotate("99.7%", xy=(2.6,0.012), ha="center", fontsize=10, fontweight="bold", color=PURPLE)

# sigma ticks
ax.set_xticks(range(-3,4))
ax.set_xticklabels([f"{i}σ" for i in range(-3,4)])
titlecard(ax, "The Normal Distribution",
          "Within 1σ lies ~68% of data · 2σ ~95% · 3σ ~99.7%")
ax.set_ylabel("Probability density"); ax.set_yticks([])
plt.tight_layout(); plt.show()

DEMO 3 · INFERENTIAL STATISTICS

🎯 From a sample to the whole population

We rarely measure everyone. Instead we take a sample and infer. The Central Limit Theorem is what makes this work: averages of samples form a tidy normal distribution centered on the true value.

# A "population" of 20,000 adult heights (cm) we pretend we cannot fully measure
population = rng.normal(170, 10, 20_000)
true_mean = population.mean()

# Take 2,000 independent samples of 50 people each; record each sample mean
sample_means = np.array([rng.choice(population, size=50, replace=False).mean()
                         for _ in range(2000)])

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11,4.6))

# Left: the population
ax1.hist(population, bins=45, color=CYAN, alpha=0.8, edgecolor="white", linewidth=0.4)
ax1.axvline(true_mean, color=NAVY, linewidth=2.4, linestyle="--")
ax1.text(true_mean, ax1.get_ylim()[1]*0.92, f"  true mean = {true_mean:.1f} cm",
         fontweight="bold", color=NAVY)
titlecard(ax1, "The Population", "20,000 individuals · wide spread")
ax1.set_xlabel("Height (cm)"); ax1.set_ylabel("Count")

# Right: distribution of the 2,000 sample means
ax2.hist(sample_means, bins=35, color=AMBER, alpha=0.85, edgecolor="white", linewidth=0.6)
ax2.axvline(true_mean, color=NAVY, linewidth=2.4, linestyle="--")
ax2.text(true_mean, ax2.get_ylim()[1]*0.92, f"  centered on {true_mean:.1f} cm",
         fontweight="bold", color=NAVY)
titlecard(ax2, "Distribution of Sample Means", "2,000 samples of n=50 · narrow & normal!")
ax2.set_xlabel("Sample mean height (cm)"); ax2.set_ylabel("Count")

plt.tight_layout(); plt.show()

print(f"Population mean      : {true_mean:.2f} cm")
print(f"Mean of sample means : {sample_means.mean():.2f} cm   <- almost identical!")
print(f"Std of sample means  : {sample_means.std():.2f} cm   (the \"standard error\")")
print(f"Theory (10/sqrt(50)) : {10/np.sqrt(50):.2f} cm")

Population mean      : 170.07 cm
Mean of sample means : 170.07 cm   <- almost identical!
Std of sample means  : 1.46 cm   (the "standard error")
Theory (10/sqrt(50)) : 1.41 cm

The big idea: even though individual heights vary a lot (left), the average of a 50-person sample is a remarkably reliable estimate of the true population mean (right). This is why a poll of 1,000 people can speak about millions, and it is exactly the principle behind train/test splits and bootstrapping in machine learning.

DEMO 4 · LAW OF LARGE NUMBERS

🎲 More data → less uncertainty

As we collect more observations, our estimate settles down toward the truth. Watch the running average of dice rolls converge to the true mean of 3.5.

rolls = rng.integers(1, 7, size=3000)          # 3,000 fair-die rolls
running_mean = np.cumsum(rolls) / np.arange(1, len(rolls)+1)
true_value = 3.5

fig, ax = plt.subplots(figsize=(9,4.6))
ax.plot(running_mean, color=GREEN, linewidth=1.8)
ax.axhline(true_value, color=PINK, linewidth=2.2, linestyle="--")
ax.text(len(rolls)*0.7, true_value+0.12, "true mean = 3.5",
        color=PINK, fontweight="bold")
ax.set_xscale("log")
titlecard(ax, "Running Average of Dice Rolls",
          "The estimate is noisy at first, then locks onto the truth")
ax.set_xlabel("Number of rolls (log scale)"); ax.set_ylabel("Running average")
ax.set_ylim(1, 6)
plt.tight_layout(); plt.show()

print(f"After 10 rolls   : {running_mean[9]:.3f}")
print(f"After 100 rolls  : {running_mean[99]:.3f}")
print(f"After 3000 rolls : {running_mean[-1]:.3f}")

After 10 rolls   : 3.800
After 100 rolls  : 3.630
After 3000 rolls : 3.507

DEMO 5 · STATISTICS → MACHINE LEARNING

🤖 A regression line is a machine-learning model

We fit a least-squares line to predict salary from years of experience. Least squares is a 200-year-old statistical method, and simultaneously one of the most-used ML models on earth.

# Generate data: salary tends to rise with experience (+ real-world noise)
experience = rng.uniform(0, 15, 80)
salary = 30 + 4.2*experience + rng.normal(0, 6, 80)   # $000s

# --- Fit a least-squares line: the core statistical/ML step ---
slope, intercept = np.polyfit(experience, salary, deg=1)
pred = slope*experience + intercept

# R^2 — how much variation the model explains (a statistics metric used everywhere in ML)
ss_res = np.sum((salary - pred)**2)
ss_tot = np.sum((salary - salary.mean())**2)
r2 = 1 - ss_res/ss_tot

# Predict for a new, unseen employee with 10 years experience
new_x = 10
new_pred = slope*new_x + intercept

fig, ax = plt.subplots(figsize=(9,5))
ax.scatter(experience, salary, color=CYAN, s=55, alpha=0.7,
           edgecolor="white", linewidth=1, label="Observed employees")
xs = np.linspace(0, 15, 100)
ax.plot(xs, slope*xs+intercept, color=PURPLE, linewidth=2.8,
        label=f"Model: salary = {intercept:.1f} + {slope:.1f}·exp")

# Highlight a prediction
ax.scatter([new_x], [new_pred], color=AMBER, s=180, zorder=5, marker="*",
           edgecolor="white", linewidth=1.2, label=f"Prediction @ 10 yrs = ${new_pred:.0f}k")
ax.plot([new_x,new_x],[0,new_pred], color=AMBER, linestyle=":", linewidth=1.5)

titlecard(ax, "Predicting Salary from Experience",
          f"Least-squares regression · R² = {r2:.2f} (explains {r2*100:.0f}% of the variation)")
ax.set_xlabel("Years of experience"); ax.set_ylabel("Salary ($000s)")
ax.set_xlim(0,15.5); ax.set_ylim(bottom=0)
ax.legend(loc="upper left", fontsize=9.5)
plt.tight_layout(); plt.show()

print(f"Learned model : salary = {intercept:.2f} + {slope:.2f} * experience")
print(f"R-squared     : {r2:.3f}")
print(f"Prediction    : a 10-year employee earns about ${new_pred:.0f}k")

Learned model : salary = 29.29 + 4.08 * experience
R-squared     : 0.898
Prediction    : a 10-year employee earns about $70k

Why this is the punchline of the chapter: what we just did has two names.

• A statistician calls it linear regression, estimating parameters and explaining variance (R²).
• An ML engineer calls it training a model, learning weights from data to predict unseen cases.

They are the same thing. Scale this idea up, more features, more data, nonlinear functions, millions of parameters, and you arrive at modern machine learning and AI. The statistics never left; it just got bigger.

DEMO 6 · WHY HOW YOU SAMPLE MATTERS

🎯 A biased sample misses the truth

Sample size is not everything. If a sample is not random, even a large one can land far from the real population value.

# A population whose true mean we know
pop = rng.normal(50, 12, 100_000)
true_mean = pop.mean()

# A fair RANDOM sample vs a BIASED sample (only values above 60)
fair   = rng.choice(pop, 500, replace=False)
biased = rng.choice(pop[pop > 60], 500, replace=False)

print(f"True population mean : {true_mean:.1f}")
print(f"Random sample mean   : {fair.mean():.1f}   (close to the truth)")
print(f"Biased sample mean   : {biased.mean():.1f}   (far off, despite n=500)")

fig, ax = plt.subplots(figsize=(9,4.4))
ax.hist(pop, bins=50, color="#d7dbe8", alpha=0.7, edgecolor="white")
ax.axvline(true_mean,    color=NAVY, lw=2.4, ls="--", label=f"true mean {true_mean:.1f}")
ax.axvline(fair.mean(),  color=CYAN, lw=2.6, label=f"random sample {fair.mean():.1f}")
ax.axvline(biased.mean(),color=PINK, lw=2.6, label=f"biased sample {biased.mean():.1f}")
titlecard(ax, "A biased sample misses the population mean", "n=500 either way; only randomness keeps it honest")
ax.set_xlabel("value"); ax.set_ylabel("count"); ax.legend(fontsize=9)
plt.tight_layout(); plt.show()

True population mean : 50.0
Random sample mean   : 49.5   (close to the truth)
Biased sample mean   : 66.8   (far off, despite n=500)

Takeaway: more data does not fix bias. A non-random sample (here, only high values) lands far from the true mean no matter how large it is. How you sample matters as much as how much.

---

🎓 Recap

• Descriptive statistics summarized our exam scores (center & spread).
• The normal distribution and 68–95–99.7 rule describe how data clusters.
• Inferential statistics + the Central Limit Theorem let a small sample speak for a whole population.
• The Law of Large Numbers showed why more data means more certainty.
• A least-squares regression revealed that statistics is the engine inside machine learning.

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

←Back to the bookChapter 1 · What Is Statistics?