Statistics for Data Science: Lecture Notes

DATA11007

Authors

Affiliations

Luigi Acerbi

Associate Professor, Department of Computer Science, University of Helsinki

Antti Honkela

Professor, Department of Computer Science, University of Helsinki

Preface

Welcome to the lecture notes for the Statistics for Data Science (DATA11007) course offered by the Master’s Programme in Data Science at the University of Helsinki.

These notes are designed to be an accessible companion to the course, providing multiple perspectives on statistical concepts for students with diverse backgrounds.

Note

Please refer to the official Course Description and Moodle page for specific information about current implementations of the course.

About These Notes

These lecture notes complement the main course textbook, All of Statistics by Larry Wasserman; Wasserman (2013). While Wasserman’s book provides a comprehensive and rigorous treatment, these notes aim to:

• Provide gentler introductions to complex topics
• Offer multiple perspectives (intuitive, practical, mathematical, computational)
• Include extensive code examples in Python (and occasionally R)
• Bridge prerequisite gaps for students from different backgrounds

Formats and Feedback

These notes are available in two formats:

• HTML (recommended): Interactive version with tabs, code folding, and better navigation.
• PDF: Traditional document format for printing or offline reading.

The content is almost identical in both formats. The PDF version is automatically generated from the source using Quarto’s LaTeX conversion and we are aware it contains some formatting artifacts (e.g., additional page breaks), but this should not affect readability. Interactive code elements and end-of-chapter code blocks are not included in the PDF version. The HTML version of the notes is the officially recommended version.

Your Feedback is Welcome!

These lecture notes are new and under active development. We welcome feedback, corrections, and suggestions! If you spot any errors, unclear explanations, or have ideas for improvements, please let your instructor know. Your feedback will contribute directly to the development of these materials.

Work in Progress

These lecture notes are being developed throughout the course. While the initial chapters are currently available, additional chapters will be released progressively as they are prepared. Each chapter will be made available before its corresponding lecture, ensuring you have the materials when you need them.

We appreciate your patience as we continue to build this comprehensive resource!

How to Use These Notes

Throughout these notes, you’ll encounter sections with multiple perspectives on the same concept. These are presented in tabbed panels to help you explore different viewpoints:

Intuitive

This perspective provides conceptual understanding through analogies, visualizations, and real-world examples. It focuses on building intuition without heavy mathematical formalism.

Mathematical

This perspective explains concepts using mathematical language and notation. It may provide alternative formulations or connect to other mathematical ideas that some readers find clearer or more familiar.

Computational

This perspective shows how to implement and explore concepts through code. It includes simulations, numerical experiments, and practical algorithms.

Important: These perspectives often contain more advanced material than required for the course, or teasers for material which will be covered later. You’re not expected to understand everything in every panel! Think of them as:

• Additional resources for deeper exploration
• Different entry points to the same concept
• Teasers for the content of future lectures
• Optional enrichment beyond the core material

Choose the perspective that best matches your learning style, or explore all three to build a richer understanding. The core course content is always presented in the main text outside these panels.

Example: Understanding Variance

Here’s a brief example of how these perspectives work. Consider the concept of variance (\text{Var}), which measures how spread out data values are:

Intuitive

Variance measures how far data points typically are from their average.

Imagine test scores in two classes:

• Class A: Everyone scores between 75-85 (low variance)
• Class B: Scores range from 40-100 (high variance)

Both might have the same average (say, 80), but Class B has much more spread. Variance captures this spread numerically.

Mathematical

The variance of a random variable \(X\) is defined as:

\[\text{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2\]

For a discrete distribution:

\[\text{Var}(X) = \sum_i (x_i - \mu)^2 \cdot \mathbb{P}(X = x_i)\]

Key properties:

• \(\text{Var}(aX + b) = a^2\text{Var}(X)\)
• If \(X, Y\) are independent: \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)\)

Computational

import numpy as np
import matplotlib.pyplot as plt

# Two datasets with same mean but different variance
class_a = [75, 78, 80, 82, 85, 77, 83, 79, 81, 80]
class_b = [40, 95, 100, 65, 85, 90, 45, 98, 70, 82]

print(f"Class A: mean = {np.mean(class_a):.1f}, variance = {np.var(class_a):.1f}")
print(f"Class B: mean = {np.mean(class_b):.1f}, variance = {np.var(class_b):.1f}")

# Visualize to see the spread
plt.figure(figsize=(7, 3))

# Use same bins for both histograms to ensure consistent bar width
bins = np.linspace(30, 110, 17)  # 16 bins from 30 to 110

plt.hist(class_a, alpha=0.5, label='Class A (low variance)', bins=bins, edgecolor='black')
plt.hist(class_b, alpha=0.5, label='Class B (high variance)', bins=bins, edgecolor='black')
plt.xlabel('Test Score')
plt.ylabel('Count')
plt.title('Comparison of Test Score Distributions')
plt.legend(loc='upper left')

# Set y-axis to show only integers
plt.gca().yaxis.set_major_locator(plt.MaxNLocator(integer=True))

plt.grid(True, alpha=0.3, axis='y')
plt.tight_layout()
plt.show()

Class A: mean = 80.0, variance = 7.8
Class B: mean = 77.0, variance = 413.8

Notice how each perspective offers something different: intuition through examples, mathematical rigor through formulas, and hands-on exploration through code.

Prerequisites

The course assumes:

• Basic programming ability (Python or R)
• Calculus (derivatives and integrals)
• Linear algebra (matrices and vectors)
• Basic probability (sample spaces, events)

Don’t worry if you’re rusty on some topics – we’ll review key concepts as needed.

Course Structure

1. Foundations: Probability, random variables, distributions
2. Inference: Estimation, confidence intervals, hypothesis testing
   
3. Methods: Regression, resampling, Bayesian approaches
4. Advanced Topics: Missing data, causal inference, experimental design

Available Chapters

The following chapters are currently available:

1. Probability Foundations
2. Expectation
3. Convergence and Statistical Inference
4. Nonparametric Methods and Bootstrap
5. Parametric Inference I: Finding Estimators
6. Parametric Inference II: Properties of Estimators
7. Hypothesis Testing and p-values
8. Bayesian Inference and Decision Theory
9. Linear and Logistic Regression

Programming

In this course, we’ll mainly use Python with occasional usages of R. Python is the main language for modern machine learning, and both Python and R are widely used in data science. Feel free to choose the environment you’re most comfortable with.

Acknowledgments

These notes build upon materials from previous course iterations and incorporate feedback from many students. Special thanks to the teaching assistants and students who have helped improve these materials.

The course content is largely adapted from Wasserman (2013), which remains the main course textbook.

The course slides and materials in early course iterations (2021-2024) were fully developed by Antti Honkela. Starting from 2025, we introduced the lecture notes that you are currently reading, which have been developed mainly by Luigi Acerbi expanding on the existing course slides and materials.

These notes were compiled with the assistance of Gemini 2.5 Pro, GPT-5 Thinking, and the Claude 4 family of AI models (largely via Claude Code), particularly when it comes to the Quarto layout, coding content, spotting mistakes, and brainstorming examples.

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Last updated: September 23, 2025

Wasserman, Larry. 2013. All of Statistics: A Concise Course in Statistical Inference. Springer Science & Business Media.