Statistics

Core statistical thinking for turning data into summaries, uncertainty statements, and defensible conclusions in CS, AI, and engineering.

Keywords

statistics, descriptive statistics, inference, experimental design, regression

1 Why This Module Matters

Statistics is the layer where probability meets data.

Probability tells you how random quantities behave in a model. Statistics asks the harder practical question: given finite, noisy, biased, or incomplete data, what are you justified in saying about the world?

That shift matters everywhere in CS, AI, and engineering. Experimental results, model evaluation, uncertainty intervals, regression fits, A/B tests, benchmark tables, and scientific claims all depend on statistical reasoning, not just probability formulas.

This module is the first pass through that reasoning. It begins with how to describe data responsibly, then moves toward estimation, uncertainty, model fitting, and experimental judgment.

Prerequisites Probability, algebra, and basic comfort reading formulas

Unlocks Inference, regression, experimental design, learning theory

Research Use Benchmarking, uncertainty reporting, study design, result interpretation

2 First Pass Through This Module

1. Descriptive Statistics and Data Models
2. Estimation and Bias-Variance
3. Maximum Likelihood and Bayesian Basics
4. Confidence Intervals and Hypothesis Testing
5. Regression and Classification Basics
6. Experimental Design and Model Evaluation

The full six-page first-pass statistics spine is now live.

3 Navigation Rule

Use the module overview and core concept pages as the main first-pass route.

Each concept page should be strong enough for first-pass learning on its own. Any proof, application, lab, or paper-lab companions should feel optional rather than required.

4 Core Concepts

• Descriptive Statistics and Data Models: teaches how to identify units, variables, sample/population roles, and the right numerical or graphical summaries before making any claim.
• Estimation and Bias-Variance: explains how sample-based procedures target population quantities and why accuracy has both systematic and random components.
• Maximum Likelihood and Bayesian Basics: introduces two major ways to connect models, parameters, and observed data.
• Confidence Intervals and Hypothesis Testing: teaches uncertainty statements, p-values, and the basic relation between intervals and tests.
• Regression and Classification Basics: turns statistical summaries into predictive and explanatory modeling.
• Experimental Design and Model Evaluation: teaches how randomization, controls, splits, and metrics shape what conclusions you may trust.

5 Proof Patterns In This Module

• Match the summary to the variable type: counts and proportions for categorical data; center, spread, and shape for quantitative data.
• Separate description from inference: a sample summary is not yet a population claim.
• Track the data-generating story: what the units are and how they were collected matters before any formula does.

6 Applications

6.1 Experimental Science And Benchmarking

Every benchmark table quietly makes statistical choices: what the observational unit is, which runs are pooled, whether variation is shown, and which summary statistic is reported. Bad summaries can make good experiments look weak or weak experiments look stronger than they are.

6.2 Machine Learning Evaluation

Train/validation/test splits, repeated seeds, error bars, calibration plots, confusion matrices, and regression diagnostics all rely on basic statistical thinking about samples, variability, and what a reported number actually represents.

7 Go Deeper By Topic

7.1 Describing Data Correctly

Start with Descriptive Statistics and Data Models.

If the page feels slippery, revisit:

• Sample Spaces, Events, and Conditioning for the probability-side language of populations, samples, and conditioning
• Expectation, Variance, Covariance for the probability-side meaning of average and spread

8 Optional Deep Dives After First Pass

Now that the full first-pass statistics spine is live, the strongest official deeper references are:

• Penn State STAT 500 - follow how the course moves from collecting and summarizing data into probability models and inferential procedures. Checked 2026-04-24.
• MIT 18.05 Introduction to Probability and Statistics - watch the transition from probability foundations to statistical inference and modeling. Checked 2026-04-24.
• CMU OLI Probability & Statistics - useful as a practice-oriented second pass. Checked 2026-04-24.

9 Study Order

The intended first pass is the six-step sequence above, and right now the first live path is:

1. Descriptive Statistics and Data Models
2. Estimation and Bias-Variance
3. Maximum Likelihood and Bayesian Basics
4. Confidence Intervals and Hypothesis Testing
5. Regression and Classification Basics
6. Experimental Design and Model Evaluation

You are ready to move deeper into the module when you can:

• identify the observational unit in a dataset or experiment
• distinguish sample from population and statistic from parameter
• choose a reasonable summary for categorical versus quantitative variables
• explain why the collection process matters before inference

10 Sources and Further Reading

• Penn State STAT 500 - First pass - strong official open notes for the applied statistics arc from data description to inference. Checked 2026-04-24.
• MIT 18.05 Introduction to Probability and Statistics - Second pass - official MIT course with a clean probability-to-statistics bridge. Checked 2026-04-24.
• CMU OLI Probability & Statistics - Second pass - structured practice environment with beginner-friendly reinforcement. Checked 2026-04-24.
• NIST/SEMATECH e-Handbook: Exploratory Data Analysis - Paper bridge - strong official reference for how descriptive summaries and plots support real data analysis before formal modeling. Checked 2026-04-24.

Sources checked online on 2026-04-24:

• Penn State STAT 500 course page
• MIT 18.05 course page
• CMU OLI Probability & Statistics
• NIST exploratory data analysis chapter