Statistics & Probability
The mathematics of uncertainty and data. Probability gives us a calculus of "what might happen"; statistics works the other way, asking what data can tell us about an unknown reality. Together they're the tools of empirical reasoning.
Mean, Median & Mode
Three notions of "the typical value." When each is the right summary — and when each lies about the data.
Variance & Standard Deviation
How spread out the data is. The two standard ways to measure variability — and why we usually use one rather than the other.
Counting & Combinatorics
How to count outcomes without enumerating them. Permutations, combinations, the binomial theorem, Pascal's triangle, and inclusion-exclusion.
Probability Basics
Sample spaces, events, and the rules for computing probabilities. The starting point of everything that follows.
Conditional Probability & Bayes
Updating your beliefs when new information arrives. Bayes's rule, the most useful single fact in applied probability.
Random Variables
A function from outcome to number. PMF, PDF, CDF; expectation as a balance point; variance and the LOTUS shortcut.
Distributions
From discrete (binomial, Poisson) to continuous (normal, exponential). The shapes that randomness takes in the wild.
Sampling Design
How you pick observations decides whether the rest of the chapter is meaningful. SRS, stratified, cluster, systematic — and the biases that wreck the others.
The Central Limit Theorem
Why the normal distribution shows up everywhere. The deepest single result in elementary statistics.
Inferential Statistics
From sample to population. Sampling distributions, standard error, confidence intervals — and the misinterpretation everyone falls into.
Hypothesis Testing
The framework for drawing conclusions from data. p-values, significance levels, and what they really mean.
Two-Sample Hypothesis Testing
Comparing two groups. Independent vs. paired designs, two-means t-tests (pooled and Welch), two-proportion z, and the effect-size question.
Chi-Square Tests
Pearson's invention for asking "do these counts fit my model?" Goodness-of-fit, independence, and homogeneity for categorical data.
ANOVA — Analysis of Variance
Comparing three or more group means at once. Variance partitioning, the F-distribution, and why this beats running many t-tests.
Regression & Correlation
Linear fits via least squares. The Pearson coefficient, R², residuals, Anscombe's quartet, and why correlation isn't causation.
Bayesian Statistics
Beliefs that update as data arrives. Priors, likelihoods, posteriors, conjugate families, and credible intervals.