Statistics and Probability

Introduction to Statistics and Probability

Statistics and probability give you the tools to reason about uncertainty — to quantify risk, to draw conclusions from incomplete data, and to recognise when a pattern is genuine rather than coincidental. Unlike the deterministic world of algebra and calculus, where one input guarantees one output, this section deals in distributions, likelihoods, and expected values. You will learn to summarise data with numbers (mean, median, standard deviation) and pictures (histograms, box plots, scatter diagrams), to model random processes with the binomial and normal distributions, and — at HL — to update your beliefs in light of new evidence using Bayes' theorem. The analytical rigour of AA is present throughout: you will prove that expectation is linear, that variances add for independent variables, and that a correlation coefficient measures only linear association, not causation.

Section Map — What Is Covered

• Sampling and Presenting Data (4.1): simple random, stratified, systematic, and convenience sampling; bias; and data displays — histograms, box plots, cumulative frequency graphs
• Measures of Central Tendency and Dispersion (4.2): mean, median, mode; range, IQR, variance $\\sigma^2$ and standard deviation $\\sigma$; and how linear transformations $y=ax+b$ affect mean and spread differently
• Correlation and Regression (4.3): scatter diagrams, Pearson's r measuring linear association, and the least-squares regression line for prediction — with the crucial distinction between interpolation and extrapolation
• Probability Fundamentals (4.4): sample spaces, the addition rule $P(A\\cup B)=P(A)+P(B)-P(A\\cap B)$, conditional probability $P(A|B)$, tree and Venn diagrams, and the test for independence
• Discrete Random Variables and Expected Value (4.5a): probability distributions, $E(X)=\\sum x_i P(X=x_i)$, and the concept of a fair game
• The Binomial Distribution (4.5b): $X\\sim B(n,p)$, the four conditions for a binomial model, and the formulas $E(X)=np,\\ \\text{Var}(X)=np(1-p)$
• The Normal Distribution (4.6): the bell curve $X\\sim N(\\mu,\\sigma^2)$, the 68-95-99.7 rule, z-scores, and inverse normal calculations for percentile problems
• Bayes' Theorem (4.7, HL): reversing conditional probabilities $P(A|B)=\\frac{P(B|A)P(A)}{P(B)}$ — the mathematical engine behind medical diagnosis and the surprising result that a "95% accurate" test can still leave a low probability of actually having a rare disease
• Further Expectation and Variance (4.8, HL): linear combinations of random variables, the fact that $\\text{Var}(X-Y)=\\text{Var}(X)+\\text{Var}(Y)$ (variances add even for a difference), and the variance of a sample mean