IB Mathematics: AA SL topic guide

Statistics and Probability

Statistics and Probability is a core part of IB Mathematics: AA SL. This guide connects the syllabus ideas behind Introduction to Statistics and Probability, Sampling and Presenting Data, Measures of Central Tendency and Dispersion, Correlation and Regression, Probability Fundamentals and 3 more units, shows how they appear in worked problems, and points you to the formulas and full lessons needed for exam revision.

What you will learn

Statistics and Probability syllabus outline

The units below follow the structure used in the full Study to Learn course. Use the outline to identify exactly which idea needs attention, then work through the public example before continuing to the complete lesson path.

4.0

Introduction to Statistics and Probability

What You Will Learn in Statistics and Probability

4.1

Sampling and Presenting Data

Sampling Techniques · Presenting Data

4.2

Measures of Central Tendency and Dispersion

Mean, Median & Mode · Range, IQR, Variance & Standard Deviation

4.3

Correlation and Regression

Scatter Diagrams & Pearson's r · The Regression Line

4.4

Probability Fundamentals

Sample Space & Basic Probability · Venn/Tree Diagrams & Conditional Probability

4.5a

Discrete Random Variables and Expected Value

Discrete Random Variables & Expected Value

4.5b

The Binomial Distribution

The Binomial Distribution · Binomial Distribution — Applications and Calculations

4.6

The Normal Distribution

Standardization & Probabilities · Inverse Normal and Applied Problems

Free worked preview

What You Will Learn in Statistics and Probability

This complete preview comes from the Introduction to Statistics and Probability unit. It introduces the core language, shows the method in context, and gives you a real example of the lesson quality before you create an account.

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 sigma2\\sigma^2 and standard deviation sigma\\sigma; and how linear transformations y=ax+by=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(AcupB)=P(A)+P(B)−P(AcapB)P(A\\cup B)=P(A)+P(B)-P(A\\cap B), conditional probability P(A∣B)P(A|B), tree and Venn diagrams, and the test for independence
  • Discrete Random Variables and Expected Value (4.5a): probability distributions, E(X)=sumxiP(X=xi)E(X)=\\sum x_i P(X=x_i), and the concept of a fair game
  • The Binomial Distribution (4.5b): XsimB(n,p)X\\sim B(n,p), the four conditions for a binomial model, and the formulas E(X)=np,textVar(X)=np(1−p)E(X)=np,\\ \\text{Var}(X)=np(1-p)
  • The Normal Distribution (4.6): the bell curve XsimN(mu,sigma2)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)=fracP(B∣A)P(A)P(B)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 textVar(X−Y)=textVar(X)+textVar(Y)\\text{Var}(X-Y)=\\text{Var}(X)+\\text{Var}(Y) (variances add even for a difference), and the variance of a sample mean
How to Approach This SectionStatistics is as much about interpretation as calculation. A computed mean of 72 means nothing until you know whether the distribution is symmetric or skewed. A correlation of r = 0.9 means nothing until you have looked at the scatter plot to confirm the relationship is actually linear (and that no lurking variable explains both quantities). Always ask: what does this number actually say about the real-world situation it describes? In probability, draw the tree diagram or Venn diagram before reaching for a formula — a visual representation catches errors that algebra alone might miss. The binomial and normal distributions are not just two more formulas to memorise; they are models, and the single most important skill is recognising which model fits a described scenario before applying its formulas. For Bayes' theorem, the tree-diagram approach — "path through the state you want, divided by the sum of all paths ending at the observation" — is almost always clearer than plugging directly into the formula.

Use the formulas

Keep the formula booklet beside you

Revision becomes faster when you practise locating the right equation and understand what every variable means. The full reference is public and searchable on any device.

Open IB Math AA SL Formula Booklet →

Reviewed by the Study to Learn editorial team · Updated 2026-07-24