Unit Overview

Statistics & Probability

IB Mathematics Analysis and Approaches Higher Level· 5 min read 📊 12-14% of total exam

1. Unit at a Glance

This unit follows the natural flow of a full statistical investigation, building incrementally from foundational concepts to advanced inference. You will start by learning how data is collected and described, then model randomness with probability, before moving to the HL-only content that lets you draw formal conclusions about populations from sample data.

Core content required for all IB AA HL students comes first, followed by HL-only extension topics that feature heavily in exam questions. All sub-topics build on the previous ones, so it is best to complete them in order.

This unit is split into the following sub-topics:

Data types and sampling

Distinguish between different data types and evaluate sampling methods for sources of bias.

★⏱ 5 min

Data presentation and summary statistics

Learn to visualize data and calculate key summary measures like mean, variance and standard deviation.

★★⏱ 7 min

Basic probability concepts and rules

Master fundamental probability definitions, set notation and core rules for combined events.

★★⏱ 6 min

Conditional probability and Bayes' theorem

Extend probability to conditional events and use Bayes' theorem to update probabilities with new information.

★★★⏱ 7 min

Discrete probability distributions

Learn the properties of discrete random variables and calculate their expected value and variance.

★★⏱ 6 min

Binomial and Poisson distributions

Explore two common discrete distributions, their properties, and real-world applications.

★★★⏱ 8 min

Continuous probability distributions and PDFs

Understand continuous random variables, probability density functions and how to calculate probabilities.

★★★⏱ 7 min

Normal distribution

Master the most widely used continuous distribution, including standardization and quantile calculation.

★★★⏱ 8 min

Bivariate data: correlation and regression

Analyze relationships between two variables, calculate correlation and fit linear regression lines.

★★★⏱ 8 min

t-distribution and confidence intervals (HL only)

Construct and interpret confidence intervals for unknown population means using the t-distribution.

★★★★⏱ 9 min

Hypothesis testing (HL only)

Learn the hypothesis testing framework and conduct one/two-tailed tests for population means.

★★★★⏱ 10 min

Chi-squared tests (HL only)

Apply chi-squared tests for goodness of fit, independence and homogeneity of categorical data.

★★★★⏱ 9 min

2. Common Pitfalls

Wrong move:

Confusing population parameters with sample statistics

Why:

This leads to incorrect interpretation of inference results and misstates the goal of statistical analysis

Correct move:

Use consistent notation: Greek letters for population parameters, Latin letters for sample statistics

Wrong move:

Assuming correlation implies causation in bivariate analysis

Why:

Correlation only measures association, not a causal relationship between two variables

Correct move:

Interpret correlation cautiously, and only conclude causation for properly designed experiments

Wrong move:

Treating PDF values as probabilities for continuous distributions

Why:

PDF values can be greater than 1, which is often confusing for new learners

Correct move:

Remember that only areas (integrals) under the PDF curve equal probabilities

3. Quick Reference Cheatsheet

Concept	Key Formula/Rule
Bayes' Theorem	$P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}$
Expected Value (Discrete)	$E [X] = \sum x P (X = x)$
Variance of a Random Variable	$V a r (X) = E [X^{2}] - (E [X])^{2}$
Binomial Distribution (Mean/Variance)	$μ = n p, σ^{2} = n p (1 - p)$
Poisson Distribution (Mean/Variance)	$μ = λ, σ^{2} = λ$
Pearson Correlation Coefficient	$r = \frac{\sum ( x - x ˉ ) ( y - y ˉ )}{s _{x} s _{y} ( n - 1 )}$
t-confidence Interval for Mean	$\overset{x}{ˉ} \pm t^{*} \frac{s}{n}$
Chi-squared Test Statistic	$χ^{2} = \sum \frac{( O - E ) ^{2}}{E}$

What's Next

Start with the first sub-topic of this unit to build your foundational knowledge of statistics and probability, following the sequence in the unit index above to master concepts incrementally. Once you complete all sub-topics in this unit, you can move on to the next core unit on differential calculus for IB AA HL.