Unit Overview

Statistics and probability

IB Mathematics Applications & Interpretation HL· 5 min read 📊 15-20% of overall exam

1. Unit at a Glance

This unit follows a logical learning arc, starting from foundational concepts of data collection and summary statistics, before moving into bivariate analysis, probability, probability distributions, and finally inferential statistics and graph theory. All topics emphasize real-world applications, aligned with the core focus of the IB AI HL syllabus.

Statistical topics build sequentially: you will first learn how to collect and describe data, then model random processes, then test hypotheses about populations using sample data. The final section on graph theory introduces discrete optimization tools widely used in operations research and data science.

Below are all sub-topics in this unit, ordered to follow the official syllabus sequence:

Data types, sampling, bias

Learn to classify data types and identify common sources of sampling bias.

★⏱ 5 min

Descriptive statistics: measures of center and spread

Calculate and interpret measures of center, spread, and visualize data distributions.

★★⏱ 6 min

Correlation and linear regression

Explore linear relationships between bivariate data and fit simple linear regression models.

★★⏱ 7 min

Probability concepts, conditional probability, independent events

Master core probability rules, conditional probability, and tests for independent events.

★★★⏱ 6 min

Probability distributions: expected value and variance

Calculate expected value and variance for discrete and continuous probability distributions.

★★★⏱ 6 min

Binomial, Poisson and normal distributions

Apply three common probability distributions to model real-world random processes.

★★★⏱ 8 min

Non-linear regression models

Transform non-linear relationships to fit linear models to bivariate data.

★★★⏱ 7 min

Chi-squared tests: goodness of fit and independence

Conduct chi-squared tests for goodness of fit and association between categorical variables.

★★★★⏱ 8 min

t-tests and confidence intervals for means

Construct confidence intervals and conduct t-tests for population means.

★★★★⏱ 8 min

Spearman's rank correlation coefficient

Calculate and interpret Spearman's rho for ranked bivariate data.

★★★⏱ 5 min

Graph theory: basic concepts

Learn core terminology and representations for graphs and networks.

★★⏱ 5 min

Graph theory: adjacency matrices, paths and cycles

Use adjacency matrices to count paths and identify cycles in graphs.

★★★⏱ 7 min

Spanning trees and minimum spanning tree algorithms

Apply Prim's and Kruskal's algorithms to find minimum spanning trees.

★★★★⏱ 8 min

Graph theory applications: CPP and TSP

Solve the Chinese Postman Problem and Traveling Salesman Problem using graph theory methods.

★★★★★⏱ 9 min

2. Common Pitfalls

Wrong move:

Confusing correlation with causation in regression analysis

Why:

A strong correlation does not prove that changes in one variable cause changes in the other

Correct move:

Always interpret correlation within context and acknowledge potential confounding variables

Wrong move:

Forgetting to check conditions before conducting hypothesis tests

Why:

Most hypothesis tests require specific conditions to produce valid results

Correct move:

Always verify conditions (expected frequency > 5, normality, independence) before interpreting test results

Wrong move:

Mixing up node degrees and adjacency matrix entries in graph theory

Why:

Adjacency matrix entries count edges between nodes, not node degrees

Correct move:

Remember that the degree of a node equals the sum of entries in its corresponding row/column in the adjacency matrix

3. Quick Reference Cheatsheet

Concept	Key Formula/Result
Pearson correlation coefficient	$r = \frac{\sum ( x _{i} - x ˉ ) ( y _{i} - y ˉ )}{\sum ( x _{i} - x ˉ ) ^{2} \sum ( y _{i} - y ˉ ) ^{2}}$
Linear regression line	$\overset{y}{^} = a x + b$ , where $a$ = slope, $b$ = y-intercept
Expected value (discrete distribution)	$E (X) = \sum x_{i} P (X = x_{i})$
Variance of a random variable	$V a r (X) = E (X^{2}) - [E (X)]^{2}$
Chi-squared test statistic	$χ^{2} = \sum \frac{( O _{i} - E _{i} ) ^{2}}{E _{i}}$
Spearman's rank correlation	$r_{s} = 1 - \frac{6 \sum d _{i}^{2}}{n ( n ^{2} - 1 )}$
Handshaking lemma	Sum of all node degrees = $2 \times$ number of edges

What's Next

Start with the first sub-topic of this unit to build foundational knowledge of data collection and bias, which is required for all subsequent statistical topics. Once you complete all sub-topics in this unit, move on to the next unit on calculus to continue your progress through the IB AI HL syllabus.