# Statistics and probability

> IB Mathematics Applications & Interpretation HL · IB AI HL
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u5-overview/
> Weight: 15-20% of overall exam

This unit covers all core descriptive and inferential statistics, probability, and discrete graph theory for IB AI HL. Topics are heavily tested on both papers and form the foundation for applied data analysis.

**Prerequisites:** [Algebra and functions for IB AI HL](https://www.owlsprep.com/study/ib-math-ai-hl-u2-overview/)

## Learning objectives

- Classify data types and identify sources of bias in sampling methods for real-world datasets
- Calculate and interpret descriptive statistics, correlation, and regression models for bivariate data
- Apply core probability rules and common probability distributions to model real-world random processes
- Construct confidence intervals and conduct hypothesis tests for statistical inference on population parameters
- Use graph theory concepts and algorithms to solve discrete network optimization problems

## 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](https://www.owlsprep.com/study/ib-math-ai-hl-u5-data-types-sampling-bias/) — Learn to classify data types and identify common sources of sampling bias.
- [Descriptive statistics: measures of center and spread](https://www.owlsprep.com/study/ib-math-ai-hl-u5-descriptive-statistics-measures-of-center/) — Calculate and interpret measures of center, spread, and visualize data distributions.
- [Correlation and linear regression](https://www.owlsprep.com/study/ib-math-ai-hl-u5-correlation-and-linear-regression/) — Explore linear relationships between bivariate data and fit simple linear regression models.
- [Probability concepts, conditional probability, independent events](https://www.owlsprep.com/study/ib-math-ai-hl-u5-probability-concepts-conditional-probability-independent/) — Master core probability rules, conditional probability, and tests for independent events.
- [Probability distributions: expected value and variance](https://www.owlsprep.com/study/ib-math-ai-hl-u5-probability-distributions-expected-value-and/) — Calculate expected value and variance for discrete and continuous probability distributions.
- [Binomial, Poisson and normal distributions](https://www.owlsprep.com/study/ib-math-ai-hl-u5-binomial-poisson-and-normal-distributions/) — Apply three common probability distributions to model real-world random processes.
- [Non-linear regression models](https://www.owlsprep.com/study/ib-math-ai-hl-u5-non-linear-regression-models/) — Transform non-linear relationships to fit linear models to bivariate data.
- [Chi-squared tests: goodness of fit and independence](https://www.owlsprep.com/study/ib-math-ai-hl-u5-chi-squared-tests-goodness-of/) — Conduct chi-squared tests for goodness of fit and association between categorical variables.
- [t-tests and confidence intervals for means](https://www.owlsprep.com/study/ib-math-ai-hl-u5-t-tests-and-confidence-intervals/) — Construct confidence intervals and conduct t-tests for population means.
- [Spearman's rank correlation coefficient](https://www.owlsprep.com/study/ib-math-ai-hl-u5-spearman-s-rank-correlation-coefficient/) — Calculate and interpret Spearman's rho for ranked bivariate data.
- [Graph theory: basic concepts](https://www.owlsprep.com/study/ib-math-ai-hl-u5-graph-theory-basic-concepts/) — Learn core terminology and representations for graphs and networks.
- [Graph theory: adjacency matrices, paths and cycles](https://www.owlsprep.com/study/ib-math-ai-hl-u5-graph-theory-adjacency-matrices-paths/) — Use adjacency matrices to count paths and identify cycles in graphs.
- [Spanning trees and minimum spanning tree algorithms](https://www.owlsprep.com/study/ib-math-ai-hl-u5-spanning-trees-and-minimum-spanning/) — Apply Prim's and Kruskal's algorithms to find minimum spanning trees.
- [Graph theory applications: CPP and TSP](https://www.owlsprep.com/study/ib-math-ai-hl-u5-graph-theory-applications-cpp-and/) — Solve the Chinese Postman Problem and Traveling Salesman Problem using graph theory methods.

## Common pitfalls

- **Wrong:** Confusing correlation with causation in regression analysis
  - Why it fails: A strong correlation does not prove that changes in one variable cause changes in the other
  - Correct: Always interpret correlation within context and acknowledge potential confounding variables
- **Wrong:** Forgetting to check conditions before conducting hypothesis tests
  - Why it fails: Most hypothesis tests require specific conditions to produce valid results
  - Correct: Always verify conditions (expected frequency > 5, normality, independence) before interpreting test results
- **Wrong:** Mixing up node degrees and adjacency matrix entries in graph theory
  - Why it fails: Adjacency matrix entries count edges between nodes, not node degrees
  - Correct: Remember that the degree of a node equals the sum of entries in its corresponding row/column in the adjacency matrix

## Cheatsheet

| Concept | Key Formula/Result |
| --- | --- |
| Pearson correlation coefficient | $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$ |
| Linear regression line | $\hat{y} = ax + b$, where $a$ = slope, $b$ = y-intercept |
| Expected value (discrete distribution) | $E(X) = \sum x_i P(X=x_i)$ |
| Variance of a random variable | $Var(X) = E(X^2) - [E(X)]^2$ |
| Chi-squared test statistic | $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ |
| 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.

- [Data types, sampling, bias](https://www.owlsprep.com/study/ib-math-ai-hl-u5-data-types-sampling-bias/)
- [Descriptive statistics: measures of center and spread](https://www.owlsprep.com/study/ib-math-ai-hl-u5-descriptive-statistics-measures-of-center/)
- [Correlation and linear regression](https://www.owlsprep.com/study/ib-math-ai-hl-u5-correlation-and-linear-regression/)

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