# Statistics and Probability

> IB Mathematics AI SL · IB AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u4-overview/
> Weight: 24-26% of overall exam

This unit covers core data analysis and probability concepts, the largest weighted section of the IB Mathematics AI SL syllabus, with heavy focus on real-world applied problems for both papers.

**Prerequisites:** Foundational arithmetic and algebraic manipulation skills

## Learning objectives

- Classify data types and select appropriate methods to represent and summarise real-world data sets
- Calculate and interpret key measures of central tendency and spread for ungrouped and grouped data
- Apply probability rules to solve problems involving combined events, conditional probability and independence
- Model real-world scenarios with binomial and normal distributions and calculate relevant probabilities and values

## Unit at a Glance

This unit follows a clear logical learning arc: we start with foundational data classification, move to visualising and summarising data, explore relationships between variables, build up core probability rules, and end with the two most common probability distributions tested at AI SL.

Nearly all questions in this unit are context-based, and many require interpreting output from your GDC, so building confidence with technological tools for statistical calculations is a key theme throughout your learning.

This unit is split into 10 focused sub-topics to learn step-by-step:
- [Discrete and continuous data types](https://www.owlsprep.com/study/ib-math-ai-sl-u4-discrete-and-continuous-data-types/) — Learn the core classification of data types, the foundation for all subsequent statistical analysis.
- [Sampling methods and bias](https://www.owlsprep.com/study/ib-math-ai-sl-u4-sampling-and-bias/) — Distinguish populations from samples, apply the main sampling methods, and recognise sources of bias.
- [Data representation: histograms, box plots, cumulative frequency](https://www.owlsprep.com/study/ib-math-ai-sl-u4-data-representation-histograms-box-plots/) — Create and interpret common data visualisations for both discrete and continuous data sets.
- [Measures of central tendency](https://www.owlsprep.com/study/ib-math-ai-sl-u4-measures-of-central-tendency/) — Calculate and compare mean, median and mode for ungrouped and grouped data.
- [Measures of spread: variance, standard deviation](https://www.owlsprep.com/study/ib-math-ai-sl-u4-measures-of-spread-variance-standard/) — Quantify spread in data using range, interquartile range, variance and standard deviation.
- [Correlation and linear regression](https://www.owlsprep.com/study/ib-math-ai-sl-u4-correlation-and-linear-regression/) — Measure linear correlation between two variables and fit regression lines for prediction.
- [Basic probability concepts and combined events](https://www.owlsprep.com/study/ib-math-ai-sl-u4-basic-probability-concepts-and-combined/) — Learn core probability rules and solve problems for combined events using Venn diagrams.
- [Conditional probability and independence](https://www.owlsprep.com/study/ib-math-ai-sl-u4-conditional-probability-and-independence/) — Calculate conditional probability and test for independence of two events.
- [Discrete probability distributions](https://www.owlsprep.com/study/ib-math-ai-sl-u4-discrete-probability-distributions/) — Understand properties of discrete distributions and calculate expected value.
- [Binomial distribution](https://www.owlsprep.com/study/ib-math-ai-sl-u4-binomial-distribution/) — Identify binomial conditions and calculate probabilities, expectation and variance.
- [Normal distribution and applications](https://www.owlsprep.com/study/ib-math-ai-sl-u4-normal-distribution-and-applications/) — Apply the normal distribution to calculate probabilities and inverse values for real problems.

## Common pitfalls

- **Wrong:** Confusing discrete and continuous data when choosing the correct graph or calculation method.
  - Why it fails: Exam questions often penalise incorrect classification leading to wrong approaches.
  - Correct: Always check if data can only take specific values (discrete) or any value in a range (continuous) first.
- **Wrong:** Using sample standard deviation instead of population standard deviation for GDC calculations.
  - Why it fails: AI SL questions almost always require population standard deviation unless explicitly stated otherwise.
  - Correct: Confirm which standard deviation value your question asks for before reporting your answer.
- **Wrong:** Applying the binomial distribution without checking its required conditions.
  - Why it fails: You will lose method marks for not verifying the model is appropriate for the problem.
  - Correct: Always check for fixed trials, two outcomes, independent trials and constant probability before using binomial.

## Cheatsheet

| Concept / Formula | Description |
| --- | --- |
| Mean (ungrouped data) | $\bar{x} = \frac{\sum x}{n}$ |
| Mean (grouped data) | $\bar{x} = \frac{\sum fx}{\sum f}$ |
| Pearson's $r$ | Measures strength of linear correlation between $-1$ and $1$ |
| Combined events probability | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ |
| Conditional probability | $P(A\|B) = \frac{P(A \cap B)}{P(B)}$ |
| Binomial expectation | $E(X) = np$ |
| Binomial variance | $Var(X) = np(1-p)$ |
| Normal distribution notation | $X \sim N(\mu, \sigma^2)$ where $\mu$ = mean, $\sigma^2$ = variance |

## What's next

Start your learning of this unit with the first sub-topic, which introduces core data classification that all later statistical work builds on. Once you complete all 10 sub-topics in this unit, you will move on to the next unit covering core calculus concepts for applications.

- [Discrete and continuous data types](https://www.owlsprep.com/study/ib-math-ai-sl-u4-discrete-and-continuous-data-types/)
- [Unit 5: Calculus](https://www.owlsprep.com/study/ib-math-ai-sl-u5-overview/)
- [Data representation: histograms, box plots, cumulative frequency](https://www.owlsprep.com/study/ib-math-ai-sl-u4-data-representation-histograms-box-plots/)

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From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-sl-u4-overview/