# Statistics & Probability

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u4-overview/
> Weight: 15-18% of overall IB AA SL examination

This unit covers core descriptive statistics, probability theory, and key common probability distributions for IB AA SL, building foundational data analysis skills that feature heavily across all exam sections.

**Prerequisites:** Basic algebraic manipulation and function properties; Introductory descriptive statistics at IGCSE/GCSE level

## Learning objectives

- Classify data types and identify sources of bias in common sampling methods
- Represent and summarize numerical data using appropriate graphs and measures of center and dispersion
- Apply core probability rules to calculate probabilities for conditional and independent events
- Model random processes using binomial and normal probability distributions to solve real-world problems

## Unit at a Glance

This unit progresses logically from foundational descriptive statistics to applied probability modelling. We start with how data is collected and classified, then cover how to represent and summarize data numerically, before moving into core probability rules and finally modelling random processes with the two key distributions required for SL.

All sub-topics build directly on prior content in this unit: understanding data types is required to select correct graphs and summary measures, while core probability fundamentals underpin all work on discrete and continuous probability distributions.

This unit is split into 8 core sub-topics:
- [Data types, sampling and bias](https://www.owlsprep.com/study/ib-math-aa-sl-u4-data-types-sampling-and-bias/) — Learn to classify data types and evaluate sampling methods for common sources of bias.
- [Data representation: histograms, box plots, cumulative frequency](https://www.owlsprep.com/study/ib-math-aa-sl-u4-data-representation-histograms-box-plots/) — Master constructing and interpreting common statistical graphs for grouped and ungrouped data.
- [Measures of central tendency and dispersion](https://www.owlsprep.com/study/ib-math-aa-sl-u4-measures-of-central-tendency-and/) — Calculate and compare mean, median, mode, variance, standard deviation and interquartile range.
- [Cumulative distributions and percentiles](https://www.owlsprep.com/study/ib-math-aa-sl-u4-cumulative-distributions-and-percentiles/) — Use cumulative frequency graphs to find percentiles, quartiles and medians for any dataset.
- [Probability, conditional probability and independent events](https://www.owlsprep.com/study/ib-math-aa-sl-u4-probability-conditional-probability-and-independent/) — Apply Venn diagrams, tree diagrams and probability rules to solve combined event problems.
- [Discrete probability distributions, expectation and variance](https://www.owlsprep.com/study/ib-math-aa-sl-u4-discrete-probability-distributions-expectation-and/) — Calculate expectation, variance and probabilities for general discrete probability distributions.
- [Binomial probability distribution](https://www.owlsprep.com/study/ib-math-aa-sl-u4-binomial-probability-distribution/) — Identify binomial conditions and calculate probabilities, mean and variance for binomial distributions.
- [Normal probability distribution](https://www.owlsprep.com/study/ib-math-aa-sl-u4-normal-probability-distribution/) — Calculate probabilities and inverse normal values for continuous normally distributed data.

## Common pitfalls

- **Wrong:** Skipping data classification before selecting a graph or summary measure
  - Why it fails: Using methods for continuous data on discrete categorical data leads to incorrect interpretation
  - Correct: Always classify your data type first before choosing how to analyze or represent it
- **Wrong:** Mixing up the denominator in conditional probability calculations
  - Why it fails: Dividing by P(A) instead of P(B) for $P(A|B)$ leads to wrong results even with correct intersection values
  - Correct: Explicitly write the condition into the formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$ to avoid confusion
- **Wrong:** Applying binomial or normal distributions without checking their conditions
  - Why it fails: Using the wrong distribution leads to incorrect results even if your calculation steps are correct
  - Correct: Always verify conditions (fixed trials for binomial, continuous symmetric data for normal) before proceeding

## Cheatsheet

| Concept / Formula | Key Note for IB SL |
| --- | --- |
| Sample Standard Deviation | $s = \sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}$ is used for sample data in IB |
| Conditional Probability | $P(A\|B) = \frac{P(A \cap B)}{P(B)}$ for $P(B) > 0$ |
| Independent Events | $P(A \cap B) = P(A) \times P(B)$ and $P(A\|B) = P(A)$ |
| Expectation of Discrete $X$ | $E(X) = \mu = \sum x \cdot P(X=x)$ |
| Variance of Discrete $X$ | $Var(X) = \sigma^2 = E(X^2) - [E(X)]^2$ |
| Binomial Distribution | $X \sim B(n, p)$, $E(X) = np$, $Var(X) = np(1-p)$ |
| Standardized Normal Value | $Z = \frac{X - \mu}{\sigma}$ for $X \sim N(\mu, \sigma^2)$ |
| Interquartile Range | $IQR = Q_3 - Q_1$, resistant to outlier values |

## What's next

Begin this unit with the first sub-topic on data classification, sampling and bias, which establishes core vocabulary for all subsequent statistics work in this unit. After you complete all 8 sub-topics in this Statistics & Probability unit, you can progress to the first topic of the next unit on calculus for IB AA SL.

- [Data types, sampling and bias (First sub-topic of this unit)](https://www.owlsprep.com/study/ib-math-aa-sl-u4-data-types-sampling-and-bias/)
- [Data representation: histograms, box plots, cumulative frequency](https://www.owlsprep.com/study/ib-math-aa-sl-u4-data-representation-histograms-box-plots/)
- [Measures of central tendency and dispersion](https://www.owlsprep.com/study/ib-math-aa-sl-u4-measures-of-central-tendency-and/)

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