Syllabus map

AA SL Statistics and Probability syllabus

The checklist below covers all Standard Level syllabus subtopics for Topic 4. Each interactive guide includes full notes, custom diagrams, 10 tiered exercises, formula summaries, and exam-style tests.

  1. 4.1Concepts of population, sample, random sample, discrete and continuous data; sampling techniques
  2. 4.2Presentation of data: frequency tables, histograms, cumulative frequency tables and ogives, box plots
  3. 4.3Measures of central tendency ($\bar{x}$, median, mode) and dispersion ($\text{IQR}$, $\sigma$, $\sigma^2$); linear transformations
  4. 4.4Correlation ($r$), scatter diagrams, least-squares line of regression of $y$ on $x$, interpolation and extrapolation
  5. 4.5Probability axioms, sample spaces, complementary events $P(A') = 1 - P(A)$, addition rule
  6. 4.6Venn diagrams, tree diagrams, conditional probability $P(A|B)$, and tests for independent events
  7. 4.7Discrete random variables, probability distribution functions, expected value $E(X)$, and fair games
  8. 4.8Binomial distribution $X \sim B(n, p)$, BINS assumptions, mean $np$, and variance $np(1-p)$
  9. 4.9Normal distribution $X \sim N(\mu, \sigma^2)$, bell curve symmetry, points of inflection, and empirical rule
  10. 4.10Equation of the regression line of $x$ on $y$ and prediction directionality
  11. 4.11Formal formulae for conditional probability and independence
  12. 4.12Standardization of normal variables: $Z = \frac{X - \mu}{\sigma} \sim N(0, 1)$ and inverse normal calculations

Study by topic

Statistics and Probability lessons

Work through focused interactive lesson guides. Each contains complete study notes, semantic diagrams, 10 tiered problems with collapsible solutions, formula booklets, and mini tests.

4.1–4.2Sampling and Data DisplaysSampling techniques, selection bias, frequency density in histograms, cumulative frequency ogives, box plots, and outlier boundaries.Open lesson 4.3Measures of Central Tendency and DispersionMean, median, mode, IQR, standard deviation, grouped variance, skewness ordering, and linear transformations $y = ax + b$.Open lesson 4.4, 4.10Correlation and Regression LinesPearson's $r$, scatter diagrams, least-squares line of $y$ on $x$, mean point $(\bar{x}, \bar{y})$, interpolation vs extrapolation, and causation.Open lesson 4.5, 4.6, 4.11Probability and Conditional EventsVenn diagrams, tree diagrams, addition rule $P(A \cup B)$, conditional probability $P(A|B)$, and mathematical tests for independence.Open lesson 4.7, 4.8Discrete Random Variables & Binomial DistributionProbability distributions, expected value $E(X)$, fair games, binomial model $X \sim B(n, p)$, BINS assumptions, and cumulative tails.Open lesson 4.9, 4.12Normal Distribution and Z-ValuesGaussian bell curve $N(\mu, \sigma^2)$, $z$-score standardization $Z = \frac{X - \mu}{\sigma}$, inverse normal percentiles, and finding unknown $\mu$ and $\sigma$.Open lesson

Learn the ideas

Key ideas and methods

Sampling, displays and summary statistics

  • A sample must represent the targeted population. Stratified sampling selects proportional random samples from mutually exclusive strata: $n_i = \frac{N_i}{N} \times n$. Convenience sampling is prone to severe selection bias.
  • In histograms with unequal class widths, vertical height represents frequency density ($\text{frequency} / \text{class width}$), ensuring bar area is proportional to frequency.
  • The mean $\bar{x}$ incorporates every value but is pulled by outliers; the median is resistant. Under $y = ax + b$, the mean shifts to $a\bar{x} + b$, standard deviation scales to $|a|\sigma$, while adding $b$ leaves dispersion completely unchanged!

Correlation, regression and probability

  • Pearson's $r$ ($-1 \le r \le 1$) quantifies linear correlation. Correlation measures association, never causation.
  • The line of regression of $y$ on $x$ passes through $(\bar{x}, \bar{y})$ and must ONLY be used to predict $y$ from $x$. Interpolating within $[x_{\min}, x_{\max}]$ is reliable; extrapolating beyond is unreliable.
  • The addition rule is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. For independent events, $P(A \cap B) = P(A)P(B)$ and $P(A|B) = P(A)$.

Random variables and distributions

  • For discrete random variables, $\sum P(X = x) = 1$ and $E(X) = \sum x P(X = x)$. A game is fair when $E(\text{Net Gain}) = 0$.
  • A Binomial variable $X \sim B(n, p)$ models $n$ independent binary trials with constant probability $p$: $P(X = r) = \binom{n}{r} p^r (1 - p)^{n - r}$, with mean $np$ and variance $np(1 - p)$.
  • A Normal variable $X \sim N(\mu, \sigma^2)$ is continuous and symmetrical about $\mu$. Standardize using $Z = \frac{X - \mu}{\sigma} \sim N(0, 1)$. When finding unknown $\mu$ and $\sigma$, set up simultaneous equations from standardized percentiles.

Follow the reasoning

Worked examples

These original examples show common reasoning steps and standard IB notation.

Worked example 1

Find a binomial probability

Question. A component is defective with probability $p = 0.1$, independently for each component. In a sample of $5$, find the probability that exactly one is defective.

  1. $X \sim B(5, 0.1)$ because trials are independent with constant $p = 0.1$.
  2. $P(X = 1) = \binom{5}{1} (0.1)^1 (0.9)^4 = 5(0.1)(0.6561) = 0.32805$.

Answer: The probability is $0.32805$ (or $\approx 0.328$).

Worked example 2

Standardize a normal value

Question. A normal variable has mean $\mu = 70$ and standard deviation $\sigma = 8$. Find the $z$-value for $x = 82$.

  1. Use $z = \frac{x - \mu}{\sigma}$.
  2. $z = \frac{82 - 70}{8} = \frac{12}{8} = 1.5$.

Answer: $z = 1.5$ (1.5 standard deviations above the mean).

Worked example 3

Use conditional probability

Question. In a cohort, $60\%$ study French ($F$), $40\%$ study Spanish ($S$), and $25\%$ study both. Find $P(F|S)$.

  1. Apply conditional formula: $P(F|S) = \frac{P(F \cap S)}{P(S)}$.
  2. Substitute: $\frac{0.25}{0.40} = 0.625$.

Answer: $P(F|S) = 0.625$.

Try it yourself

Practice with worked answers

  1. Question 1. A fair coin is tossed $6$ times. Find the probability of obtaining exactly $4$ heads.

    Show worked answer

    $X \sim B(6, 0.5)$. $P(X = 4) = \binom{6}{4}(0.5)^6 = 15 \times \frac{1}{64} = \frac{15}{64} \approx 0.234$.

  2. Question 2. Exam scores follow $X \sim N(68, 100)$ ($\sigma = 10$). Find the $z$-score for a mark of $83$.

    Show worked answer

    $z = \frac{83 - 68}{10} = \frac{15}{10} = 1.5$.

  3. Question 3. If $P(A) = 0.3$, $P(B) = 0.4$, and $A$ and $B$ are independent, calculate $P(A \cap B)$ and $P(A \cup B)$.

    Show worked answer

    $P(A \cap B) = P(A)P(B) = 0.3(0.4) = 0.12$. Addition rule: $P(A \cup B) = 0.3 + 0.4 - 0.12 = 0.58$.

Avoid these slips

Common mistakes

  • Assuming correlation proves a causal relationship.
  • Rearranging the regression line of $y$ on $x$ to predict $x$ from $y$.
  • Confusing mutually exclusive ($P(A \cap B) = 0$) with independent ($P(A \cap B) = P(A)P(B)$).
  • Forgetting that in $X \sim N(\mu, \sigma^2)$, the second parameter is variance, so you must take the square root to get $\sigma$.
  • Treating cumulative binomial inequalities carelessly (e.g. confusing $X < k$ with $X \le k$).

Before you move on

Revision checklist

  • Can I evaluate sampling methods for bias and interpret histograms using frequency density?
  • Can I calculate mean, median, IQR, and standard deviation and evaluate linear data transformations?
  • Can I interpret Pearson's $r$, verify that $(\bar{x}, \bar{y})$ lies on the regression line, and distinguish interpolation from extrapolation?
  • Can I solve conditional probability problems using Venn diagrams, tree diagrams, and independence tests?
  • Can I apply the binomial model $B(n, p)$ and calculate expected values for fair games?
  • Can I standardize normal random variables $Z = \frac{X - \mu}{\sigma}$ and solve for unknown $\mu$ and $\sigma$ using simultaneous equations?

About this guide

This is an independent SJMaths study resource, not an official IB publication. It follows the current AA guide first assessed in 2021 (final assessment November 2028).

Check the IB Mathematics programme page and the official AA subject guide for the authoritative syllabus.