IB Math RevisionStatistics and probability: IB Maths AI HL knowledge organiser
Everything to know about statistics and probability on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.
Key definitions
- Poisson distribution
- Counts random events in a fixed interval at a constant mean rate m; its mean and variance are both m.
- Transition matrix
- Holds the probabilities of moving from each state to each state in one step of a Markov chain.
- Confidence interval
- A range of values, worked out from a sample, that is likely to contain the population parameter.
- Type I error
- Rejecting H₀ when it is in fact true; its probability is the significance level.
Key formulas
Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.
| IQR; meanIn the formula booklet | \(IQR=Q_3-Q_1,\) \(\bar x=\frac{\sum f_ix_i}{n}\) |
| Outliers | \( |
| ProbabilityIn the formula booklet | \(P(A)=\frac{n(A)}{n(U)},\) \(P(A)+P(A')=1\) |
| Combined eventsIn the formula booklet | \(P(A\cup B)=P(A)+P(B)\) \(-P(A\cap B)\) |
| IndependentIn the formula booklet | \(P(A\cap B)=P(A)P(B)\) |
| ConditionalIn the formula booklet | \(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\) |
| Expected valueIn the formula booklet | \(E(X)=\sum xP(X=x)\) |
| Variance | \(\mathrm{Var}(X)=E(X^2)-[E(X)]^2\) |
| Binomial pmf | \(P(X=r)=\tbinom nrp^r(1-p)^{n-r}\) |
| Binomial mean, varianceIn the formula booklet | \(np,\) \(np(1-p)\) |
| Poisson \(X\sim\mathrm{Po}(m)\) | \(P(X=x)=\frac{m^xe^{-m}}{x!}\) |
| Poisson mean, varianceIn the formula booklet | \(E(X)=m,\) \(\mathrm{Var}(X)=m\) |
| Sum of independent Poissons | \(\mathrm{Po}(m_1)+\mathrm{Po}(m_2)=\mathrm{Po}(m_1+m_2)\) |
| Linear combinationsIn the formula booklet | \(E(a_1X_1\pm a_2X_2)=a_1E(X_1)\pm a_2E(X_2)\) |
| … independent \(X_i\)In the formula booklet | \(\mathrm{Var}(a_1X_1\pm a_2X_2)=a_1^2\mathrm{Var}(X_1)+a_2^2\mathrm{Var}(X_2)\) |
| Unbiased variance estimateIn the formula booklet | \(s^2_{n-1}=\frac{n}{n-1}s^2_n\) |
More formulas are on the full IB Maths AI HL formula sheet.
Worked example
X ~ Po(3). Find P(X = 2).
- P(X = 2) = e⁻³ × 3² / 2!
- = 4.5e⁻³
Answer: P(X = 2) = 0.224 (3 s.f.)
Common mistakes
- Hypothesis tests: hypotheses in words, the wrong test, and conclusions that don't follow
- Distributions: variances add, the normal is continuous, the binomial coefficient
- Treating a convenience sample as if it were random
- Using the class boundary instead of the midpoint when estimating the mean from grouped data
More on what examiners see students get wrong: Examiner Insights.
You should be able to…
- Distinguish populations from samples and discrete from continuous data, compare sampling techniques and identify sources of bias.
- Find conditional probabilities from Venn diagrams, trees and tables, and decide whether two events are independent.
- Decide whether a binomial, Poisson or normal model fits a situation, justify the choice and use it to calculate probabilities in context.
- Find the mean and variance of linear combinations of independent random variables, such as the total weight of several items packed together.
- Carry out hypothesis tests for a population mean using z or t with technology, stating hypotheses and concluding in context.
- Carry out χ² goodness-of-fit tests to given binomial, Poisson and normal models with technology and conclude in context.
The printable sheet
Revise it next
- IB Maths AI HL revision notes: Statistics and probability
- Practise statistics and probability questions
- Skill Builders
- IB Maths AI HL formula sheet (PDF)
Other IB Maths AI HL topics: Number and algebra · Functions · Geometry and trigonometry · Calculus · All IB Maths AI HL organisers