Knowledge organisers

Statistics and probability: IB Maths AA 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.

Download the A4 sheet (PDF)

Key definitions

Bayes' theorem: Reverses a conditional probability: P(B | A) = P(B)P(A | B) / P(A).
Probability density function: For a continuous variable, f(x) ≥ 0 and the total area under f is 1.
Expected value: E(X) is the long-run mean of X: Σx P(X = x), or ∫x f(x) dx for a continuous variable.
Variance: Var(X) = E(X²) − [E(X)]² measures spread around the mean.

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	\( \(\text{or}\) \(>Q_3+1.5\,IQR\)
Variance	\(\sigma^2=\frac{\sum f(x-\mu)^2}{n}=\frac{\sum fx^2}{n}-\mu^2\)
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	\(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\)
BayesIn the formula booklet	\(P(B\mid A)=\frac{P(B)P(A\mid B)}{P(B)P(A\mid B)+P(B')P(A\mid B')}\)
Bayes, partition \(B_1,B_2,B_3\)In the formula booklet	\(P(B_i\mid A)=\frac{P(B_i)P(A\mid B_i)}{\sum_jP(B_j)P(A\mid B_j)}\)
Discrete \(X\)In the formula booklet	\(E(X)=\sum xP(X=x),\) \(\mathrm{Var}(X)=E(X^2)-[E(X)]^2\)
Binomial pmf	\(P(X=r)=\tbinom nrp^r(1-p)^{n-r}\)
Binomial mean, varianceIn the formula booklet	\(np,\) \(np(1-p)\)
Standardised normalIn the formula booklet	\(z=\frac{x-\mu}\sigma\)
Continuous \(X\)In the formula booklet	\(E(X)=\int x\,f(x)\,dx,\) \(\mathrm{Var}(X)=\int x^2f(x)\,dx-\mu^2\)
pdf; median \(m\)	\(\int f(x)\,dx=1;\) \(\int_{-\infty}^mf(x)\,dx=\tfrac12;\) \(\text{mode: max of }f\)
Linear transformationIn the formula booklet	\(E(aX+b)=aE(X)+b,\) \(\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)\)

More formulas are on the full IB Maths AA HL formula sheet.

Worked example

X has probability density function f(x) = kx for 0 ≤ x ≤ 2. Find k and E(X).

∫₀² kx dx = 2k = 1, so k = ½
E(X) = ∫₀² x × ½x dx = [x³/6]₀² = 8/6

Answer: k = ½ and E(X) = 4/3

Common mistakes

Continuous random variables: mode, median and expectation from a pdf
Discrete probability: independence assumed, E(X²) and Var misused
Using the class boundary instead of the midpoint when estimating the mean from grouped data
Plotting cumulative frequency at class midpoints instead of upper class boundaries

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 simple random, systematic, stratified, quota and convenience sampling, and identify sources of bias.
Find least-squares regression lines of y on x and x on y with technology, choose the right one for a prediction and explain why correlation is not causation.
Find the variance of a discrete random variable, model situations with B(n, p) and find binomial probabilities, mean and variance with technology.
Standardise normal values with z = (x − μ)/σ and use the inverse normal to find an unknown mean or standard deviation from given probabilities.
Work with piecewise probability density functions and use E(aX + b) and Var(aX + b) to find the mean and variance of transformed variables.
Use Bayes' theorem with tree diagrams in contexts such as medical testing and interpret the result.

The printable sheet

Statistics and probability knowledge organiser for IB Maths AA HL: one A4 page of key definitions, formulas, a worked example and common mistakes — Statistics and probability knowledge organiser (IB Maths AA HL), A4. Download the PDF.

Revise it next

Other IB Maths AA HL topics: Number and algebra · Functions · Geometry and trigonometry · Calculus · All IB Maths AA HL organisers