Knowledge organisers

Statistics and probability: IB Maths AA SL 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

Independent events: A and B are independent when P(A ∩ B) = P(A) × P(B): one happening does not change the chance of the other.
Conditional probability: P(A | B) is the probability of A given that B has happened: P(A ∩ B) / P(B).
Discrete random variable: A variable that takes separate values, each with a probability; the probabilities add to 1.
Normal distribution: A symmetric, bell-shaped distribution set by its mean μ and standard deviation σ.

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.

Interquartile rangeIn the formula booklet	\(IQR=Q_3-Q_1\)
Mean (\(n=\sum f_i\))In the formula booklet	\(\bar x=\frac{\sum f_ix_i}{n}\)
Outliers	\(x \(\text{or}\) \(x>Q_3+1.5\,IQR\)
Variance (s.d. \(\sigma\))	\(\sigma^2=\frac{\sum f(x-\mu)^2}{n}=\frac{\sum fx^2}{n}-\mu^2\)
Data \(\times a\) then \(+b\)	\(\text{mean}\to a\bar x+b,\) \(\text{s.d.}\to\|a\|\sigma\)
Regression: use \(y\) on \(x\) to predict \(y\), \(x\) on \(y\) to predict \(x\); interpolate, don't extrapolate
ProbabilityIn the formula booklet	\(P(A)=\frac{n(A)}{n(U)},\) \(P(A)+P(A')=1\)
Combined eventsIn the formula booklet	\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Mutually exclusiveIn the formula booklet	\(P(A\cup B)=P(A)+P(B)\)
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	\(E(X)=\sum x\,P(X=x)\)
Valid distribution; fair game	\(\sum P(X=x)=1;\) \(E(\text{gain})=0\)
Binomial \(X\sim B(n,p)\)	\(P(X=r)=\binom nrp^r(1-p)^{n-r}\)
Binomial mean, varianceIn the formula booklet	\(E(X)=np,\) \(\mathrm{Var}(X)=np(1-p)\)
Standardised normalIn the formula booklet	\(z=\frac{x-\mu}{\sigma}\)

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

Worked example

X ~ B(10, 0.3). Find P(X = 2).

P(X = 2) = ¹⁰C₂ × 0.3² × 0.7⁸
= 45 × 0.09 × 0.0576…

Answer: P(X = 2) = 0.233 (3 s.f.)

Common mistakes

Probability: assuming independence, and conditional probability with complements
Binomial and normal: "at least", inequalities, rounding bounds and which distribution
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.
Draw a line of best fit through the mean point, find the regression line of y on x with technology and use it to predict within the data range.
Use tree diagrams and sample space diagrams to find probabilities of combined events, with and without replacement, using P(A ∪ B) for overlapping events.
Find normal probabilities with technology and use the inverse normal to find the value with a given probability below it, such as the top 5% boundary.
Use P(A | B) = P(A ∩ B)/P(B) and test whether events are independent by checking P(A ∩ B) = P(A)P(B).
Combine standardisation and the binomial distribution in multi-part questions, such as the number of items outside a tolerance.

The printable sheet

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

Revise it next

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