Knowledge organisers

Statistics and probability: IB Maths AI 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

Pearson's r: A number from −1 to 1 measuring the strength and direction of a linear relationship.
Regression line: The line of best fit of y on x, used to predict y from a value of x inside the data range.
Null hypothesis: H₀, the statement of no effect or no association that a test assumes until the evidence says otherwise.
p-value: The probability, assuming H₀ is true, of a result at least as extreme as the one observed.

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 \(\text{or}\) \(x>Q_3+1.5\,IQR\)
Standard deviation	\(\sigma=\sqrt{\frac{\sum f(x-\bar x)^2}{n}}\)
Data \(\times a\) then \(+b\)	\(\text{mean}\to a\bar x+b,\) \(\text{s.d.}\to\|a\|\sigma\)
Regression: \(y=ax+b\) (GDC); use to predict \(y\) within the data range only
\(\|r\|\) near 1: strong linear correlation; Spearman's \(r_s\) = PMCC of the ranks
ProbabilityIn the formula booklet	\(P(A)=\frac{n(A)}{n(U)},\) \(P(A)+P(A')=1\)
Expected number of occurrences	\(n\times P(A)\)
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 \(X\sim B(n,p)\)	\(P(X=r)=\binom nrp^r(1-p)^{n-r}\)

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

Worked example

X ~ N(50, 4²). Find P(X < 56).

z = (56 − 50)/4 = 1.5
P(X < 56) = P(Z < 1.5)

Answer: P(X < 56) = 0.933 (3 s.f.)

Common mistakes

χ² and t-tests: hypotheses, the comparison and the conclusion in context
Probability: adding instead of multiplying, conditional sample spaces, without replacement
Using the class boundary instead of the midpoint when estimating the mean from grouped data
Calling a convenience sample random, or labelling a value an outlier without checking the Q₁ − 1.5×IQR and Q₃ + 1.5×IQR fences

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 and outliers in data collection.
Find probabilities from sample spaces and Venn diagrams, use the complement rule and calculate the expected number of times an event occurs.
Set up a probability distribution for a discrete random variable, use the total probability of 1, find E(X) and decide whether a game is fair.
Organise data in contingency tables and calculate expected frequencies under the assumption of independence, ready for a χ² test.
Read the test statistic and p-value from technology and write the hypotheses, the decision and the conclusion in context.
Solve multi-part questions that combine normal probabilities with a binomial model.

The printable sheet

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

Revise it next

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