Everyone or everything you want information about.
Sample
The part of the population you actually collect data from.
Discrete data
Countable values (shoe size, number of goals).
Continuous data
Measured values (height, time).
Bias
A systematic error that makes the sample unrepresentative.
Outlier
More than 1.5 × IQR below Q₁ or above Q₃.
Simple random: every member has an equal chance (random number generator). Systematic: every \(k\)th member from a random start. Stratified: split into groups and sample each in proportion. Quota: fill a fixed number per group, not randomly. Convenience: whoever is easiest to reach.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: When asked to criticise a sample, name the type of sampling and give a specific group that would be over- or under-represented.
Lists each value or class with how often it occurs.
Histogram
Bars with no gaps for continuous data (AI uses equal class widths).
Cumulative frequency
Running total of frequencies, plotted at the upper class boundary.
Box-and-whisker plot
Shows min, Q₁, median, Q₃, max (and outliers as crosses).
On a cumulative frequency graph with \(n\) values: median at \(\frac n2\), \(Q_1\) at \(\frac n4\), \(Q_3\) at \(\frac{3n}{4}\); percentiles in the same way.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Comparing two box plots? Two sentences: “The median of A is higher, so A … on average. The IQR of B is larger, so B is more varied.”
4.3Measures of central tendency and dispersion
Key terms
Mean
Sum of values ÷ number of values.
Median
The middle value when data are in order.
Mode / modal class
The most frequent value / class.
Standard deviation σ
Typical distance of values from the mean.
Variance
σ²
IQR
Q₃ − Q₁: spread of the middle half.
For grouped data, use class mid-points to estimate the mean (it is only an estimate).
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Write “estimated mean” for grouped data. Using the word “estimate” shows you understand the mid-point method.
4.4Linear correlation and regression
Key terms
Scatter diagram
Plots pairs (x, y) to show a relationship.
Pearson's r
Measures the strength of a linear relationship: −1 ≤ r ≤ 1.
Line of best fit
Passes through the mean point (x̄, ȳ).
Regression line of y on x
y = ax + b found by the GDC; used to predict y from x.
Interpolation / extrapolation
Predicting inside / outside the range of the data.
Interpret \(r\): \(0\lt|r|\le0.25\) very weak, \(0.25\lt|r|\le0.5\) weak, \(0.5\lt|r|\le0.75\) moderate, \(0.75\lt|r|\le1\) strong. The sign gives the direction.
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
Exam tip: For any prediction, comment on two things: the strength of r and whether you are interpolating or extrapolating.
4.5Probability: basic concepts
Key terms
Trial / outcome
One repetition of an experiment / one possible result.
Sample space U
The set of all possible outcomes.
Event
A set of outcomes.
Relative frequency
Number of times an event happened ÷ number of trials.
Expected number
n × P(A) for n trials.
Use the theoretical formula when outcomes are equally likely; otherwise use relative frequency from experiments.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Probabilities can be fractions, decimals or percentages. Give exact fractions when they are simple, otherwise 3 s.f.
4.6Venn diagrams, tree diagrams and combined events
Key terms
Mutually exclusive
A and B cannot both happen: P(A ∩ B) = 0.
Independent
One event does not change the probability of the other.
Conditional probability
P(A | B): probability of A given that B has happened.
Without replacement
Probabilities on the second branch change.
On a tree diagram, multiply along branches and add between different branches.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: In “one of each” questions, list both orders. Forgetting one order halves the answer.
4.7Discrete random variables
Key terms
Discrete random variable
Takes separate values, each with a probability.
Probability distribution
A table or formula giving P(X = x) for every x; the probabilities add to 1.
Expected value E(X)
The long-run average: Σ x P(X = x).
Fair game
Expected gain = 0.
Use \(\sum P=1\) to find an unknown probability.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Write the distribution as a table before calculating. It makes every probability visible for method marks.
4.8The binomial distribution
Key terms
Binomial conditions
Fixed number of trials n, two outcomes, constant p, independent trials.
X ~ B(n, p)
X counts the successes.
Mean
np
Variance
np(1 − p)
GDC: binompdf for \(P(X=r)\), binomcdf for \(P(X\le r)\).
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
Exam tip: Write the inequality you are calculating, e.g. P(X ≥ 3) = 1 − P(X ≤ 2), before using the GDC.
4.9The normal distribution
Key terms
X ~ N(μ, σ²)
Normal with mean μ and variance σ² (standard deviation σ).
Properties
Bell-shaped, symmetric about μ; mean = median = mode.
68–95–99.7
About 68%, 95%, 99.7% of values lie within 1, 2, 3 standard deviations of the mean.
Inverse normal
Finds the value x for a given probability.
GDC: normalcdf(lower, upper, μ, σ) for probabilities; invNorm(area to the left, μ, σ) for values.
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
Exam tip: The GDC asks for σ, not σ². If the question gives the variance, take the square root first.
4.10Spearman's rank correlation
Key terms
Rank
The position of a value when the data are ordered.
Spearman's rₛ
Pearson's r applied to the ranks: measures a monotonic relationship.
Monotonic
Always increasing or always decreasing, not necessarily linear.
Tied ranks
Equal values share the average of their ranks.
Rank each variable separately, then find \(r\) of the ranks on the GDC.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Show your table of ranks. If you rank in the wrong direction for one variable, the sign of rₛ flips.
4.11Hypothesis testing: χ² and t-tests
Key terms
Null hypothesis H₀
The “no effect / no difference / independent” statement.
Alternative hypothesis H₁
What you accept if there is enough evidence against H₀.
Significance level
The threshold probability, e.g. 5%.
p-value
The probability of results at least this extreme if H₀ is true.
Degrees of freedom
For a contingency table: (rows − 1)(columns − 1).
Decision rule: if p-value \(\lt\) significance level, reject \(H_0\); otherwise do not reject \(H_0\). Then write the conclusion in context.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Write “do not reject H₀”, never “accept H₀”. A test can't prove the null hypothesis true.
4.12Designing data collection HL
Key terms
Questionnaire
A set of questions used to collect data.
Reliability
Repeating the test gives consistent results.
Validity
The test measures what it is supposed to measure.
Test–retest
The same test given twice to check reliability.
Parallel forms
Two equivalent versions of a test, to check reliability.
Good questions are clear, neutral (not leading), have options that cover every answer without overlapping, and don't ask two things at once.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: In your IA, describe how you checked the reliability and validity of your data. It is a frequent missing point.
4.13Non-linear regression HL
Key terms
Residual
Observed value − predicted value.
SSᵣₑₛ
Sum of the squares of the residuals: smaller means a better fit.
Least squares regression
Chooses the curve that minimises SSᵣₑₛ.
Coefficient of determination R²
The proportion of the variation in y explained by the model (0 to 1).
GDC regressions: quadratic, cubic, exponential, power, sine, logistic. The GDC gives \(R^{2}\) for most models.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Don't choose a model on R² alone: check the shape at the ends of the domain and whether it makes sense in context.
4.14Linear transformations and unbiased estimators HL
Key terms
E(aX + b)
aE(X) + b
Var(aX + b)
a²Var(X): adding b does not change the spread.
Linear combination
aX ± bY of independent variables.
Unbiased estimator
On average equals the parameter it estimates.
s²ₙ₋₁
Unbiased estimate of the population variance.
The variance rule for combinations needs \(X\) and \(Y\) to be independent. Variances always add, even for \(X-Y\).
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: “Variances add, even when you subtract” is the single most tested fact here.
4.15Combinations of normal variables and the central limit theorem HL
Key terms
Sum of normals
If X and Y are independent normals, aX + bY is also normal.
Sample mean X̄
The mean of a random sample of size n.
Central limit theorem (CLT)
For large n (n ≥ 30), X̄ is approximately normal, whatever the original distribution.
Standard error
σ ÷ √n: the standard deviation of X̄.
If the population is normal, \(\bar X\) is normal for any \(n\). If not, the CLT makes it approximately normal for \(n\ge30\).
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Enter σ/√n (here 2) as the standard deviation on the GDC, not σ.
4.16Confidence intervals for the mean HL
Key terms
Confidence interval
A range of values that probably contains the true population mean.
Confidence level
e.g. 95%: in the long run, 95% of such intervals contain μ.
Known σ
Use the normal (z) interval.
Unknown σ
Use the t-interval with sₙ₋₁ and n − 1 degrees of freedom.
Use the GDC: ZInterval (σ known) or TInterval (σ unknown).
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Interpret in context: “We are 95% confident that the mean … lies between 70.6 and 74.4.”
4.17The Poisson distribution HL
Key terms
X ~ Po(m)
Counts events in a fixed interval of time or space.
Conditions
Events occur singly, independently, at a constant average rate m.
Mean and variance
Both equal m.
Sum
Independent Po(m₁) + Po(m₂) = Po(m₁ + m₂).
Scale \(m\) to the interval: 3 per hour gives \(m=6\) for 2 hours.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Always adjust m to the time interval in the question before calculating.
4.18Critical regions, tests for means and proportions, errors HL
Key terms
Critical region
The values of the test statistic that lead to rejecting H₀.
Critical value
The boundary of the critical region.
Type I error
Rejecting H₀ when it is true.
Type II error
Not rejecting H₀ when it is false.
Test for ρ = 0
Uses the sample r to test whether there is correlation in the population.
Tests for a mean: z-test (σ known) or t-test (σ unknown), one- or two-tailed. Paired data: test the differences.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: For a discrete critical region, show the two probabilities either side of the boundary. That's the method mark.
4.19Markov chains and transition matrices HL
Key terms
State
One of the possible conditions (e.g. sunny / rainy).
Transition matrix T
Entry in row i, column j = P(moving to state i from state j); columns add to 1.
State vector sₙ
Probabilities of each state after n steps: sₙ = Tⁿs₀.
Steady state s
The long-run distribution: Ts = s.
Regular chain
Some power of T has all entries positive, so a unique steady state exists.
Draw a transition diagram (directed graph) first, then build \(T\) column by column.
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
Exam tip: Label the rows and columns of T with the states. It prevents the classic row/column mix-up.