A student investigates the relationship between the shoe size of primary school children and their reading speed in words per minute. The student calculates a Pearson’s correlation coefficient of $r = 0.89$ . State the direction and strength of the linear correlation. The student concludes that having larger feet causes children to read faster. Explain why this conclusion is invalid, suggesting a possible lurking variable.

There is a strong, positive, linear correlation. A1A1 Correlation does not imply causation. R1 The relationship is likely caused by a lurking (confounding) variable, such as the age of the child. As children get older, their feet grow larger, and simultaneously their cognitive abilities develop, causing them to read faster. R1

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IB Maths AI SL · Unit 4: Statistics and Probability

Correlation & Regression (IB Maths AI SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

Bivariate data pairs two variables, such as hours revised and test score. Plot a scatter diagram, measure the strength of a linear relationship with Pearson's $r$ (or Spearman's $r_{s}$ for ranks), and if the correlation is strong use the regression line of $y$ on $x$ from your GDC to predict $y$ , but only inside the range of the data.

45 questions
Paper 1: 21
Paper 2: 24
3 worked examples

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Notes
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Worked examples
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FAQ

Correlation & Regression notes

Scatter diagrams

Describe a scatter diagram by its direction (positive or negative), its form (linear or not) and its strength (strong, moderate, weak). Outliers and clusters are worth naming. Correlation is not causation: a third factor can drive both variables.

Pearson's correlation coefficient

$r$ runs from $- 1$ to $1$ . Close to $1$ or $- 1$ means a strong linear relationship; close to 0 means little linear relationship. Use the GDC's two-variable statistics, and quote $r$ , not $r^{2}$ , when $r$ is asked for.

Spearman's rank correlation

$r_{s}$ is Pearson's $r$ calculated on the ranks of the data. It measures whether one variable goes up when the other does (a monotonic relationship), even if the pattern is curved. Rank each variable separately; tied values share the average of their ranks.

The regression line

The line of $y$ on $x$ , $y = a x + b$ , comes from the GDC. Interpret the gradient in context with units: "each extra hour of revision adds about $a$ marks". For example, with $y = 4.2 x + 31$ , 6 hours predicts $4.2 \times 6 + 31 = 56.2$ marks.

Is the prediction reliable?

A prediction is reliable when the correlation is strong and the value you substitute is inside the range of the data (interpolation). Outside the range (extrapolation) the pattern may not continue, so say the prediction is unreliable.

Key formulas

Regression: $y = a x + b$ (GDC); use to predict $y$ within the data range only Learn it
$| r |$ near 1: strong linear correlation; Spearman's $r_{s}$ = PMCC of the ranks Learn it
Mean ( $n = \sum f_{i}$ ) In the booklet: $\bar{x} = \frac{\sum f_{i} x_{i}}{n}$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AI SL formulas →

Correlation & Regression worked examples

Worked example 1: Pearson's correlation coefficient · easy

A researcher records hours of sunshine $x$ and ice creams sold $y$ over $5$ days: $(2, 12), (4, 20), (6, 26), (8, 36), (10, 46)$ . Find $r$ to 3 s.f. and interpret it.

Solution

5 steps · 1 to fill in

Examiner tip: If your GDC doesn't display

r

, enable "Diagnostics" in the catalogue or mode settings.

Worked example 2: Regression line and prediction · medium

Using the same data, find the regression line $y = a x + b$ , then estimate ice creams sold with $7$ hours of sunshine.

Solution

5 steps · 1 to fill in

Examiner tip: Always use the regression line of

y

x

when predicting a

y

-value from an

x

-value.

Worked example 3: Reliability and extrapolation · hard

The kiosk owner uses $y = 4.2 x + 2.8$ to predict $103.6$ sales for $x = 24$ hours. Explain mathematically why this is unreliable.

Solution

4 steps

Examiner tip: Even with

r = 1.00

, any prediction outside the poles of the data must be classified as unreliable.

Correlation & Regression exam-style questions

3 questions from our AI SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Each has a full mark scheme in the practice area.

Question 1 · easy · 4 marks · Paper 1

A set of bivariate data has a regression line $y$ on $x$ with the equation $y = 3.5 x + 12$ . The mean value of the $x$ -data is $\overset{―}{x} = 8$ .

State the exact coordinates of the mean point, $M (\overset{―}{x}, \overset{―}{y})$ , for this data set.
Explain the mathematical significance of the mean point in relation to the line of best fit.

Attempt it and see the mark scheme →

Question 2 · medium · 4 marks · Paper 1

A student investigates the relationship between the shoe size of primary school children and their reading speed in words per minute. The student calculates a Pearson’s correlation coefficient of $r = 0.89$ .

State the direction and strength of the linear correlation.
The student concludes that having larger feet causes children to read faster. Explain why this conclusion is invalid, suggesting a possible lurking variable.

Full worked solution Free, no sign-in

There is a strong, positive, linear correlation. A1A1
Correlation does not imply causation. R1
The relationship is likely caused by a lurking (confounding) variable, such as the age of the child. As children get older, their feet grow larger, and simultaneously their cognitive abilities develop, causing them to read faster. R1

Every question here has a full worked solution. See them all →

Question 3 · hard · 4 marks · Paper 1

The weights of 10 athletes and the weights of their corresponding bicycles are recorded. The Pearson’s correlation coefficient is found to be $r = 0.76$ . The researcher realises that the bicycle weights were recorded in kilograms, but they should have been recorded in grams (by multiplying every bicycle weight by 1000).

State the new value of $r$ after the bicycle weights are converted to grams.
Give a mathematical reason for your answer to part (a).

Attempt it and see the mark scheme →

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All 45 correlation & regression questions →

Where students lose marks on correlation & regression

Examiner's noteInterpreting in context: parameters, derivatives and justifications

Where marks are lost: Weaker candidates could not say what a gradient or intercept means in context, gave only one of the two when comparing costs, described a regression gradient loosely, explained sampling weaknesses too briefly to show understanding, and justified a choice with reasons that did not use the model or the numbers.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Correlation & Regression on your calculator

A GDC is allowed on every AI paper. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Correlation & Regression FAQ

What is the difference between Pearson's and Spearman's correlation?

Pearson's r measures how close the data are to a straight line. Spearman's r_s is Pearson's r applied to the ranks, so it measures whether the variables increase together even when the pattern is curved.

When is a regression prediction reliable?

When the correlation is strong and you predict y for an x inside the range of the data. Predicting outside the range is extrapolation, and the pattern may not continue.

How many IB Maths AI SL bivariate data questions are there?

There are 45 exam-style bivariate data questions in the AI SL question bank (Paper 1: 21 · Paper 2: 24), graded 12 easy, 15 medium, 12 hard, 6 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is bivariate data on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 21 · Paper 2: 24. Practise with your GDC — AI papers expect calculator methods throughout.

Next steps

More AI SL Unit 4: Descriptive Statistics · Probability · Sampling Techniques · Chi-Squared Tests · Two-Sample t-Test · Venn Diagrams & Hard Probability

Teaching this topic? Free ready-to-teach lesson (slides and teacher notes): Least-squares regression line.

Keep revising AI SL: AI SL revision notes · IB Maths AI SL past papers · AI SL common exam mistakes · AI SL syllabus checklist · AI SL statistics and probability knowledge organiser

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Correlation & Regression (IB Maths AI SL): notes, exam questions and worked examples

Try it first

Correlation & Regression notes

Scatter diagrams

Pearson's correlation coefficient

Spearman's rank correlation

The regression line

Is the prediction reliable?

Key formulas

Correlation & Regression worked examples

Worked example 1: Pearson's correlation coefficient · easy

Solution

Worked example 2: Regression line and prediction · medium

Solution

Worked example 3: Reliability and extrapolation · hard

Solution

Correlation & Regression exam-style questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 4 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 4 marks · Paper 1

Where students lose marks on correlation & regression

Correlation & Regression on your calculator

Correlation & Regression FAQ

Next steps

What else is free here