PMCC and Correlation (AA HL)
Pearson's product-moment correlation coefficient, \(r\), tells you how strong a straight-line relationship is between two variables, and in which direction. This page covers how to read and interpret \(r\), plus the trickiest idea in the topic - that correlation, however strong, never proves causation - with worked examples and the mistakes that cost marks. It's part of the broader Bivariate Statistics topic.
14 questions on this sub-topic.
Interpreting r
Covered under IB syllabus reference SL4.4: Pearson's product-moment correlation coefficient \(r\), found using technology, and the distinction between correlation and causation.
Pearson's correlation coefficient
\(r=\dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\)
On the formula booklet, but always found using technology in practice: \(r\) close to \(\pm1\) means a strong linear correlation, \(r\) close to \(0\) means little or none.
Correlation vs causation
A strong \(r\) never proves one variable causes the other - a third factor could be driving both, or the link could be coincidental.
Need the full syllabus wording and the wider statistics formula table? See Bivariate Statistics. For calculator steps, see the parent topic's GDC guidance.
Worked examples
Two quantities have Pearson's product-moment correlation coefficient \(r=-0.85\).
(a) Describe the linear relationship.
(b) State what \(r=0\) would indicate.
Worked solution
(a) Strong A1
negative linear correlation. A1
(b) No linear correlation between the variables. A1
A study finds a strong positive correlation between ice-cream sales and drowning incidents.
(a) Suggest a confounding variable.
(b) Explain how it produces the correlation without direct causation.
Worked solution
(a) Hot weather (temperature). A1
(b) Higher temperatures increase both ice-cream sales and swimming (hence drownings), R1
This produces a correlation between the variables without either one causing the other. A1
Two quantities have Pearson’s product-moment correlation coefficient \(r=-0.85\).
(a) Describe the linear relationship.
(b) State what \(r=0\) would indicate.
Worked solution
(a) Strong A1
negative linear correlation. A1
(b) No linear correlation between the variables. A1
Common mistakes
- Treating correlation as causation. A strong \(r\) shows two variables move together, not that one causes the other - a third factor, or coincidence, could explain the pattern.
- Confusing the sign of \(r\) with its strength. \(r=-0.9\) is a stronger correlation than \(r=0.3\), even though it's negative - strength is about how close \(|r|\) is to \(1\), not about the sign.
- Ignoring outliers when judging \(r\). A single far-off point can pull \(r\) noticeably towards \(0\) even when the rest of the data shows a tight linear trend, so a low \(r\) doesn't always mean "no relationship".
Ready to practise properly?
14 correlation questions, marked instantly like the real exam.
Quick answers
What does Pearson's correlation coefficient r measure?
\(r\) measures the strength and direction of the linear relationship between two variables, ranging from \(-1\) (perfect negative) to \(+1\) (perfect positive), with \(0\) meaning no linear correlation.
Does a strong correlation mean one variable causes the other?
No. A strong \(r\) only shows the two variables move together - it never proves causation, since a third (confounding) factor could be driving both, or the pattern could be coincidental.