Pearson's Correlation Coefficient
Master Pearson's correlation coefficient \(r\) for NSW Year 12 Mathematics Standard 2. You will learn that \(r\) is a single number between \(-1\) and \(1\) whose sign gives the direction of a linear correlation and whose size gives its strength — strong, moderate or weak.
This topic shows you how to read \(r\) from a scientific calculator's statistics mode, estimate it from a scatterplot, classify a correlation as positive or negative and strong, moderate or weak, and explain why a strong correlation does not prove causation — core Standard 2 skills for the bivariate data section of the HSC.
Theory
Pearson's correlation coefficient \(r\) measures how strongly two numerical variables are linearly related. This Year 12 Standard 2 (NSW) guide shows how to find \(r\) with a scientific calculator, read its sign as the direction and its size as the strength of the correlation, estimate \(r\) from a scatterplot, and why correlation does not prove causation.
Pearson's correlation coefficient \(r\) is a single number that measures the strength and direction of the linear relationship between two numerical variables. It always lies between \(-1\) and \(1\).
The sign of \(r\) gives the direction: \(r>0\) is a positive correlation (both variables tend to rise together) and \(r<0\) is a negative correlation (one rises as the other falls). The size of \(|r|\) gives the strength: values near \(1\) are strong, values near \(0\) are weak.
\(r=1\) or \(r=-1\) means the points lie exactly on a straight line (a perfect correlation), while \(r=0\) means there is no linear correlation. In this Year 12 Standard 2 (NSW) course you read \(r\) straight from a scientific calculator's statistics mode.
From a data set, the calculator computes \(r\) as
where \(S_{xy}\) measures how \(x\) and \(y\) vary together and \(S_{xx}, S_{yy}\) measure how each varies on its own. You never compute this by hand in Standard 2 — you enter the pairs and read \(r\) — but it shows why \(r\) is a pure number with
How to find and interpret \(r\)
- Enter the paired data into your calculator's statistics (or regression) mode as \((x,y)\) pairs.
- Read off the value labelled \(r\), rounded to \(2\) decimal places.
- Direction from the sign: \(r>0\) positive, \(r<0\) negative.
- Strength from the size: \(|r|\ge 0.75\) strong, \(0.5\le|r|<0.75\) moderate, \(|r|<0.5\) weak.
- Interpret in context, and remember a correlation does not prove causation.
Read the direction from the trend, then the strength from how close the points are to a line.
| \(\text{trend rises}\) | \(\Rightarrow\) | \(r>0\ (\text{positive})\) |
| \(\text{close to a line}\) | \(\Rightarrow\) | \(\text{strong}\) |
| \(r\) | \(\approx\) | \(0.9\) |
A strong positive correlation: more training hours go with more laps.
Enter the five pairs, read \(r\), then classify the sign and size.
| \(x\) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \(y\) | 6 | 5 | 8 | 6 | 9 |
| \(r\) | \(=\) | \(\dfrac{S_{xy}}{\sqrt{S_{xx}\,S_{yy}}}\) |
| \(r\) | \(=\) | \(\dfrac{7}{\sqrt{10\times 10.8}}=\dfrac{7}{\sqrt{108}}\) |
| \(r\) | \(\approx\) | \(0.67\) |
Since \(0.5\le 0.67<0.75\) and \(r>0\), it is a moderate positive correlation.
Direction from the falling trend; strength from \(|r|\), compared as sizes.
| \(\text{trend falls}\) | \(\Rightarrow\) | \(r<0\) |
| \(\text{close to a line}\) | \(\Rightarrow\) | \(\text{strong}\) |
| \(r\) | \(\approx\) | \(-0.89\) |
| \(|-0.89|=0.89\) | \(>\) | \(0.7\) |
A strong negative correlation; and \(0.89>0.7\), so this group is the stronger of the two.
Read \(r\) from the calculator, classify it, then judge the causation claim.
| \(I\) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \(S\) | 4 | 3 | 6 | 6 | 8 | 9 |
| \(r\) | \(\approx\) | \(0.94\) |
| \(r>0,\ |r|=0.94\) | \(\ge\) | \(0.75\) |
| \(\Rightarrow\) | \(\) | \(\text{strong positive}\) |
(iii) No — correlation does not prove causation. Hot weather lifts both sales, so it is the common cause.
Common pitfalls
Frequently asked questions
What does Pearson's correlation coefficient r tell you?
It tells you two things about the linear relationship between two numerical variables: the direction (its sign, positive or negative) and the strength (how close its size is to 1). It is always a number between -1 and 1.
What value of r counts as a strong correlation?
A common guide is that |r| of 0.75 or more is strong, between 0.5 and 0.75 is moderate, and below 0.5 is weak. The sign is read separately: it only tells you the direction, not the strength.
Can the correlation coefficient be greater than 1?
No. Pearson's r can never be more than 1 or less than -1, so a value like 1.4 is impossible and usually means a calculation or data-entry error. The extremes r = 1 and r = -1 are perfect straight-line relationships.
Is r the same as the gradient of the line of best fit?
No. The gradient can be any size and carries the units of the data, while r is a pure number in the range -1 to 1. They always share the same sign, but a gradient of 2 does not mean r = 2.
Does a high correlation mean one variable causes the other?
No. A high r shows the variables tend to change together, but correlation does not prove causation. A third factor can drive both variables, so you cannot conclude cause and effect from r alone.
How do you find r on a calculator?
Put the calculator into statistics or regression mode, enter the data as (x, y) pairs, then read off the value labelled r. Round it to two decimal places and interpret its sign and size in context.