Coaching Centre
Pearson’s correlation coefficient
What Pearson's correlation coefficient r measures, the scale from -1 to 1, classifying strength, finding r with a calculator, and what r cannot tell you.
"Strong", "moderate" and "weak" are judgement calls, and two people looking at the same scatterplot can disagree. Pearson's correlation coefficient, written $r$, puts a number on it. It measures how closely the points follow a straight line, and in which direction.
The scale from $-1$ to $1$
- $r$ is always between $-1$ and $1$.
- The sign gives the direction: $r \gt 0$ is positive, $r \lt 0$ is negative.
- The size (ignoring the sign) gives the strength. The closer $r$ is to $1$ or $-1$, the closer the points are to a straight line.
- $r = 1$ or $r = -1$ means every point sits exactly on a line. $r = 0$ means no linear association at all.
| Value of $r$ | Strength of linear association |
|---|---|
| $0.75 \le |r| \lt 1$ | strong |
| $0.5 \le |r| \lt 0.75$ | moderate |
| $0.25 \le |r| \lt 0.5$ | weak |
| $|r| \lt 0.25$ | no linear association (or very weak) |
These cut-offs are a guideline used in NSW textbooks, not a law of nature. In an exam, use the guideline the question gives you if it gives one.
Finding $r$ with technology
You will never calculate $r$ by hand in Standard 2. You enter the pairs and let the calculator (or a spreadsheet) do it. On most scientific calculators the steps are:
- Switch to statistics mode and choose the linear regression option (often labelled $y = a + bx$ or
A+BX). - Enter the $x$-values in one column and the matching $y$-values in the other. Check each row matches its pair.
- Open the regression results and read off $r$.
In a spreadsheet, put $x$-values in column A and $y$-values in column B, then use =CORREL(A2:A9,B2:B9).
A teacher records the number of days each of eight students was absent in Term 2 and their mark on a test out of $70$.
| Days absent ($x$) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Mark ($y$) | 62 | 55 | 58 | 49 | 47 | 41 | 44 | 36 |
Find $r$ to two decimal places and describe the association.
What $r$ can't tell you
$r$ only measures linear association. A perfect curve, like the height of a thrown ball against time, can have an $r$ close to $0$ even though the two variables are completely connected. That's why you always look at the scatterplot first. A number on its own can mislead you.
$r$ is also sensitive to outliers. One wild point can drag a strong $r$ down to moderate, or create a correlation that isn't really there.
Play ten rounds. Drag the slider to your guess for $r$, then press Reveal. Your average error is tracked above the scatterplot.
Notice which scatterplots fool you. Do you overestimate the strength when there are only a few points? What happens to your guesses for the curved plots?
Your turn
Five datasets have $r = 0.12$, $r = -0.58$, $r = 0.81$, $r = -0.97$ and $r = 0.43$. Which shows the strongest linear association, and how would you describe $r = -0.58$?
Show the answer
The strongest is $r = -0.97$, because $|-0.97| = 0.97$ is the closest to $1$. $r = -0.58$ is a moderate negative linear association, since $0.5 \le 0.58 \lt 0.75$.
Which value of $r$ shows the strongest linear association?
Compare sizes: $|0.7| = 0.7$, $|-0.5| = 0.5$, $|-0.88| = 0.88$ and $|0.85| = 0.85$. The largest is $0.88$, so $r = -0.88$.
$r = 0.62$ describes a:
$0.62$ is positive and between $0.5$ and $0.75$, so it is moderate and positive.
Every point on a scatterplot lies exactly on a line that slopes down. The value of $r$ is:
A perfect linear association gives $|r| = 1$. Sloping down means negative, so $r = -1$.
Which value is impossible for $r$?
$r$ always lies between $-1$ and $1$, so $1.2$ is impossible. If you get this from a calculator, you've entered something wrong.
Use your calculator: find $r$, to two decimal places, for the absence data in Example 1 (days absent $1$ to $8$; marks $62, 55, 58, 49, 47, 41, 44, 36$).
Entering the eight pairs gives $r = -0.9558\ldots \approx -0.96$.
Hours of study per week ($x$) and exam mark ($y$) for eight students are: $(2, 48)$, $(4, 55)$, $(5, 52)$, $(7, 63)$, $(8, 66)$, $(10, 70)$, $(12, 79)$, $(13, 81)$. Find $r$ to two decimal places.
Entering the pairs in stats mode gives $r = 0.9868\ldots \approx 0.99$, a strong positive linear association.
A scatterplot shows points forming a clear U shape. A student calculates $r = 0.04$ and says "there is no relationship". The best response is:
A U shape is a clear pattern, just not a straight line. $r$ near $0$ means no linear association, which is why you always check the scatterplot.
Removing one outlier from a dataset changes $r$ from $0.45$ to $0.82$. This shows that:
One point dragged $r$ from strong down to weak. $r$ is sensitive to outliers, so check whether an outlier is a mistake before reporting $r$.
Exam style: For a set of data, $r = -0.79$. Which statement is correct?
$|-0.79| = 0.79 \ge 0.75$, so the association is strong. The sign is negative. $r$ is not a percentage of points on the line.
Exam style: Two studies find $r = 0.65$ (study A) and $r = -0.81$ (study B). Which statement is correct?
$|-0.81| = 0.81$ is bigger than $0.65$, so B is stronger (strong), while A is moderate. B's association is negative and A's is positive.