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Module 5 · L03 of 8 ~30 min MST-12-S2-08 ⚡ +70 XP available

Pearson's Correlation Coefficient

CSIRO researchers use Pearson's r to confirm whether rainfall patterns and crop yields in regional NSW are related enough to build prediction models. A single number between −1 and +1 captures both direction and strength of a linear relationship.

Think first, What would a single number that measures the strength and direction of correlation look like? What range of values would it need to cover all possibilities?
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Orient to Pearson's r

Connect visual correlation with a numerical measure from -1 to 1.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Think First, recall from memory
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What would a single number that measures the strength and direction of correlation look like? What range of values would it need to cover all possible situations, from perfect positive to perfect negative? Write your thoughts before reading on.

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The big idea, one number captures everything
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Pearson's correlation coefficient $r$ is a number that measures the strength and direction of a linear relationship between two variables.

The range: $r$ is always between $-1$ and $+1$ inclusive. $r = +1$ is perfect positive; $r = -1$ is perfect negative; $r = 0$ means no linear correlation.

Sign gives direction; magnitude gives strength. The further $r$ is from 0 (closer to ±1), the stronger the correlation.

$$-1 \le r \le +1$$
Sign = direction
Positive $r$ → positive correlation. Negative $r$ → negative correlation.
Magnitude = strength
$|r|$ close to 1 = strong. $|r|$ close to 0 = weak. $|r|$ around 0.5–0.8 = moderate.
Not calculated by hand
You interpret $r$, not calculate it by hand. The formula is given if needed, focus on what the value means.
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What you will learn
Know

Key facts

  • $r$ ranges from $-1$ to $+1$
  • $r = +1$, $r = -1$, and $r = 0$ each have specific meanings
  • Guidelines for classifying strength from $r$
Understand

Concepts

  • How the sign of $r$ indicates direction
  • How the magnitude of $r$ indicates strength
  • Why $r$ only measures linear (not curved) relationships
Can do

Skills

  • Interpret a given $r$ value in context
  • Choose which $r$ value matches a described scatterplot
  • State the limitations of $r$
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Key terms, Pearson's r
Pearson's correlation coefficient ($r$)A numerical measure of the strength and direction of a linear relationship, always between $-1$ and $+1$.
Perfect correlation$r = +1$ (all points on an upward line) or $r = -1$ (all points on a downward line).
No linear correlation$r \approx 0$, no consistent linear pattern, though a curved pattern may still exist.
MagnitudeThe size (absolute value) of $r$, ignoring the sign. Magnitude close to 1 = strong; close to 0 = weak.
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Read sign and magnitude

Use the sign for direction and the magnitude for strength.

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What $r$ tells us, sign and magnitude
MST-12-S2-08 core

Pearson's $r$ packages both direction and strength into one number:

  • Sign (+/−): Positive $r$ → points slope upward. Negative $r$ → points slope downward.
  • Magnitude (distance from zero): The closer $|r|$ is to 1, the stronger (tighter) the relationship.

Strength guidelines:

Range of $|r|$Strength
$0.8$ to $1.0$Strong
$0.5$ to $0.8$Moderate
$0.3$ to $0.5$Weak
$0$ to $0.3$Very weak / no correlation
Note: These are guidelines, not strict rules, and other sources draw the lines in slightly different places. Use these boundaries throughout this course: they are the ones the later lessons and the module quiz mark against.

Pearson's r measures linear correlation strength and direction: r = +1 is perfect positive, r = −1 is perfect negative, r = 0 is no linear correlation. The sign gives direction; the magnitude (|r| close to 1) gives strength.

Pause, copy the r scale from −1 to +1, the sign rule (negative = negative direction; positive = positive direction), and the magnitude rule (closer to ±1 = stronger association; closer to 0 = weaker) into your book.

Quick check: Which $r$ value indicates the strongest correlation?

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Interpret r in context

Write a complete contextual interpretation of a correlation coefficient.

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Interpreting $r$ in context
MST-12-S2-08 core

Pearson's r ranges from −1 (perfect negative linear) to +1 (perfect positive linear), with 0 meaning no linear association. The sign matches the direction you read from the scatterplot. For strength, HSC uses two thresholds: |r| ≥ 0.8 is strong, 0.5 ≤ |r| < 0.8 is moderate, and |r| < 0.5 is weak.

When interpreting $r$, always state: (1) the direction, (2) the strength, and (3) what it means in context of the two variables.

Examples:

  • $r = 0.87$: strong positive linear correlation, as [x variable] increases, [y variable] tends to increase strongly.
  • $r = -0.92$: strong negative linear correlation, as [x variable] increases, [y variable] tends to decrease strongly.
  • $r = 0.41$: weak positive linear correlation, as [x variable] increases, [y variable] shows a slight tendency to increase, but the relationship is not consistent.
  • $r = -0.05$: essentially no linear correlation, knowing [x variable] tells us almost nothing about [y variable].

Real example: For a study of age (x) and resting heart rate (y), $r = -0.68$ means "there is a moderate negative linear correlation between age and resting heart rate, as age increases, resting heart rate tends to decrease moderately."

Interpreting r in context: |r| ≥ 0.8 is strong, 0.5 ≤ |r| < 0.8 is moderate, |r| < 0.5 is weak. Always state the interpretation in terms of the actual variables, not just the number.

Pause, copy the three classification thresholds: |r| ≥ 0.8 strong, 0.5 ≤ |r| < 0.8 moderate, |r| < 0.5 weak, and the rule that every r description must use the actual variable names into your book.

Which does NOT belong? Things you can tell from Pearson's $r$ alone:

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Recognise limitations

Explain linearity, outlier sensitivity and why r does not prove causation.

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Limitations of $r$, what it cannot tell you
MST-12-S2-08 core

The thresholds |r| ≥ 0.8 (strong), 0.5–0.8 (moderate), and < 0.5 (weak) let you classify any r value, but r has three important limitations: it only measures linear association (a perfect curve could give r near 0), it is sensitive to outliers (one extreme point can dramatically change r), and a strong r does not mean one variable causes the other.

Pearson's $r$ has important limitations that examiners test:

  • $r$ only measures linear relationships. Two variables can have a perfect curved (non-linear) relationship with $r \approx 0$. Low $r$ does not mean no relationship, just no linear one.
  • $r$ does not imply causation. A high $r$ tells you the variables are strongly associated, but it does not prove that one causes the other. (We will explore this in Lesson 4.)
  • Outliers can distort $r$. A single outlier can pull $r$ toward 0 or toward ±1, making the relationship look weaker or stronger than it is for the main cluster of data.
Key exam point: If a student says "$r = 0$ means the variables are not related," this is incorrect. It means they are not linearly related. A curved pattern can give $r = 0$.

Limitations of r: it only measures linear association (not curved relationships), is sensitive to outliers, and never proves causation, a high r value means association only, not that one variable causes the other.

Pause, copy the three limitations of Pearson's r: it only detects linear (not curved) relationships, it is sensitive to outliers, and a high r value does not establish that one variable causes the other into your book.

Interactive · Correlation Explorer

Try this: the ten points start at $r \approx 0.99$. Change the exam score of the student who studies 2 hours to 95, and watch what one observation does to $r$.

Use the explorer. Change the exam score of the student who studies 2 hours from 45 to 95, and read the Strength box. One point takes the correlation from very strong down to positive.

Complete: A value of $r = -0.92$ indicates a linear correlation.

5

Calculate r

Use scientific-calculator statistics mode to produce the coefficient.

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Producing $r$ yourself, on a scientific calculator
MST-12-S2-08 core

Every $r$ so far has been handed to you. In the exam it usually is not: you are given the raw pairs and you have to produce $r$ before you can say a word about it. Nobody computes $r$ by hand at this level, the calculator does it, and the marks are for entering the data correctly and reading the right value off the screen.

Eight Year 12 students recorded their paid work hours in a typical week and their mark in the same exam:

Work hours ($x$)035810121519
Exam mark ($y$)8976838274696950

On a Casio fx-82AU PLUS II, the most common calculator in NSW exam rooms:

  1. MODE then 2 to enter STAT mode.
  2. 2 to choose A+BX . This is the two-variable setting; choosing the one-variable setting is why some students never see an $r$ at all.
  3. Type the eight $x$ values down the X column, each followed by = , then the eight $y$ values down the Y column. Every pair must stay on its own row.
  4. AC , then SHIFT 1 to reopen the STAT menu, 5 for Reg , then 3 for $r$ .

The screen gives $r = -0.8995554\ldots$ , which you report as $r = -0.90$ to two decimal places.

Other models. Menus differ, the three things you are hunting never do: a two-variable or linear-regression mode, a place to type the pairs, and a results screen listing $A$ , $B$ and $r$ . Find those three on your own calculator before the exam, not during it.
Two checks that catch a data-entry error. First, $r$ can never leave the interval from $-1$ to $1$ . A screen showing $r = 1.4$ means the pairs are misaligned, usually one column has more entries than the other. Second, the sign of $r$ must match the direction you can already see: these points fall from left to right, so a positive $r$ here would mean the columns were swapped or a value was mistyped.

Reading it back in context: $r = -0.90$ is a strong negative linear correlation, students who worked more paid hours tended to score lower marks. Note the word "tended": this is Lesson 4's warning arriving early, $r$ measures association and settles nothing about cause.

To get r on a scientific calculator, enter two-variable STAT mode, type the pairs into the X and Y columns, then read r from the regression results. Report it to two decimal places. Check that r lies between −1 and 1 and that its sign matches the direction of the scatterplot.

Pause, copy your own calculator's exact key sequence for reaching $r$ into your book, then copy the two checks: $r$ must lie between $-1$ and $1$ , and the sign of $r$ must match the direction of the scatterplot.

True or false: A student enters the work-hours and exam-mark data above and their calculator displays $r = 1.12$ . This can be trusted, because the correlation is very strong.

6

Apply Pearson's r

Work examples, classify coefficients and revisit the opening estimate.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · INTERPRET r = 0.72

For a dataset of weekly exercise hours (x) and body mass index (y), $r = 0.72$. Interpret this value in context.

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Sign: $r = +0.72$ is positive → positive correlation.
Always check the sign first, it gives direction. Here it means that as exercise hours increase, BMI tends to increase too — which is plausible, because training builds muscle and BMI counts muscle mass the same way it counts fat.
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PROBLEM 2 · INTERPRET r = −0.95

For daily screen time (x) and hours of sleep (y), $r = -0.95$. Interpret this value in context.

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Sign: negative → as screen time increases, sleep hours tend to decrease.
Negative r: the two variables move in opposite directions.
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PROBLEM 3 · MATCH r TO A SCATTERPLOT

Three scatterplots are described: (A) tightly grouped upward, (B) widely scattered downward, (C) random scatter. Match each to the most likely r value: $r = 0.95$, $r = -0.45$, $r = 0.02$.

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Scatterplot A (tightly grouped upward): positive (upward) and strong (tightly grouped) → $r = 0.95$.
Tightly grouped + upward = high positive r.
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Activity, interpret and classify r values

For each $r$ value below, state the direction, classify the strength, and write a sentence of interpretation. Assume x = advertising spend ($\$000$s) and y = monthly sales ($\$000$s).

  1. $r = 0.88$
  2. $r = -0.31$
  3. $r = 0.05$
  4. $r = -0.97$
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Revisit your thinking

At the start you thought about what range a single correlation number would need. The answer is $-1 \le r \le +1$: negative values capture negative correlation, positive values capture positive correlation, and the size (magnitude) captures the strength. $r = 0$ sits in the middle, meaning no linear relationship. This elegant range makes $r$ easy to interpret consistently.

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Answer Pearson questions

Interpret and evaluate correlation coefficients in short-answer questions.

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Short answer
ApplyBand 33 marks

Q1. For a study of daily exercise (minutes) and resting heart rate (bpm), $r = -0.84$. (a) What is the direction of this correlation? (b) What is the strength? (c) Write a full interpretation in context. (3 marks)

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AnalyseBand 42 marks

Q2. A researcher finds $r = 0.03$ for the relationship between a person's favourite colour and their reaction time. A student concludes "there is no relationship between these variables." Is the student correct? Explain. (2 marks)

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Answers (click to reveal)

Activity: (1) $r=0.88$: strong positive, as advertising increases, sales tend to increase strongly. (2) $r=-0.31$: weak negative, slight tendency for higher advertising to associate with lower sales (unusual, suggests confounding). (3) $r=0.05$: no linear correlation, knowing advertising spend tells us almost nothing about sales. (4) $r=-0.97$: strong negative linear correlation.

Q1 (3 marks): (a) Negative, as exercise increases, heart rate decreases [1]. (b) $|r|=0.84$, which is at least $0.8$ → strong [1]. (c) "There is a strong negative linear correlation between daily exercise and resting heart rate ($r=-0.84$). As exercise time increases, resting heart rate tends to decrease." [1]

Q2 (2 marks): The student is not fully correct. $r = 0.03$ indicates no linear relationship [1], but there could still be a non-linear (curved) relationship between the variables that $r$ cannot detect [1].