Pearson’s Correlation Coefficient (r) (VCE SSCE General Mathematics): Revision Notes

Pearson's Correlation Coefficient (r)

Introduction

When examining relationships between two numerical variables, we often want to measure not just whether an association exists, but how strong that association is. Pearson's correlation coefficient, denoted by $r$, provides exactly this measurement for linear associations.

This coefficient gives us a numerical measure that tells us how closely the points in a scatterplot cluster around a straight line. The tighter the clustering, the stronger the relationship, and the higher the value of $r$.

Key assumptions

Properties of Pearson's correlation coefficient

Understanding the properties of $r$ helps us interpret its value correctly.

Range and interpretation

Pearson's correlation coefficient has several important properties:

• $r$ has a value between -1 and +1 – These are the minimum and maximum possible values.
• Larger values indicate stronger associations – The closer $r$ is to $1$ or $-1$, the stronger the linear relationship.
• The sign indicates direction:
  • Positive $r$ means a positive linear association (as one variable increases, the other tends to increase)
  • Negative $r$ means a negative linear association (as one variable increases, the other tends to decrease)
• $r$ close to zero indicates no linear association – The variables don't have a linear relationship.

Extreme values

Understanding the extreme values helps us recognise different patterns:

• $r = 0$: No linear association – points are scattered randomly with no clear linear pattern
• $r = +1$: Perfect positive linear association – all points lie exactly on an upward-sloping straight line
• $r = -1$: Perfect negative linear association – all points lie exactly on a downward-sloping straight line

Real-world values

In practice, we rarely see $r$ values of exactly $0$, $+1$, or $-1$. Most real data gives us values somewhere in between. Here are some examples:

These scatterplots illustrate an important point: the stronger the association, the larger the magnitude of Pearson's correlation coefficient. Notice how:

• Strong correlations (like $r = 0.915$ or $r = -0.874$) show points tightly clustered around a line
• Moderate correlations (like $r = 0.767$ or $r = -0.501$) show more scatter but still a clear trend
• Weak correlations (like $r = 0.551$ or $r = 0.150$) show considerable scatter with less obvious patterns

Summary of properties

Estimating correlation from scatterplots

Before calculating $r$ precisely using technology, it's useful to estimate its value by examining a scatterplot. This helps us check whether our calculated value makes sense and catch any potential errors.

Worked example: estimating $r$ values

Tips for estimation

Calculating Pearson's correlation coefficient

The formula

The formula for calculating $r$ is:

$r = \frac{1}{n-1} \sum \left(\frac{x - \bar{x}}{s_x}\right)\left(\frac{y - \bar{y}}{s_y}\right)$

Where:

• $n$ is the number of data pairs
• $\bar{x}$ and $s_x$ are the mean and standard deviation of the $x$ values
• $\bar{y}$ and $s_y$ are the mean and standard deviation of the $y$ values

This formula is quite tedious to calculate by hand, so we typically use technology instead. However, understanding the formula helps us appreciate that $r$ depends on how the variables vary together relative to their individual variations.

Important notes about calculation

Using technology to calculate $r$

We'll use an example to demonstrate the calculation process.

Example data:

Income ($'000)	8.9	23.0	7.5	8.0	18.0	16.7	5.2	12.8	19.1	16.4	21.7
CO₂ (tonnes)	7.5	12.0	6.0	1.8	7.7	5.7	3.8	5.7	11.0	9.7	9.9

This data shows the per capita income and carbon dioxide emissions for 11 countries.

TI-Nspire CAS calculator

ClassPad calculator

Worked example: test scores

Classifying the strength of linear associations

Once we've calculated $r$, we need to interpret what the value means. We use standard guidelines to classify the strength of the association.

Classification guidelines

Here's the complete classification system:

Value of $r$	Strength of association
$0.75 \leq r \leq 1$	strong positive association
$0.5 \leq r < 0.75$	moderate positive association
$0.25 \leq r < 0.5$	weak positive association
$-0.25 < r < 0.25$	no association
$-0.5 < r \leq -0.25$	weak negative association
$-0.75 < r \leq -0.5$	moderate negative association
$-1 \leq r \leq -0.75$	strong negative association

Worked example: classification

Practice classifications

Here are some additional examples:

• $r = 0.807$ → strong, positive (between 0.75 and 1)
• $r = -0.818$ → strong, negative (between -1 and -0.75)
• $r = 0.224$ → no association (between -0.25 and 0.25)
• $r = -0.667$ → moderate, negative (between -0.75 and -0.5)

Correlation and causation

Understanding the difference

A strong correlation between two variables means they vary together:

• If the correlation is positive, both variables tend to increase together
• If the correlation is negative, one tends to decrease as the other increases

However, even a strong correlation is not sufficient evidence that changing one variable will cause a change in the other. It only suggests that this might be a possible explanation.

Example: smoking and heart disease

Suppose we find a high correlation between smoking rates and incidence of heart disease across different countries. Can we conclude that smoking causes heart disease based solely on this correlation?

No, we cannot. Here's why:

Alternative explanations might exist. For example:

• People who smoke might also neglect other lifestyle factors like exercise and diet
• It could be lack of exercise that actually causes heart disease
• Smoking and heart disease might both be related to a third factor we haven't measured

Correct vs incorrect interpretations

Worked example: income and emissions

Why this matters

Being careful about the distinction between correlation and causation is essential because:

• Incorrect causal interpretations can lead to poor decisions
• Many factors can create correlations without causal relationships
• Establishing causation requires more than just observing correlation (such as controlled experiments)

Remember: Correlation shows that variables are associated, but not that one causes the other.

Remember!