Strength of a Linear Relationship: The Correlation Coefficient Revision Notes for VCE SSCE General Mathematics

Strength of a Linear Relationship: The Correlation Coefficient

What is the correlation coefficient?

The correlation coefficient is a numerical measure that tells us how closely the points in a scatterplot cluster around a straight line. When we want to understand the strength of a linear relationship between two variables, we use a specific type of correlation coefficient called Pearson's correlation coefficient, represented by the symbol $r$ .

This important statistical measure was developed by a statistician named Carl Pearson. It provides a precise, numerical way to describe the strength of a linear association, rather than relying only on visual inspection of scatterplots.

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Carl Pearson (1857-1936) was an English mathematician and biostatistician who made significant contributions to the field of statistics. His correlation coefficient remains one of the most widely used statistical tools today for measuring the strength of linear relationships between variables.

Understanding strength

The strength of a linear association refers to how closely the data points fit a straight line pattern:

If the points lie exactly on a straight line, we have a perfect linear association
If the points show no linear pattern at all, we have no association
Most real-world data falls somewhere in between, showing an imperfect fit

Values of the correlation coefficient

Pearson's correlation coefficient $r$ always has a value between $-1$ and $+1$ , inclusive. Understanding what these values mean is crucial for interpreting the strength and direction of linear relationships.

Perfect correlations

There are three special cases where the correlation takes exact values:

Perfect positive linear association: When all points fall exactly on an upward-sloping straight line, $r = +1$
Perfect negative linear association: When all points fall exactly on a downward-sloping straight line, $r = -1$
No linear association: When the points show a random scatter with no linear pattern, $r = 0$

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The Three Perfect Values to Remember:

$r = +1$ means all points lie exactly on an upward-sloping straight line
$r = -1$ means all points lie exactly on a downward-sloping straight line
$r = 0$ means there is no linear pattern in the data

These are the only values where we can describe the relationship as "perfect"!

Imperfect correlations

Most real-world data shows imperfect linear associations, where the correlation coefficient falls between the perfect values. This means $-1 \< r \< +1$ . The closer $r$ is to $+1$ or $-1$ , the stronger the linear relationship. The closer $r$ is to $0$ , the weaker the relationship.

The scatterplots above show examples of different correlation strengths:

$r = -0.7$ shows a moderately strong negative relationship
$r = +0.5$ shows a moderate positive relationship
$r = -0.3$ shows a weak negative relationship
$r = +0.9$ shows a very strong positive relationship

Properties of Pearson's correlation coefficient

The Pearson's correlation coefficient $r$ has several important properties that make it useful for analysing linear relationships:

Measures strength: It quantifies how strong a linear association is, with larger absolute values (closer to $1$ or $-1$ ) indicating stronger relationships
Range: The value is always between $-1$ and $+1$ , inclusive
Sign indicates direction:
- $r$ is positive when the linear association has a positive direction (as one variable increases, the other tends to increase)
- $r$ is negative when the linear association has a negative direction (as one variable increases, the other tends to decrease)
Zero indicates no linear relationship: When $r$ is close to zero, there is no linear association between the variables

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The magnitude (absolute value) of $r$ tells you about the strength of the relationship, while the sign tells you about the direction. For example, $r = -0.8$ and $r = +0.8$ both indicate equally strong relationships, but in opposite directions!

The correlation coefficient formula

Pearson's correlation coefficient can be calculated using a specific formula. While you'll typically use a calculator to find $r$ , understanding the formula helps you appreciate what the correlation coefficient actually measures.

If we call our two variables $x$ and $y$ , and we have $n$ observations (data points), then:

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

In this formula:

$\bar{x}$ is the mean (average) of the $x$ -values
$s_x$ is the standard deviation of the $x$ -values
$\bar{y}$ is the mean of the $y$ -values
$s_y$ is the standard deviation of the $y$ -values
$n$ is the number of observations
$\sum$ means "sum of"

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Understanding the Formula:

The numerator $\sum(x - \bar{x})(y - \bar{y})$ measures how $x$ and $y$ vary together. When both variables tend to be above or below their means together, this sum is large and positive. When one is above its mean while the other is below, the sum becomes negative.

The denominator $(n-1)s_x s_y$ standardises this measure by accounting for the spread in both variables, producing a value that always falls between $-1$ and $+1$ .

Calculating the correlation coefficient manually

Whilst calculators make finding correlation coefficients quick and easy, working through a manual calculation once helps you understand what the correlation coefficient represents.

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Worked Example: Manual Calculation of Correlation Coefficient

Let's calculate the correlation coefficient for this dataset:

Table

Given information:

$\bar{x} = 4$ , $s_x = 2.236$
$\bar{y} = 5$ , $s_y = 3.082$
$n = 5$

Step 1: Write down all the given values

We have: $\bar{x} = 4$ , $s_x = 2.236$ , $\bar{y} = 5$ , $s_y = 3.082$ , and $n = 5$

Step 2: Set up a table to calculate $\sum(x - \bar{x})(y - \bar{y})$

For each data point, we:

Calculate $(x - \bar{x})$ by subtracting the mean from each $x$ -value
Calculate $(y - \bar{y})$ by subtracting the mean from each $y$ -value
Multiply these deviations together to get $(x - \bar{x}) \times (y - \bar{y})$
Sum all the products

From the table: $\sum(x - \bar{x})(y - \bar{y}) = 23$

Step 3: Substitute into the formula and calculate

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

$r = \frac{23}{(5 - 1) \times 2.236 \times 3.082}$

$r = \frac{23}{27.516...}$

$r = 0.834...$

$r = 0.83 \text{ (to 2 decimal places)}$

Answer: The correlation coefficient is $r = 0.83$ , indicating a strong positive linear association between the variables.

Using technology to find the correlation coefficient

Modern calculators can compute correlation coefficients quickly and accurately, saving time and reducing calculation errors. Here's how to use a TI-Nspire CAS calculator to find the correlation coefficient for real-world data.

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Worked Example: Income and CO₂ Emissions Using Technology

The following data shows the per capita income (in thousands of dollars) and per capita carbon dioxide emissions (in tonnes) for 11 countries:

Table

Using a TI-Nspire CAS calculator:

Step 1: Start a new document by pressing ctrl + N

Step 2: Select "Add Lists & Spreadsheet" and enter the data into lists named income and co2

Step 3: Press ctrl + I and select "Add Calculator"

Step 4: Use the correlation matrix command by typing:

corrmat(income, co2)

Then press enter.

Alternatively, press ctrl + T then C to access the Catalogue, scroll to corrMat( and complete the command.

Result: The correlation coefficient is $r = 0.8342...$ or $r = 0.83$ (to 2 decimal places)

Interpretation: This strong positive correlation suggests that countries with higher per capita incomes tend to have higher carbon dioxide emissions.

Classifying the strength of a linear association

Once you've calculated a correlation coefficient, you need to interpret what it means. We classify correlations based on their value to describe the strength of the relationship in words.

Classification table

The following table provides a standard system for classifying the strength and direction of linear associations based on the correlation coefficient value.

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Correlation Classification Guidelines

Range of $r$	Classification
$0.75 \leq r \leq 1$	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

Key points:

Values closer to $\pm 1$ indicate stronger relationships
The sign (positive or negative) indicates the direction
Values near zero suggest little to no linear relationship

Worked examples of classification

Let's practice classifying correlation coefficients using the classification table.

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Worked Examples: Classifying Correlation Strength

Example a: $r = 0.35$

The value $0.35$ falls in the range $0.25 \leq r < 0.5$

Classification: Weak, positive association

Example b: $r = -0.507$

The value $-0.507$ falls in the range $-0.75 < r \leq -0.5$

Classification: Moderate, negative association

Example c: $r = 0.992$

The value $0.992$ falls in the range $0.75 \leq r \leq 1$

Classification: Strong, positive association

Example d: $r = -0.159$

The value $-0.159$ falls in the range $-0.25 < r < 0.25$

Classification: No association

Note: Even though this value is negative, it's so close to zero that we classify it as having no meaningful linear association.

Important considerations and warnings

The correlation coefficient is a powerful tool, but it must be used appropriately. There are important conditions that must be met before using $r$ to measure the strength of an association.

When to use the correlation coefficient

You should only use Pearson's correlation coefficient as a measure of association strength when the following three conditions are met:

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Three Essential Conditions for Using Correlation Coefficient

1. The variables are numeric

The correlation coefficient requires numerical data for both variables. It cannot be used with categorical data (like colours or types of animals).

2. The association is linear

The correlation coefficient specifically measures linear relationships. If the relationship between variables follows a curve or other non-linear pattern, $r$ will not accurately reflect the strength of that relationship.

3. There are no outliers in the data

Outliers (data points that are very different from the rest) can dramatically affect the correlation coefficient. The presence of outliers can give a misleading indication of the strength of the linear association. Always check for outliers in your scatterplot before relying on the correlation coefficient.

Warning: Using the correlation coefficient when these conditions aren't met can lead to incorrect conclusions about the relationship between variables!

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Exam Tip: Finding and Interpreting Correlation Coefficients

When asked to find and interpret a correlation coefficient in an exam:

Always check the scatterplot first to ensure the relationship appears linear
Calculate $r$ using your calculator
Use the classification table to describe the strength
Always include both the strength (weak/moderate/strong) and direction (positive/negative) in your answer
If there are outliers, mention that they may be affecting the correlation coefficient

A complete answer might look like: "The correlation coefficient is $r = 0.78$ , indicating a strong positive linear association between the variables."

Remember!

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Key Points to Remember:

The correlation coefficient $r$ measures the strength of a linear relationship between two variables, with values ranging from $-1$ to $+1$
Perfect correlations occur at the extremes: $r = +1$ (perfect positive), $r = -1$ (perfect negative), and $r = 0$ (no linear relationship)
The sign of $r$ tells you the direction: positive values mean positive associations (both variables increase together), whilst negative values mean negative associations (one variable increases as the other decreases)
The magnitude (absolute value) tells you the strength: values closer to $1$ or $-1$ indicate stronger linear relationships, whilst values closer to $0$ indicate weaker relationships
Only use correlation coefficients when:
- Variables are numeric
- The association is linear
- There are no significant outliers Otherwise the correlation coefficient can be misleading
Use the classification table to describe correlation as weak, moderate, or strong, and always include both strength and direction in your interpretation

Strength of a Linear Relationship: The Correlation Coefficient (VCE SSCE General Mathematics): Revision Notes