Pearson’s Correlation Coefficient Revision Notes for HSC SSCE Mathematics Standard

Pearson's Correlation Coefficient

What is Pearson's correlation coefficient?

When we analyse bivariate data (data with two variables), we often want to know how closely the two variables are related. Pearson's correlation coefficient, denoted by the symbol $r$ , is a statistical measure that tells us the strength of a linear relationship between two variables.

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The correlation coefficient helps us understand whether two variables move together in a predictable pattern, and if so, how strong that pattern is.

Understanding the correlation coefficient

The range of values

Pearson's correlation coefficient always falls between $-1$ and $+1$ :

$-1 \leq r \leq +1$

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The closer the value of $r$ is to $-1$ or $+1$ , the stronger the linear relationship between the variables.

Values near $+1$ indicate a strong positive relationship
Values near $-1$ indicate a strong negative relationship
Values near $0$ indicate little to no linear relationship

Types of correlation

There are three main types of correlation:

Positive correlation ( $0$ to $+1$ )

When one variable increases, the other variable also increases
Similarly, when one variable decreases, the other decreases as well
Both variables move in the same direction
Example: As foot length increases, height tends to increase

Zero or no correlation ( $r = 0$ )

There is no linear relationship between the variables
Knowing the value of one variable doesn't help predict the other
The data points on a scatterplot appear randomly scattered

Negative correlation ( $-1$ to $0$ )

When one variable increases, the other variable decreases
The variables move in opposite directions
Example: As the amount of exercise increases, body weight might decrease

Visual representation of correlation

Looking at scatterplots can help us understand different correlation values. The pattern of points on the graph shows the strength and direction of the relationship.

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Understanding Scatterplot Patterns:

The way data points cluster on a scatterplot tells us about the strength and direction of the correlation.

Perfect correlations:

When $r = +1$ : all points lie exactly on a straight line sloping upward
When $r = 0$ : points are scattered randomly with no clear pattern
When $r = -1$ : all points lie exactly on a straight line sloping downward

Varying strengths of positive correlation:

Weak positive ( $r = +0.3$ ): points show a slight upward trend but are quite scattered
Moderate positive ( $r = +0.5$ ): points show a clear upward trend with moderate spread
Strong positive ( $r = +0.9$ ): points cluster tightly around an upward sloping line

Classifying the strength of associations

We can classify the strength of a linear association using the value of Pearson's correlation coefficient. The table below provides guidelines for interpretation:

Association	Pearson's correlation coefficient
Strong positive	$r = +0.75$ to $+0.99$
Moderate positive	$r = +0.50$ to $+0.74$
Weak positive	$r = +0.25$ to $+0.49$
No linear association	$r = -0.24$ to $+0.24$
Weak negative	$r = -0.25$ to $-0.49$
Moderate negative	$r = -0.50$ to $-0.74$
Strong negative	$r = -0.75$ to $-0.99$

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Key points about classification:

Values closer to 0 indicate little or no linear relationship
Values closer to +1 or -1 indicate stronger relationships
The sign (positive or negative) tells us the direction of the relationship
The magnitude (size) tells us the strength of the relationship

Calculating Pearson's correlation coefficient

Modern scientific and graphing calculators can compute Pearson's correlation coefficient quickly and accurately. Here's the general process:

Using a scientific calculator

Step 1: Enter statistics mode

Access the statistics or data analysis mode on your calculator

Step 2: Select the appropriate regression type

Choose the linear regression option (often shown as "A + BX" or similar)

Step 3: Enter your data

Input the first variable's data into List 1
Input the second variable's data into List 2
Ensure data pairs are entered in the correct order

Step 4: Calculate the correlation coefficient

Select the regression calculation option
Choose to display the correlation coefficient ( $r$ )

Worked example: foot length and height

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Worked Example: Finding and Classifying Pearson's Correlation Coefficient

Problem: The table below shows foot length and height in centimetres for a group of students.

Foot length (cm)	24.5	25.6	26.1	25.9	26.2	27.0	27.4	27.5	28.1
Height (cm)	167	168	170	174	176	180	181	185	187

Part a) Find Pearson's correlation coefficient.

Part b) Classify the strength of the association.

Solution:

Part a) Finding the correlation coefficient:

Following the calculator steps:

Enter statistics mode and select linear regression
Enter foot length data into List 1:

$24.5, 25.6, 26.1, 25.9, 26.2, 27.0, 27.4, 27.5, 28.1$
Enter height data into List 2:

$167, 168, 170, 174, 176, 180, 181, 185, 187$
Calculate regression statistics and display $r$

The calculator gives us:

$r = 0.94781831$

We can round this to r = 0.95 (to two decimal places).

Part b) Classifying the strength:

The value $r = 0.95$ falls in the range $+0.75$ to $+0.99$
According to our classification table, this indicates a strong positive linear association

Interpretation: There is a strong positive relationship between foot length and height. This means that as foot length increases, height tends to increase as well, and the relationship is very strong (the points would cluster tightly around a line).

Important: correlation vs causation

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Critical Point: Correlation Does NOT Mean Causation

A high correlation between two variables does not mean that one variable causes changes in the other.

Even if two variables are strongly correlated, this doesn't prove a cause-and-effect relationship. There could be other explanations:

Both variables might be influenced by a third factor
The relationship might be coincidental
The correlation might only be true for the specific data set examined

Example: Ice cream sales and drowning incidents are positively correlated, but ice cream doesn't cause drowning. Both increase in summer due to hot weather (the third factor).

Always be careful when interpreting correlation results. Statistical association is different from causation.

Remember!

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Key Points to Remember:

Pearson's correlation coefficient ( $r$ ) measures the strength and direction of a linear relationship between two variables, with values ranging from $-1$ to $+1$
Positive correlation means both variables move in the same direction, negative correlation means they move in opposite directions, and zero correlation means no linear relationship exists
Strong associations have $|r|$ values between $0.75$ and $0.99$ , moderate associations between $0.50$ and $0.74$ , and weak associations between $0.25$ and $0.49$
You can calculate $r$ using a scientific or graphing calculator by entering your bivariate data and selecting the regression analysis function
Correlation does not imply causation – even if two variables are strongly correlated, this doesn't prove that one causes changes in the other

Pearson’s Correlation Coefficient (HSC SSCE Mathematics Standard): Revision Notes