Pearson’s r: Measuring the Strength and Direction of Linear Relationships

Understanding the relationship between two variables is crucial in data analysis.
While scatter plots help us visualize this relationship, they don’t give us an objective measure of strength or direction.

This is where Pearson’s r comes in. It quantifies the strength and direction of a linear relationship.

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📚 This post is part of the "Intro to Statistics" series

🔙 Previously: Correlation Between Variables: Contingency Tables and Scatter Plots

🔜 Next: Regression: Predicting Relationships Between Variables

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🎯 Types of Correlation

There are four main types of correlation you may encounter in data:

1. Strong Positive Linear Relationship
   • As one variable increases, the other increases in a perfect straight line.
2. Weak Positive Linear Relationship
   • The variables increase together, but not in a perfect linear fashion.
3. Strong Negative Linear Relationship
   • As one variable increases, the other decreases in a perfect straight line.
4. Perfect Negative Linear Relationship
   • A perfect inverse relationship, where every increase in one variable exactly corresponds to a decrease in the other.

However, not all relationships are linear. Many datasets show curves or non-linear relationships, where Pearson’s r would not be applicable.

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Visualizing the Types of Correlation and Linear vs Non-Linear Relationships:

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📊 Why Scatter Plots Alone Aren’t Enough

A scatter plot helps visualize the relationship between two variables, but it doesn’t quantify how strong or weak the correlation is.

The scatter plot might show a weak or strong relationship, but it doesn’t give you a precise measurement. For that, we use Pearson’s r, which quantifies both the strength and direction of the relationship.

🧮 The Equation for Pearson’s r

Pearson’s r is calculated as:

\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \]

Where:

• \( x_i \) = data points of variable X
• \( y_i \) = data points of variable Y
• \( \bar{x} \) = mean of variable X
• \( \bar{y} \) = mean of variable Y

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📈 How Pearson’s r Works

Pearson’s r produces a value between -1 and +1:

• r = +1 → Perfect positive correlation
• r = -1 → Perfect negative correlation
• r = 0 → No correlation
• 0 < r < 1 → Positive relationship (but not perfect)
• -1 < r < 0 → Negative relationship (but not perfect)

Interpretation:

• Strength: The closer Pearson’s r is to +1 or -1, the stronger the relationship.
• Direction:
  • Positive (+) = as one variable increases, the other increases
  • Negative (-) = as one variable increases, the other decreases

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📉 Visualizing Pearson’s r: How the Data Points Collect Around the Line

Let’s consider how the points align along the line. A strong correlation means the data points are tightly grouped along a straight line.

• Perfect Correlation: Data points lie exactly along the line.
• Weak Correlation: Data points are spread out, but still follow the general trend.
• No Correlation: Data points are scattered randomly.

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🧪 Example Calculation

Let’s walk through a small example:

Data:

X	Y
1	3
2	4
3	6
4	8
5	10

Now calculate Pearson’s r:

1. Calculate the means:
   • \( \bar{x} = 3 \)
   • \( \bar{y} = 6.2 \)
2. Calculate the numerator of the formula:

\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (1-3)(3-6.2) + (2-3)(4-6.2) + (3-3)(6-6.2) + (4-3)(8-6.2) + (5-3)(10-6.2) \]

\[ = 4.4 + 2.2 + 0 + 2.2 + 2.4 = 11.2 \]

1. Calculate the denominator of the formula (the squared deviations for both variables):

\[ \sum (x_i - \bar{x})^2 = (1 - 3)^2 + (2 - 3)^2 + (3 - 3)^2 + (4 - 3)^2 + (5 - 3)^2 = 4 + 1 + 0 + 1 + 4 = 10 \]

\[ \sum (y_i - \bar{y})^2 = (3 - 6.2)^2 + (4 - 6.2)^2 + (6 - 6.2)^2 + (8 - 6.2)^2 + (10 - 6.2)^2 = 10.24 + 4.84 + 0.04 + 3.24 + 14.44 = 32.8 \]

1. Now calculate Pearson’s r:

\[ r = \frac{11.2}{\sqrt{10 \times 32.8}} = \frac{11.2}{\sqrt{328}} = \frac{11.2}{18.1} \approx 0.62 \]

This means the correlation is moderately strong positive, but not perfect.

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🧑‍💻 Python Code for Pearson’s r

You can also calculate Pearson’s r using Python’s numpy:

import numpy as np

# Data
x = np.array([1, 2, 3, 4, 5])
y = np.array([3, 4, 6, 8, 10])

# Calculate Pearson's r
r = np.corrcoef(x, y)[0, 1]
print("Pearson's r:", r)

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🧠 Level Up: When Not to Use Pearson’s r

Pearson’s r assumes a linear relationship between two variables. But what if the data forms a curve ?

• 📊 If your data is non-linear , Pearson’s r won’t give you an accurate measure of correlation. The relationship might look like a U-shape or exponential curve, which Pearson’s r can’t capture.
• 🔄 For non-linear relationships , try using Spearman’s rank correlation , which assesses monotonic relationships (whether increasing or decreasing).
• 🎯 Always visualize the data first with a scatter plot to check if the relationship is linear before calculating Pearson’s r.

Understanding when Pearson’s r applies — and when it doesn’t — is key to reliable data analysis.

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📌 Try It Yourself

Q: Given the following data:

X	Y
1	3
2	4
3	6
4	8
5	10

Calculate the Pearson’s r for this data.

💡 Show Answer

✅ Step 1: Calculate the means for X and Y: - \( \bar{x} = 3 \), \( \bar{y} = 6.2 \)
Step 2: Apply the Pearson's r formula: - Compute the numerator and denominator as shown in the example above, which results in \( r = 0.62 \).

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🔁 Summary

Relationship Type	Pearson’s r Interpretation
Strong Positive	( r = +1 )
Weak Positive	( 0 < r < 1 )
Strong Negative	( r = -1 )
No Correlation	( r = 0 )

🎯 Next: Regression - Predicting Relationships Between Variables

In our upcoming post, we will delve into regression analysis, which goes beyond correlation to help us predict the value of one variable based on another.

Key points we will cover:

1. What is regression?
2. Linear regression equation and its components
3. How to interpret the regression line in a scatter plot
4. How regression helps us predict future values

Stay tuned as we explore this essential technique that forms the backbone of predictive modeling and machine learning!