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Correlation Analysis
Measuring and interpreting the linear relationship between two numeric variables using the Pearson correlation coefficient.
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Measuring and interpreting the linear relationship between two numeric variables using the Pearson correlation coefficient.
The Pearson correlation coefficient r measures the strength and direction of a linear relationship between two numeric variables. It always lies between −1 and 1: r = 1 means a perfect positive linear relationship, r = −1 a perfect negative one, and r = 0 no linear relationship. Correlation does not imply causation — two variables can correlate without one causing the other. Scatter plots are the standard visual for detecting correlation patterns and identifying outliers that can distort r.
r = Σ(x_i − x̄)(y_i − ȳ) / √[Σ(x_i − x̄)² · Σ(y_i − ȳ)²]
- r = 0.95 → strong positive linear relationship
- r = −0.80 → strong negative linear relationship
- r ≈ 0.05 → essentially no linear relationship
How to recognize it
- The question asks you to judge whether two variables tend to increase or decrease together
- A scatter plot is usually provided or implied
- Keywords: 'correlation', 'trend', 'linear relationship', 'association'
Common mistakes
- Confusing correlation with causation
- Reporting r for a non-linear pattern (r captures only linear structure)
- Being fooled by outliers that pull r toward ±1 or toward 0