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Understanding Correlation: Analyzing Relationships Between Two Quantitative Variables

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This lecture focuses on measuring relationships over time using correlation coefficients, specifically exploring the association between two quantitative variables. We examine how to assess whether there is a positive or negative association, the shape of the relationship, and the strength of correlation. Key concepts include the response and explanatory variables, properties of the correlation coefficient (r), and how it reflects linear associations. We also discuss the implications of correlation values and provide real-world examples, such as the correlation between murder and poverty rates.

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Understanding Correlation: Analyzing Relationships Between Two Quantitative Variables

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  1. STA291 Statistical Methods Lecture 10

  2. Measuring something over time… • Like a company’s stock value: • It kind of just sits there, making your eyes glaze over. • And even though we’re measuring one variable (of interest), we need to look at two variables at a time …

  3. Time Plot

  4. Terrible, Misleading Time Plot

  5. Analyzing RelationshipsBetween Two Quantitative Variables • Is there an association between the two variables? • Positive or negative? • What is it’s shape? • How strong is the association? • Notation • Response variable: Y • Explanatory variable: X

  6. Sample Measure of Linear Relationship • Sample Correlation Coefficient: • Alternatively: • Population measures: Divide by N instead of n-1

  7. Properties of the Correlation I • The value of r does not depend on the units(e.g., changing from inches to centimeters) that X and Y are measured in • r is standardized (has no units itself) • r is always between –1 and 1 • r measures the strength and direction of the linear association between X and Y • r>0 positive linear association • r<0 negative linear association

  8. Properties of the Correlation II • r = 1 when all sample points fall exactly ona line with positive slope (perfect positive linear association) • r = – 1 when all sample points fall exactlyon a line with negative slope (perfect negative linear association) • The larger the absolute value of r, the stronger is the degree of linear association

  9. Properties of the Correlation III • If r is close to 0, this does not necessarily meanthat the variables are not associated • It only means that they are not linearly associated • The correlation treats X and Y symmetrically; that is, it does not matter which variable is explanatory (X) and which one is response (Y), the correlation remains the same

  10. All Correlation r = 0.82

  11. Scatter Diagram of Murder Rate (Y) andPoverty Rate (X) for the 50 States r = 0.63

  12. Looking back • Time plots • General two-variable analysis • Sample correlation, r

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