Correlation

Learn correlation and explained variation, correlation and units.

Lesson 2 of 32 parts Practice Flashcards

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Correlation and Explained Variation

Objective: Explain correlation and explained variation, apply the stated method, and check its conditions.

Pearson correlation r describes linear association on a scale from −1 to 1. In simple least-squares regression with an intercept, R²=r². This summary does not prove causation or guarantee an adequate model; inspect the scatterplot and residuals for structure. This r² identity has a specified regression setting.

Formulas

For a simple fitted line with intercept: R²=r²

Worked examples

Try itA synthetic fitted simple linear model with an intercept has correlation r=7/13. Find r².Show answer

Step 1: For a simple least-squares line with an intercept, R²=r². Step 2: Square 7/13. Step 3: r²=49/169; it does not establish causation.

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Correlation and Explained Variation

Objective: Explain correlation and explained variation, apply the stated method, and check its conditions.

Formulas

For a simple fitted line with intercept: R²=r²

Worked examples

Try itA synthetic fitted simple linear model with an intercept has correlation r=7/13. Find r².Show answer

Step 1: For a simple least-squares line with an intercept, R²=r². Step 2: Square 7/13. Step 3: r²=49/169; it does not establish causation.

Correlation and Units

Objective: Explain correlation and units, apply the stated method, and check its conditions.

Pearson correlation is unchanged by adding constants or multiplying a variable by a positive constant. A negative scale factor reverses the correlation sign. Unit changes preserve the pattern but affect raw slopes and intercepts; correlation is unitless. A slope’s numerical value can change even when correlation does not.

Formulas

Positive scale change preserves r

Worked examples

Try itA synthetic dataset has correlation r=7/13. If every x is converted by x*=100x, what is the new correlation?Show answer

Step 1: Multiplying one variable by a positive constant preserves correlation. Step 2: The scale factor 100 is positive. Step 3: The correlation remains 7/13; measurement units do not change it.

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