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Residuals and Coefficient of Determination (r²)

Residual = y − ŷ. r² = percent of variation in y explained by the linear model on x. r² = 1 means perfect fit.

StatisticsCommon Core DAT-1.CREGVerified correct ✓

Worked examples

A regression analysis yields a Pearson correlation coefficient of r = -0.9. Compute r² (the coefficient of determination) and state what percentage of the variation in y is explained by the linear model.
Answer: 0.81
A regression model relating temperature and ice cream sales predicts ŷ = 90.0 dollars for a given x-value. The observed value is y = 89 dollars. Compute the residual (y − ŷ).
Answer: -1

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Least-Squares Regression Line — ŷ = a + bx

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