Question

The coefficient of determination $$r^2 is $$the ratio of which two types of variations? What does $$r^2$$ measure? What does 1 - $$r^2$$ measure?

Accepted Answer

Understand that the coefficient of determination, denoted as r2, is defined as the ratio of the explained variation to the total variation in the data. Specifically, it is the ratio of the regression sum of squares (explained variation) to the total sum of squares (total variation). Express this relationship mathematically as r2 = \frac{	ext{Explained Variation}}{	ext{Total Variation}} = \frac{SS_{reg}}{SS_{tot}}, where SS_{reg} is the sum of squares due to regression and SS_{tot} is the total sum of squares. Interpret what r2 measures: it quantifies the proportion of the variance in the dependent variable that is predictable from the independent variable(s). In other words, it tells us how well the regression model explains the observed data. Understand that 1 - r2 represents the proportion of the total variation that is not explained by the regression model. This is also known as the unexplained variation or the residual variation. Summarize that r2 measures the strength of the relationship between variables in terms of explained variance, while 1 - r2 measures the amount of variation left unexplained by the model.

The coefficient of determination r^2 is the ratio of which two types of variations? What does r^2 measure? What does 1 - r^2 measure?

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Key Concepts

Coefficient of Determination (r²)

Types of Variation: Explained vs. Total Variation

Unexplained Variation and 1 - r²