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?

Verified step by step guidance

1. Understand that the coefficient of determination, denoted as $r^2$, 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).
2. Express this relationship mathematically as $r^2=\; \backslash frac\{\backslash text\{Explained\; Variation\}\}\{\backslash text\{Total\; Variation\}\}\; =\; \backslash frac\{SS\_\{reg\}\}\{SS\_\{tot\}\}$, where $SS\_\{reg\}$ is the sum of squares due to regression and $SS\_\{tot\}$ is the total sum of squares.
3. Interpret what $r^2$ 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.
4. Understand that $1\; -r^2$ 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.
5. Summarize that $r^2$ measures the strength of the relationship between variables in terms of explained variance, while $1\; -r^2$ measures the amount of variation left unexplained by the model.

Key Concepts

Coefficient of Determination (r²)

The coefficient of determination, denoted as r², measures the proportion of the total variation in the dependent variable that is explained by the independent variable(s) in a regression model. It quantifies how well the regression line fits the data.

Types of Variation: Explained vs. Total Variation

Total variation refers to the overall variability in the observed data, while explained variation is the portion of this variability accounted for by the regression model. The ratio of explained variation to total variation gives r².

Unexplained Variation and 1 - r²

The term 1 - r² represents the proportion of total variation that the regression model fails to explain, also known as unexplained or residual variation. It indicates the amount of variability due to factors outside the model.