The coefficient of determination (r²) is a key measure in linear regression that indicates how effectively an explanatory variable accounts for variation in a response variable.

The Role of r² in Least-Squares Regression

The coefficient of determination, written as r², plays a central role in interpreting a least-squares regression model because it assesses how well the model explains variability in the response variable. As the AP syllabus emphasizes, r² quantifies the proportion of variance in the response variable that is predictable from the explanatory variable, offering a clear measure of the model’s explanatory strength. Unlike general observations from scatterplots, r² gives a precise numerical value describing how closely the data follow the fitted regression line.

When working with bivariate quantitative data, understanding r² helps determine whether the linear model is useful, somewhat helpful, or largely uninformative. Because r² is computed as the square of the correlation coefficient (r), the value always falls between 0 and 1, making it easy to interpret in terms of proportions or percentages.

Understanding What r² Measures

At its core, r² describes how much of the total variation in the response variable is accounted for by differences in the explanatory variable. Variation in statistics refers to how spread out the values of a variable are around their mean. A strong linear relationship corresponds to less unexplained variation once the regression model is applied.

Because r² is expressed as a proportion, meanings are straightforward. For example, an r² of 0.72 indicates that 72% of the variability in the response variable can be predicted from the linear relationship with the explanatory variable. The remaining 28% is unexplained by the model, attributable to other influences or random variability.

Calculating the Coefficient of Determination

Although technology performs the calculation automatically in nearly all AP Statistics settings, knowing the formula deepens conceptual understanding. Since r² is built from the correlation coefficient, students first recognize that r captures the direction and strength of a linear association, and then squaring r gives the proportion of explained variation.

EQUATION

A single sentence is required here to maintain proper spacing before another definition or equation.

Because r² compares the variation left in the residuals to the total variation in y, it can be interpreted as the fraction of total variation in the response that is explained by the linear model.

This figure illustrates the coefficient of determination for a linear regression, showing how explained and unexplained variation relate to r2r^2r2. The red squares represent residual variation, while the blue squares represent explained variation. The diagram emphasizes that r2r^2r2 can be viewed as one minus the ratio of residual variation to total variation. Source.

Interpreting r² in Context

Interpreting r² always requires contextual reasoning, not just reporting a number. The AP specification emphasizes explaining the significance of r², which means articulating what proportion of variation is explained and what proportion remains unexplained. A strong r² value suggests that the explanatory variable provides meaningful predictive information about the response variable within the scope of the data.

Key Interpretation Points

• Higher r² values (close to 1) mean the model explains a large portion of the response variability, indicating a strong linear fit.

• Lower r² values (close to 0) suggest that the regression line provides little predictive insight, even if a pattern appears present.

• Moderate r² values require careful contextual interpretation, as real-world data often contain noise or complex relationships not captured by a simple linear model.

Because r² depends on the linear form of the relationship, it should never be interpreted without referencing the scatterplot and the underlying form of the data.

In terms of sums of squares, r² = SSreg / SStot = 1 − SSres / SStot, where SSreg is the explained variation and SSres is the unexplained (residual) variation around the regression line.

This plot displays observed data points with a regression line and annotations indicating explained and residual sums of squares. It illustrates how total variation is partitioned into components captured and not captured by the model. The relationship between these components forms the basis of the coefficient of determination. Source.

Using r² to Evaluate Model Quality

The coefficient of determination helps judge whether a linear model is appropriate, useful, or insufficient. While a high r² often signals that the linear model fits well, students must remember that a high r² does not guarantee causation, nor does it confirm that the model is the best choice among possible modeling strategies.

r² as a Tool for Assessing Explanatory Power

Consider r² as a measure of how much better the model predicts responses compared to simply using the mean of the response variable. By quantifying improvement over the mean-only prediction, r² offers a clear way to describe the model’s value.

Common Uses of r² in AP Statistics

• Comparing multiple linear models to determine which explains more variation

• Evaluating whether the linear association is strong enough to justify predictions

• Assessing the overall effectiveness of a least-squares regression line

What r² Cannot Tell Us

Although r² is powerful, its limitations must be acknowledged to avoid misinterpretation. A high r² does not:

• Indicate that the relationship is causal

• Guarantee that the relationship is appropriate for extrapolation

• Confirm that the association is linear without examining residuals

Visually, a larger r² corresponds to data points that cluster more tightly around the regression line, leaving little vertical scatter, whereas a smaller r² leaves the points more widely dispersed around the line.

This scatterplot shows experimental data with a fitted regression line illustrating a strong linear relationship. The close alignment of the points with the line reflects a high r2r^2r2 value. The physical context shown (current and voltage) is extra detail not required in AP Statistics, but the visual fit clearly demonstrates the concept. Source.