In simple linear regression, one key question is how much of the response variable’s variability is accounted for by the explanatory variable. The coefficient of determination gives that information directly.

Understanding the Coefficient of Determination

In AP Statistics, the coefficient of determination is written as $r^2$ and is used with a simple linear regression model. It tells you how much of the variability in the response variable can be explained by its linear relationship with the explanatory variable.

The word variation refers to how much the response values differ from one another. A regression model tries to account for part of that difference by using the explanatory variable. If the linear model fits well, a larger share of the response variable’s variation is explained. If the fit is weaker, more of the variation remains unexplained.

Three scatterplots with regression lines illustrate that $r^2$ increases as points lie closer to the least-squares line. The visual message matches the interpretation of $r^2$ as the proportion of variation in the response variable explained by its linear relationship with the explanatory variable. Source

When AP questions ask for an interpretation of $r^2$, they are asking about the response variable, not the explanatory variable. This is one of the most important points to keep clear.

Explained and unexplained variation

The coefficient of determination is a proportion, so it can be written as a decimal or expressed as a percent. A value closer to $1$ means the model explains more of the response variable’s variation. A value closer to $0$ means the model explains less.

Because $r^2$ is a proportion, it has no units.

A scatterplot with a least-squares regression line highlights one point and its vertical residual (observed $y$ minus predicted $\hat{y}$). This picture connects “unexplained variation” to the vertical deviations of data points from the regression line. Source

It does not describe individual observations one by one. Instead, it summarizes how much of the overall variation in the response variable is associated with the linear model.

Interpreting $r^2$ in Context

A correct AP interpretation of $r^2$ should include:

• the percentage or proportion explained

• the response variable

• the explanatory variable

• the idea of a linear relationship or linear model

A strong interpretation follows this pattern: About $100r^2$ of the variation in [response variable] is explained by the linear relationship with [explanatory variable].

Be careful with wording. The phrase “variation in the response variable” is essential. Saying that $r^2$ is the percent of points on the line, the percent of correct predictions, or the percent caused by the explanatory variable is not correct.

What $r^2$ does and does not tell you

A larger $r^2$ means the linear model accounts for more of the response variable’s variability in the data set. This makes $r^2$ useful for describing how well the explanatory variable helps account for changes in the response variable.

A smaller $r^2$ means the model explains less of that variability. That does not automatically make the model useless. In many real settings, response variables are influenced by many factors, so even a moderate explained proportion may still be meaningful in context.

Key Properties of $r^2$

Several properties of the coefficient of determination are important for AP Statistics:

• In simple linear regression, $r^2$ is between $0$ and $1$.

• Since it is a square, $r^2$ cannot be negative.

• It describes explained variation in the response variable, not in the explanatory variable.

• It is specifically tied to the linear regression model.

Because $r^2$ is squared, it does not show whether the relationship is positive or negative. It measures how much of the variation is explained, not the direction of the relationship.

Common Mistakes to Avoid

Students often lose points by interpreting $r^2$ too loosely. Avoid these errors:

• saying the explanatory variable explains $r^2$ of the observations

• saying $r^2$ of the response variable is caused by the explanatory variable

• treating $r^2$ as a measure of slope

• forgetting to mention the response variable

• leaving out the word variation

Another common mistake is giving a definition with no context. On the AP exam, your interpretation should name the actual variables in the problem. The wording should match the situation being studied.

Writing AP-Ready Interpretations

When you see $r^2$ in a regression setting, move through these steps:

• identify the response variable

• identify the explanatory variable

• convert $r^2$ to a percent if that makes the interpretation clearer

• state that this percent of the variation in the response variable is explained by the linear relationship with the explanatory variable

This approach keeps the interpretation focused exactly where AP Statistics wants it: on the proportion of response-variable variation explained by the explanatory variable.