Residuals & Residual Plots Cheat Sheet

Key Facts

• A residual is calculated using $e = y - \hat{y}$, where $y$ is the observed value and $\hat{y}$ is the predicted value.
• A positive residual means the observed value is above the regression line because $y > \hat{y}$.
• A negative residual means the observed value is below the regression line because $y < \hat{y}$.
• For a least-squares regression line, the residuals always have a sum of $\sum e = 0$, up to rounding error.
• A residual plot graphs each point as $(x, e)$, using the original explanatory variable $x$ and the residual $e$.
• A residual plot with random scatter around $e = 0$ supports using a linear model.
• A curved pattern in a residual plot suggests that a nonlinear model may fit the data better than a line.
• A fan-shaped residual plot shows nonconstant spread, meaning prediction errors change size as $x$ changes.

Vocabulary

Residual

A residual is the difference between an observed response value and the value predicted by a model, calculated as $e = y - \hat{y}$.

Predicted Value

A predicted value, written $\hat{y}$, is the response value estimated by a regression equation for a given $x$.

Residual Plot

A residual plot is a graph of residuals against the explanatory variable, usually shown as points $(x, e)$.

Least-Squares Regression Line

A least-squares regression line is the line that minimizes the sum of squared residuals, $\sum e^2$.

Outlier

An outlier is a data point with an unusually large residual or an unusual position compared with the rest of the data.

Nonlinear Pattern

A nonlinear pattern occurs when residuals show a curve or systematic shape instead of random scatter around $0$.