Residuals, Residual Plots, and r²

Residuals and Assessing Fit

By the end of this lesson, you will be able to construct and interpret residual plots, use $r^2$ to assess model fit, and identify when a linear model is or is not appropriate.

Residuals: Review

$$\text{Residual} = y - \hat{y}$$

Each observation has a residual — the vertical distance from the point to the regression line. Positive = above the line; negative = below.

The LSRL is defined so that the sum of all residuals = 0 and the sum of squared residuals is minimized.

The Residual Plot

A residual plot graphs the residuals ($y$-axis) against the explanatory variable or predicted values ($x$-axis). It is the primary diagnostic tool for checking whether a linear model is appropriate.

A good residual plot (linear model is appropriate):

• Points are scattered randomly above and below the zero line
• No pattern — just noise
• Spread is roughly constant across all x values

A bad residual plot (signals a problem):

• Curved pattern: the relationship is nonlinear; a linear model is not appropriate
• Fan shape (heteroscedasticity): spread increases as x increases; the linear model's predictions become less reliable for larger x values
• Systematic pattern: the model is missing something important

Interpreting Residual Plots: Examples

+2 |    .   .    .
+1 |  .    .  .
 0 |------------------
-1 |   .  .    .
-2 |       .    .

Example 1 — Good fit:

Random scatter above and below zero → linear model is appropriate.

Example 2 — Curved pattern:

+2 |
+1 |  .   .       .   .
 0 |------------------
-1 |      .  .  .
-2 |

Points curve down then up (or up then down) → nonlinear relationship; linear model is not appropriate.

Example 3 — Fan shape:

+3 |                    .    .
+1 |    . .   .  .         .
 0 |------------------
-1 |    .     .        .
-3 |                       .

Spread increases → constant variance assumption violated.