Residuals and the Appropriateness of the Linear Model

What is a Residual?

A residual is the difference between the actual (observed) value and the predicted value from the regression line:

$$\text{residual} = y - \hat{y} = \text{actual} - \text{predicted}$$

• Positive residual: actual value is above the regression line
• Negative residual: actual value is below the regression line
• Zero residual: point lies exactly on the regression line

Worked Example

Equation: $\widehat{\text{score}} = 31.5 + 9.2 \times \text{hours}$

The Residual Plot

A residual plot graphs residuals ($y - \hat{y}$) on the y-axis against the explanatory variable ($x$) on the x-axis.

Interpreting a Residual Plot

Good Residual Plot (Linear Appropriate)

Residual
  +4 |    ×          ×
  +2 |         ×
   0 |----×---------×------
  -2 |  ×      ×
  -4 |              ×
     +-------------------> x

Points randomly scattered above and below zero — no pattern.

Bad Residual Plot (Curved Pattern — Non-linear)

Residual
  +4 |  ×         ×
  +2 | ×  ×     ×  ×
   0 |--------×-----------
  -2 |          ×
  -4
     +-------------------> x

Clear U-shape — the linear model is NOT appropriate.

Using Residuals to Assess the Linear Model

Step 1: Calculate residuals for all data points
Step 2: Plot residuals against $x$
Step 3: Look for patterns
Step 4: Conclude whether linear model is appropriate

Sum of Residuals

For the least squares line, the sum of residuals always equals zero:

$$\sum(y - \hat{y}) = 0$$

This is a mathematical property of the least squares method.

KEY TAKEAWAY: A random scatter in the residual plot confirms a linear model is appropriate. Any systematic pattern (curve, fan shape) means the linear model should not be used.

EXAM TIP: VCAA commonly shows a residual plot and asks you to comment on the appropriateness of the linear model. Describe the pattern you see and state your conclusion clearly.

COMMON MISTAKE: Confusing a residual plot that looks “messy” (random scatter = good!) with one that has a clear pattern (bad). Random scatter is actually what you want to see.