Algebra
Data and statistics
Residuals
20 practice questions
2 video lessons
Theory + worked examples
Theory
A residual measures how far a data point is from the line:
\[\text{residual}=\text{actual}-\text{predicted}.\]
It is the vertical distance from the point to the line.
Small, randomly scattered residuals mean the line fits well.
Residuals are the vertical gaps to the line.
Understanding residuals.
The residual:
\[\text{residual}=y_{\text{actual}}-y_{\text{predicted}}\]
Positive above the line, negative below.
How to use residuals
- Predict \(y\) from the line.
- Subtract: actual minus predicted.
- Small residuals mean a good fit.
- A random residual plot supports a linear model.
Example 1 — Compute a residual
The actual value is \(7\); the line predicts \(5\). Find the residual.
Solution
Residual is actual minus predicted.
| \(7-5\) | \(=\) | \(2\) |
Example 2 — Negative residual
Actual \(4\), predicted \(6\). Find the residual.
Solution
Below the line gives a negative residual.
| \(4-6\) | \(=\) | \(-2\) |
Example 3 — Good fit
What do small residuals indicate?
Solution
A line that fits the data closely.
Example 4 — Residual plot
What does a random residual plot suggest?
Solution
A linear model is appropriate.
Common pitfalls
Residual is actual minus predicted, in that order.
A patterned residual plot means a poor model.
Residuals can be negative (below the line).
Frequently asked questions
What is a residual?
Actual value minus the value predicted by the line.
What do small residuals mean?
The line fits the data well.
Can a residual be negative?
Yes — when the point is below the line.
What does a residual plot show?
Whether a linear model is appropriate.
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