Definition: A residual plot graphs the differences between observed and predicted values against the predictor, showing whether a linear model fits or a pattern remains.
What is a residual?
A residual is actual minus predicted: residual = y – y-hat. Plot residuals against x (or against the predicted values). If the line is a good model, the dots scatter randomly around 0 with no pattern.
How do you read the plot?
- Random scatter: a linear model is appropriate.
- Curved pattern: a nonlinear model would fit better.
- Fan shape: the spread changes with x, which breaks a model assumption.
Worked example: a regression line is y-hat = 2.5x + 10. At x = 6 the prediction is 25. If the actual value is 27, the residual is 27 – 25 = +2, a point above the line. If the actual value is 22, the residual is -3.
Residuals from a least-squares line always add up to zero.
What mistake occurs?
Subtracting in the wrong order, which flips the sign. Mathbot says: actual minus predicted, so points above the line are positive. A high r2 alone does not prove a line is appropriate.