5.3 Assessing Model Fit

Reading the residual plot

A residual plot graphs residuals (observed − predicted) against the explanatory variable. If the linear model is appropriate, residuals should show no pattern — just random scatter around zero. A curved pattern signals the relationship is not linear, meaning a straight-line model isn't a good fit for the data.

Standard deviation of residuals (s)

The standard deviation of the residuals (s) measures the typical size of prediction error — how far, on average, actual y-values fall from the regression line, in the units of y. Smaller s means predictions are more precise.

Coefficient of determination (r²)

r² is the proportion of variation in y that is explained by the linear model with x. An r² of 0.81 means 81% of the variability in y is accounted for by the least-squares regression line on x; the rest is due to other factors or unexplained scatter.

Key terms

Residual plot

A scatterplot of residuals vs. the explanatory variable, used to check if a linear model is appropriate; should show no pattern if the fit is good.

Residual

Observed y − predicted y; measures how far a point falls from the regression line.

Standard deviation of residuals (s)

The typical size of the prediction error, in the units of y; smaller s = more precise predictions.

Coefficient of determination (r²)

The proportion of variability in y explained by the linear regression model on x.

Sign of a poor linear fit

A curved pattern in the residual plot, or a low r² combined with a large s.

Try an AP-style question

r² = 0.64 for a model predicting score from hours studied. What does this mean?

1. Scores increase by about 64 points for each extra hour
2. The slope of the regression line is 0.64 points per hour
3. 64% of students' scores are predicted exactly by the line
4. 64% of the variation in scores is explained by the linear model with hours