Residuals vs. Fitted: Plot residuals on the y-axis vs. fitted values on the x-axis. You want a random, structureless cloud centered on zero. Watch for:
- Curvature → nonlinear relationship, the linear model is wrong.
- Fan or funnel shape → heteroscedasticity (non-constant variance).
- Large individual outliers → may unduly influence the model.
Normal Q-Q: Plots sorted residuals against the quantiles you'd expect from a normal distribution. Points hugging the diagonal mean normality holds. Shapiro-Wilk formalizes this as a hypothesis test.
Scale-Location: Plots $\sqrt{|\text{residual}|}$ vs. fitted values. A flat, horizontal spread of points confirms homoscedasticity; an upward slope signals that variance is growing with the fitted value (heteroscedasticity).
Homoscedasticity test: Breusch-Pagan test. Null: residual variance is constant. Significant p → heteroscedasticity. Fix: log-transform the outcome, use weighted least squares, or switch to a model that accommodates non-constant variance.

Summary statistics to watch:

Mean residual — should be near zero. A non-zero mean indicates systematic bias; the model is consistently over- or under-predicting.
SD of residuals — the typical size of a prediction error. Smaller is better, but what counts as "small" depends on the scale of your outcome.
SSE (Sum of Squared Errors) — $\sum (y_i - \hat{y}_i)^2$. The raw total of all squared residuals. It shrinks as the model fits better, and is the quantity OLS regression directly minimizes.

Bayesian Statistics