In least squares adjustment, which statement best describes a residual?

• A. The sum of all observations.
• B. The weighted average of all residuals.
• C. The final adjusted coordinate.
• D. The difference between an observed value and its predicted value from the adjustment model.

Answer: (D)

In least squares adjustment, a residual is the discrepancy for a single observation between what was actually measured and what the model predicts for that measurement. It’s the difference e = observed value minus predicted value from the fitted model.

This difference is what the adjustment process minimizes in a squared-sense across all data, so residuals reveal how well the model captures the measurements. Small, randomly scattered residuals suggest a good fit, while systematic patterns or large residuals point to issues like model misspecification, outliers, or changing variance.

So the residual is not the sum of all observations, not a weighted average of residuals, and not the final adjusted coordinate (which is an estimate of the parameter or position). It’s the per-observation difference between what was measured and what the model predicts.