Why is weighting important in least squares adjustments, and what does weight represent?

• A. Weight reflects the relative precision (inverse variance) of observations; heavier weights for more precise measurements.
• B. Weight represents the color coding of observations.
• C. Weight determines the orientation of the coordinate axes.
• D. Weight is used to select which observations to discard.

Answer: (A)

In least squares adjustments, weighting represents how much trust we place in each observation based on its uncertainty. Observations aren’t all equally reliable, so we scale their influence accordingly. Mathematically, we minimize a weighted sum of squared residuals, where each residual is multiplied by a weight that reflects its confidence. The weight is typically the inverse of the observation’s variance, so a measurement with smaller uncertainty (higher precision) gets a larger weight and pulls the estimated parameters more strongly, while a noisier measurement gets a smaller weight and exerts less influence.

This setup also affects how precise the final parameter estimates are. In matrix form, the normal equations use a weight matrix, and the covariance of the estimated parameters depends on this weighting. Heavier weights for precise observations tighten the estimated parameters’ uncertainty because those observations contribute more information to the solution.

The other ideas don’t fit the concept of weighting in least squares: color coding has no statistical meaning in this context; the orientation of coordinate axes is a separate design choice; and choosing to discard observations is a different procedure from weighting, though in practice very unreliable data might be downweighted or excluded.