Estimate a weighted mean using independent observation units and optional known noise variances. Weights express relative importance, not replicate counts. Variances may be NaN where unknown. Empirical mean variance uses Kish sample size and the unbiased weighted dispersion; supplied noise propagates as sum of squared normalized weights times variance. The larger estimate is retained rather than counting noise twice. Unknown noise at a singleton remains unestimated. Partial supplied noise gives only its known contribution. This is a descriptive approximation, not a calibrated posterior or an anytime-valid confidence interval.

Estimate paired left-minus-right differences without assuming independent responses within a pair.