Weighted Averages with Statistical and Systematic Uncertainties

We present a methodology for analyzing data that contain both statistical and systematic uncertainties when the correlations among systematic errors are unknown. The approach employs Monte Carlo sampling over the full range of admissible correlation structures, providing realistic estimates of covariance matrices, data weights, weighted averages, and their associated total uncertainties. We further introduce a criterion for evaluating the significance of systematic effects by comparing the dispersion of ensemble results with the average variance. Application of the method to measurements demonstrates how systematic uncertainties can have either a significant or negligible impact on the inferred weighted averages and overall uncertainties. An additional analysis based on three independent determinations of the Hubble constant illustrates the practicality of the framework and indicates that incomplete knowledge of systematic-error correlations has only a minor effect on the resulting weighted average and uncertainty.