Averaging Measurements with Hidden Correlations and Asymmetric Errors

In practical applications, however, one is often confronted with the problem that the measurement errors which are quoted define a 68% confidence level interval around a Maximum Likelihood estimate, whereas one would need an unbiased estimator with its standard deviation in order to form a weighted average. In addition, one usually has to deal with statistical and systematic uncertainties, and different results may be correlated, e.g. due to common uncertainties in some theoretical prediction, while a quantitative estimate for the size of the correlation does not exist.