  The weighted average of inconsistent data is a common and tedious problem
that many scientists have encountered. The standard weighted average is not
recommended for these cases, and different alternative methods are proposed in
the literature. Here, we introduce a new method based on Bayesian statistics
for a broad application that keeps the number of assumptions to a minimum. The
uncertainty associated with each input value is considered just a lower bound
of the true unknown uncertainty. By assuming a non-informative (Jeffreys')
prior for true uncertainty and marginalising over its value, a modified
Gaussian distribution is obtained with smoothly decreasing wings, which allows
for a better treatment of scattered data and outliers. The proposed method is
tested on a series of data sets: simulations, CODATA recommended value of the
Newtonian gravitational constant, and some particle properties from the
Particle Data Group, including the proton charge radius and the mass of the W
boson. For the latter in particular, contrary to other works, our prediction
lies in good agreement with the Standard Model. A freely available Python
library is also provided for a simple implementation of our averaging method.
