Past exam of the physics A course of the University of Cambridge / 2010 / Paper 2 / Section A / 4 / Solution

Assuming independent Gaussian measurement errors with the quoted standard deviations, use the inverse-variance weighted mean:Here ${\sum }_i{w}_i=0.654514$ and ${\sum }_i{w}_i{x}_i=65.805556$, givingThe consistency statistic is ${\chi }^2=4.45$ for four degrees of freedom, so there is no evident excess scatter requiring error rescaling. A shared systematic error would require a covariance matrix and would not be reduced by this averaging rule.