 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-2/section-a/4/solution

w_{i} = σ_{i}^{- 2}, \overset{x}{ˉ}_{w} = \frac{\sum _{i} w _{i} x _{i}}{\sum _{i} w _{i}}, σ_{\overset{x}{ˉ}_{w}} = (\sum_{i} w_{i})^{- 1/2} .

Here

\sum_{i} w_{i} = 0.654514

and

\sum_{i} w_{i} x_{i} = 65.805556

\overset{x}{ˉ}_{w} = 100.54 \pm 1.24 .

χ^{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.

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