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Weighted mean
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What is the mean of a set of measurements (xi) which have different uncertainties (si)?

The more accurate values are more significant than the less accurate ones. Accordingly we calculate a mean as before but with each value weighted by the inverse of its variance: 1/(si)2.

Weighted mean

Put in 10 values for xi and si (hit Autofill, then Plot).

If the si are consistent, we expect approx two thirds of the points to lie within their si values of . Note, if the si values are equal ( i.e. si = s), the formula becomes as for the sample mean. 

The uncertainty in the weighted mean   and if si = s (equal uncertainties) this formula becomes  as for the sample mean.
 

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1999, 2000 The University of Liverpool, Department of Physics

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