4.2.2 Mean of the  4.2 Applications  4.2 Applications

4.2.1 Weighted averages

If measurements $\{x\}$ are taken from a population with a Gaussian distribution, but are made with varying measurement uncertainties $\{\sigma\}$, what is the best estimate of the sample mean? The likelihood function for this case is

$${\cal L}(a) = \prod_i {{1}\over{\sigma_i\sqrt{2\pi}}} \exp\{-{{(x_i-a)^2}\over{2\sigma_i^2}}\}$$ (4.12)

 

and

$$W(a) = -\sum_i {{(x_i-a)^2}\over{2\sigma_i^2}} + {\rm constant}.$$ (4.13)

 

The maximum occurs for

$${{\partial W(a)}\over{\partial a}}\Bigr\vert _{a^*} = \sum_i ......{x_i}\over{\sigma_i^2}}) -\sum_i {{a^*}\over{\sigma_i^2}} = 0$$ (4.14)

 

or

$$a^* = {{\sum_i(x_i /\sigma_i^2)}\over{\sum_i(1 /\sigma_i^2)}} .$$ (4.15)

 

This is a weighted average of the measurements, with weight factors inversely proportional to the square of the uncertainty.

Because

$${{\partial^2W}\over{\partial a^2}}\Bigr\vert _{a^*} = -\sum_i{{1}\over{\sigma_i^2}} ,$$ (4.16)

 

the estimated uncertainty in the weighted average, from (4.11), is

$$\sigma_{a^*} = \bigl[\sum_i {{1}\over{\sigma_i^2}}\bigr]^{-1/2} .$$ (4.17)

  

   
 4.2.2 Mean of the  4.2 Applications  4.2 Applications