Uncertainty¶

$$ \large x_1 \pm \sigma_1, \quad x_2 \pm \sigma_2, \quad x_3 \pm \sigma_3, \quad \cdots, \quad x_N \pm \sigma_N $$

Weighted Average¶

$$ \large \bar{x} = \frac{ \sum^N_{i=1} w_i x_i}{\sum^N_{i=1} w_i} $$

where

$$ \large w_i = \frac{1}{\sigma^2_i} $$

Uncertainty¶

$$ \large \begin{split} \sigma & = \left( \sum^N_{i=1} w_i \right)^{-\frac{1}{2}} \\ & = \left( \sum^N_{i=1} \frac{1}{\sigma^2_i} \right)^{-\frac{1}{2}} \end{split} $$

Time¶

$$ \large t = \frac{\sigma^2}{(\text{uncertainty})^2} $$