Physics307L:Schedule/Week 9 agenda/Weighted

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The problem: You have two separate measurements with different uncertainties. How do you combine them?

Example: Speed of Light

Measurement #1 (25 +/- 3) cm / ns

Measurement #2 29(2) cm / ns

What is the best way to combine these measurements?

What do you expect the new means, uncertainties to be?

Derivation

The formula

Wikipedia article

\bar{x} = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i},

which means:

\bar{x} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}.
w_i = \frac{1}{\sigma_i^2}.

Excel examples

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