Simple ##\chi^2## Tests for Weighted Averages and Linear Regression

[schniefen]

Homework Statement

Making a $\chi^2$ test on arbitrary measurements.

Relevant Equations

$\chi^2$ testing.

1. Suppose one has the measurements [1.20, 1.15 ,2.0 ,1.17] with uncertainties [0.2,0.1,0.8,0.07]. Then, if $E$ is the weighted average, is it correct that $\chi^2$ is simply given by

$\sum \frac{(O-E)^2}{E} \ ?$​

2. If one has

| x | y |
| -- | -- |
| 0 | 0 $\pm$ 1 |
| 1 | 1 $\pm$ 1 |
| 2 | 4 $\pm$ 1|
| 3 | 9 $\pm$ 1 |

and one would like to test if $y=ax+b$, then what is $\chi^2$?

 

[BvU]

1) yes What am I saying. No ! For weighted averaging

$$\chi^2 = \sum_i \;{(x_i - \bar x)^2\over \sigma_i^2}$$2) evaluate

(picture borrowed from Edinburgh University)

(perhaps an old post has some references)