A Summary of $\chi ^2$ Fitting Formulas

To wrap up, let us summarise what we have accomplished  so far:

 

$$S_\sigma \sum_i^N \frac{1}{\sigma_i^2}$$ = (2.1)

$$\sum_i^N \frac{x_i}{\sigma_i^2}, \qquad S_y = \sum_i^N \frac{y_i}{\sigma_i^2}$$ S_x = (2.2)

$$\sum_i^N \frac{x_i^2}{\sigma_i^2}, \qquad S_{xy} = \sum_i^N \frac{x_i y_i}{\sigma_i^2}$$ S_xx = (2.3)

$$\frac{S_{xx}S_y - S_x S_{xy}} {S_\sigma S_{xx} - \left(S_x\right)... ...uad b = \frac{S_\sigma S_{xy} - S_x S_y} {S_\sigma S_{xx} - \left(S_x\right)^2}$$ a = (2.4)

$$\sigma_a \sqrt{\sum_{k=1}^N \left(\sigma_k \frac{\partial a}{\partial y_k}... ...xx} - x_k S_x} {\sigma_k^2 \left(S_{xx} S_{\sigma} - \left(S_x\right)^2\right)}$$ = (2.5)

$$\sigma_b \sqrt{\sum_{k=1}^N \left(\sigma_k \frac{\partial b}{\partial y_k}... ...S_\sigma - S_x} {\sigma_k^2 \left(S_{xx} S_\sigma - \left(S_x\right)^2 \right)}$$ = (2.6)

$$\chi^2 \sum_{i=1}^N \left(\frac{y_i - a - b x_i}{\sigma_i}\right)^2$$ = (2.7)

$$\frac{1}{\Gamma\left(\frac{N-2}{2}\right)} \int_{\chi^2/2}^\infty e^{-t} t^{\frac{N-2}{2} - 1} \, \textrm{d} t$$ Q = (2.8)

where Q is the so called goodness of fit and $\Gamma$ is the Euler gamma  function. Q  is the probability that a value of $\chi ^2$ as poor as the one obtained in your calculations, using your values of x_i and y_i, would have occurred by chance. The value of Q is between 0 and 1. The closer Q is to 1, the more trustworthy is your theory which has assumed a linear dependence of y on x.