C2 - reduced-chi squared

models are fitted by minimising the $χ^{2}$ statistic

\begin{aligned} χ^{2} & = \sum_{i} \frac{(y_{i} - y)^{2}}{α_{i}^{2}} \\ χ^{2} & = \sum_{i} \frac{(O_{i} - E_{i})^{2}}{α_{i}^{2}} \end{aligned}

the null hypothesis being tested is that the obtained sample distribution can be described by a particular parent distribution
ie: the data, $(x_{i}, y_{i})$ , are well modelled by a particular function, $y (x_{i})$
the two-thirds rule implies that a good fit will have 2/3 of the data points within 1 error bar
$95 %$ should be between 2 error bars

$\pm σ$	$\pm 2 σ$	$\pm 3 σ$
$68 %$	$95 %$	$99.7 %$

furthermore, the normalised residuals should have a mean of $0$ and a standard deviation of $1$
the degrees of freedom, $ν$ , is the number of unconstrained parameters:

ν = N - N

where $N$ and $N$ are the numbers of data points and fit parameters, respectively

eg: when trying to fit 10 data points:
- in a straight line ( $y = m x + c$ ) : $ν = 8$
- in a parabola ( $y = a x^{2} + b x + c$ ) : $ν = 7$
so, the more datapoints are unconstrained, the more robust an estimation is
reduced chi-squared statistic:

χ_{ν}^{2} = \frac{χ_{m i n}^{2}}{ν}

for a good fit, $χ_{m i n}^{2} ≃ ν$ , so $χ_{ν}^{2} ≃ 1$