We assumed each data point is taking from a distribution with mean and variance

in which, the mean can be a function of X.

For example, we have a data , it has relation with an independent variable . We would like to know the relationship between and , so we fit a function .

After the fitting (least square method), we will have so residual for each of the data

This residual should be follow the distribution

The goodness of fit, is a measure, to see the distribution of the residual, agree with the experimental error of each point, i.e.

Thus, we would like to divide the residual with and define the chi-squared

.

we can see, the distribution of

and the sum of this distribution would be the chi-squared distribution. It has a mean of the degree of freedom . Note that the mean and the peak of the chi-squared distribution is not the same that the peak at .

In the case we don’t know the error, then, the sample variance of the residual is out best estimator of the true variance. The unbiased sample variance is

,

where is degree of freedom. In the cause of , the , because there is 1 degree o freedom used in x. And because the 1 with the b is fixed, it provides no degree of freedom.