Our conclusion?
Ideally, if the model describes the data AND the estimated uncertainty of each point has been accurately determined, the reduced chi-square value would be approximately equal to one (for large \(\nu\)) and the fractional probability of \(\chi^2\) values larger and smaller than that measured would both be approximately 50%. Statistically, however, we expect some reasonable random variation from the expected mean value of \(\nu\) for \(\chi_{\min}^2\), meaning that statisticians generally will not reject the "null hypothesis" (that the model describes the data and all deviations can be reasonably attributed to expected random variation) as long as
\(\chi_{\min}^2\) is within 2 standard deviations ( \(2\sigma\) ) of the mean value.