  Reduced chi-squared is a very popular method for model assessment, model
comparison, convergence diagnostic, and error estimation in astronomy. In this
manuscript, we discuss the pitfalls involved in using reduced chi-squared.
There are two independent problems: (a) The number of degrees of freedom can
only be estimated for linear models. Concerning nonlinear models, the number of
degrees of freedom is unknown, i.e., it is not possible to compute the value of
reduced chi-squared. (b) Due to random noise in the data, also the value of
reduced chi-squared itself is subject to noise, i.e., the value is uncertain.
This uncertainty impairs the usefulness of reduced chi-squared for
differentiating between models or assessing convergence of a minimisation
procedure. The impact of noise on the value of reduced chi-squared is
surprisingly large, in particular for small data sets, which are very common in
astrophysical problems. We conclude that reduced chi-squared can only be used
with due caution for linear models, whereas it must not be used for nonlinear
models at all. Finally, we recommend more sophisticated and reliable methods,
which are also applicable to nonlinear models.
