  Parton distributions functions (PDFs), which are essential to the
interpretation of data from high energy colliders, are measured by representing
them as functional forms containing many parameters. Those parameters are
determined by fitting a wide variety of experimental data. The best-fit PDF set
is obtained by minimizing the standard $\chi^2$ measure of fit quality. The
uncertainty range is estimated in the Hessian method by regarding as
acceptable, all fits for which $\chi^2$ lies within $\Delta\chi^2$ of its
minimum. The appropriate value of $\Delta\chi^2$ for this purpose has been
estimated by a variety of arguments to be approximately 50 - 100 for a 90%
confidence limit. This paper resolves the long-standing paradox of why that
empirical value is so much larger than the $\Delta\chi^2=2.7$ for 90%
confidence that would be expected on the basis of standard Gaussian statistics.
