Measuring and calculating free quark masses Particle data book contains masses of the free quarks. I wonder, how do experimentalists determine the masses of the free quarks even though they are trapped inside hadrons (except perhaps in quark-gluon plasma)? Can quark masses be theoretically calculated and how much do they agree with experiments?
 A: Nobody has seriously calculated theoretically a quark mass from first principles. So there is no issue of agreement with experiment. They are parameters in experimental fits, but sometimes remarkably consistent across a broad range of experiments-- and the QCD/EW calculations using them as inputs. If someone pretends to know their origin, he/she is bluffing.
Constituent quark masses have been fitted to hadron masses using simple potential models in the style of additive nuclear models. They range from 0.34Gev to 177GeV and are not too informative, as they are mostly glue for the light quarks, i.e. they arose out of chiral symmetry breaking---see below. They are not of any use in QCD/EW perturbative calculations, and, culturally, they have a life of their own, of possible assistance to diffractive scattering studies.
The more fundamental current quark masses that are used in the fundamental lagrangian of the SM are  extracted from DIS fits (deep inelastic scattering), for the heavy ones; but, mostly for the light ones, they are extracted, sometimes with the aid of lattice theories, out of chiral perturbation theory on the masses of the pseudoscalar mesons, via Dashen's formula of chiral symmetry breaking. 
They range from 2MeV to 175GeV. For example, stringing several such formulas together one finds fits such as the popular one,
$$
\frac{m_u+m_d}{2m_s}=\frac{m_\pi^2}{2m_K^2-m_\pi^2}\approx \frac{1}{25}~. 
$$
