An experimental study of the normalized three-jet rate of b quark events with respect to light quarks events (light = l equivalent to u, d, s) has been performed using the CAMBRIDGE and DURHAM jet algorithms. The data used were collected by the DELPHI experiment at LEP on the Z peak from 1994 to 2000. The results are found to agree with theoretical predictions treating mass corrections at next-to-leading order. Measurements of the b quark mass have also been performed for both the b pole mass: M-b and the b running mass: m(b)(M-Z). Data are found to be better described when using the running mass. The measurement yields: m(b)(M-Z) = 2.85 +/- 0.18(stat) +/- 0.13(exp) +/- 0.19(had) +/- 0.12(theo) GeV/c(2). for the CAMBRIDGE algorithm. This result is the most precise measurement of the b mass derived from a high energy process. When compared to other b mass determinations by experiments at lower energy scales, this value agrees with the prediction of quantum chromodynamics for the energy evolution of the running mass. The mass measurement is equivalent to a test of the flavour independence of the strong coupling constant with an accuracy of 7 parts per thousand.