  The scaling behavior of pure gauge SU(3) in the region $\beta=5.85 - 7.60$ is
examined by a Monte Carlo Renormalization Group analysis. The coupling shifts
induced by factor 2 blocking are measured both on 32$^4$ and 16$^4$ lattices
with high statistics. A systematic deviation from naive 2-loop scaling is
clearly seen. The mean field and effective coupling constant schemes explain
part, but not all of the deviation. It can be accounted for by a suitable
change of coupling constant, including a correction term ${\cal O}(g^7)$ in the
2-loop lattice $\beta$-function. Based on this improvement,
$\sqrt{\sigma}/\Lambda_{\overline {MS}}^{n_f=0}$ is estimated to be $2.2(\pm
0.1)$ from the analysis of the string tension $\sigma$.
