  Current experimental data indicate that the Koide relation for the pole
masses of charged leptons, which can be parametrized as $Q^{pole}_l = 2/3$, is
valid up to the accuracy of $O(10^{-5})$. We show that the running masses of
charged leptons fail in satisfying the Koide relation (i.e., $Q_l(\mu) \neq
2/3$), but the discrepancy between $Q_l (\mu)$ and $Q^{pole}_l$ is only about
0.2% at $\mu=M_Z$. The Koide-like relations for the running masses of neutrinos
($1/3 < Q_\nu(M_Z) < 0.6$), up-type quarks ($Q_{U}(M_Z) \sim 0.89$) and
down-type quarks ($Q_{D}(M_Z) \sim 0.74$) are also examined from $M_Z$ up to
the typical seesaw scale $M_R \sim 10^{14}$ GeV, and they are found to be
nearly stable against radiative corrections. The approximate stability of
$Q_{U}(\mu)$ and $Q_{D}(\mu)$ is mainly attributed to the strong mass hierarchy
of quarks, while that of $Q_l(\mu)$ and $Q_\nu(\mu)$ is essentially for the
reason that the lepton mass ratios are rather insensitive to radiative
corrections.
