  The ordering of weakly coupled random antiferromagnetic $S=1/2$ chains, as
relevant for recent experimentally investigated spin chain materials, is
considered theoretically. The one-dimensional isotropic Heisenberg model with
random exchange interactions is treated numerically on finite chains with the
density-matrix renormalization-group approach as well as with the standard
renormalization analysis, both within the mean-field approximation for
interchain coupling $J_{\perp}$. Results for the ordering temperature $T_N$ and
for the ordered moment $m_0$ are presented and are both reduced with the
increasing disorder agreeing with experimental observations. The most
pronounced effect of the random singlet concept appears to be a very large span
of local ordered moments, becoming wider with decreasing $J_{\perp}$,
consistent with $\mu$SR experimental findings.
