Reconstruction of a reduced density operator for weakly coupled systems of spins $\frac{1}{2}$ from fits to nuclear magnetic resonance spectra is described in detail. Particular emphasis is placed on data treatment procedures that specify fewer than the ${3}^{n}$ complete spectra that are implicitly prescribed in published references to state tomography on $n$-spin systems. It is shown that if the density operator is expanded in the so-called product-operator basis, it is always possible to estimate a desired coefficient in the expansion by measuring a single spectral multiplet. This simple observation can substantially reduce the experimental effort required for either complete density-matrix reconstruction or estimation of subsets of the coefficients in the product-operator expansion. A simple iterative algorithm can be used to produce reduced measurement procedures for experiments involving small numbers of qubits.