The measurement of the spectral power distribution (SPD) of a radiation source by array spectroradiometers is a technique that is widely used. In many applications, quantities that are derived from the SPD by a weighted integral over a wavelength interval are of interest. These integral quantities ought to be accompanied by a reliable uncertainty statement, for example, to assess conformity with prescribed limits or in order to judge the consistency of results obtained at different laboratories. We have developed a generally applicable Monte Carlo procedure for evaluating the uncertainty of spectral measurements. The procedure naturally accounts for correlations in the SPD which turn out to be crucial. Means are provided to handle and transfer these large-scale correlation matrices easily. The proposed approach is illustrated by the determination of the SPD of colored LEDs from array spectroradiometer measurements, together with the derived CIE 1931 color coordinates. MATLAB(TM) software implementing the proposed analysis procedure is made available.