CIELAB color coordinates are characteristics of visible object's spectra widely used in color science, and a reliable evaluation of the uncertainty associated with measured color coordinates is desirable. The calculation of CIELAB color coordinates requires the application of a nonlinear transformation. We consider two ways to calculate the uncertainty of CIELAB color coordinates that originates from the uncertainty of the observed spectrum. The first approach is based on a linearization of the nonlinear transformation and a propagation of variances and covariances. The second approach is a Monte Carlo procedure that completely accounts for the nonlinearity of the transformation. While the approach based on linearization results in analytical formulas, the adequacy of linearization ought to be checked and the proposed Monte Carlo method can serve as a reference method for this purpose. We illustrate such proceeding in a large range of scenarios and explore those cases for which the method based on linearization may safely be applied.