  Based on Monte Carlo approach and conventional error analysis theory, taking
the heaviest doubly magic nucleus $^{208}$Pb as an example, we firstly evaluate
the propagated uncertainties of universal potential parameters for three
typical types of single-particle energies in the phenomenological Woods-Saxon
mean field. Accepting the Woods-Saxon modeling with uncorrelated model
parameters, we find that the standard deviations of single-particle energies
obtained by the Monte Carlo simulation and the error propagation rules are in
good agreement with each other. It seems that the energy uncertaintis of the
single-particle levels regularly evoluate with some quantum numbers to a large
extent for the given parameter uncertainties. Further, the correlation
properties of the single-particle levels within the domain of input parameter
uncertainties are analyzed using the method of statistical analysis, e.g., with
the aid of Pearson correlation coefficients. It is found that the positive,
negative or unrelated relationship may appear between two selected
single-particle levels, which will be very helpful for evaluating the
theoretical uncertainty related to the single-particle levels (e.g., $K$
isomer) in nuclear structural calculations.
