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

Uncertainty evaluation and correlation analysis of single-particle  energies in phenomenological nuclear mean field: An investigation of  propagating uncertainties for independent model parameters