  We derive the general analytical expressions for the statistical
uncertainties of cumulants up to fourth order including an efficiency
correction. The analytical expressions have been tested with a toy Monte Carlo
model analysis. An application to the study of particle multiplicity
fluctuations in heavy-ion collisions is investigated. In this derivation, a
mathematical proof is given that the validity of an averaged efficiency
correction and the fluctuations induced by the non-uniformity of efficiency can
be eliminated. The estimation of statistical uncertainties using the analytical
formulas is found to be significantly faster than the commonly used bootstrap
method. The simplicity and efficiency of using the analytical formulas may be
useful for massive data analysis in many fields.
