  Sensitivity analysis is concerned with understanding how the model output
depends on uncertainties (variances) in inputs and then identifies which inputs
are important in contributing to the prediction imprecision. Uncertainty
determination in output is the most crucial step in sensitivity analysis. In
the present paper, an analytic expression, which can exactly evaluate the
uncertainty in output as a function of the output's derivatives and inputs'
central moments, is firstly deduced for general multivariate models with given
relationship between output and inputs in terms of Taylor series expansion. A
$\gamma$-order relative uncertainty for output, denoted by
$\mathrm{R^{\gamma}_v}$, is introduced to quantify the contributions of input
uncertainty of different orders. On this basis, it is shown that the widely
used approximation considering the first order contribution from the variance
of input variable can satisfactorily express the output uncertainty only when
the input variance is very small or the input-output function is almost linear.
Two applications of the analytic formula are performed to the power grid and
economic systems where the sensitivity of both actual power output and Economic
Order Quantity models are analyzed. The importance of each input variable in
response to the model outputs is quantified by the analytic formula.
