  The statistics of the average height fluctuation of the one-dimensional
Kardar-Parisi-Zhang(KPZ)-type surface is investigated. Guided by the idea of
local stationarity, we derive the scaling form of the characteristic function
in the early-time regime, $t\ll N^{3/2}$ with $t$ time and $N$ the system size,
from the known characteristic function in the stationary state ($t\gg N^{3/2}$)
of the single-step model derivable from a Bethe Ansatz solution, and thereby
find the scaling properties of the cumulants and the large deviation function
in the early-time regime. These results, combined with the scaling analysis of
the KPZ equation, imply the existence of the universal scaling functions for
the cumulants and an universal large deviation function. The analytic
predictions are supported by the simulation results for two different models.
