  In this article, we obtain explicit bounds on the uniform distance between
the cumulative distribution function of a standardized sum $S_n$ of $n$
independent centered random variables with moments of order four and its
first-order Edgeworth expansion. Those bounds are valid for any sample size
with $n^{-1/2}$ rate under moment conditions only and $n^{-1}$ rate under
additional regularity constraints on the tail behavior of the characteristic
function of $S_n$. In both cases, the bounds are further sharpened if the
variables involved in $S_n$ are unskewed. We also derive new Berry-Esseen-type
bounds from our results and discuss their links with existing ones. We finally
apply our results to illustrate the lack of finite-sample validity of one-sided
tests based on the normal approximation of the mean.
