  The zero bias distribution $W^*$ of $W$, defined though the characterizing
equation $\mathit{EW}f(W)=\sigma^2Ef'(W^*)$ for all smooth functions $f$,
exists for all $W$ with mean zero and finite variance $\sigma^2$. For $W$ and
$W^*$ defined on the same probability space, the $L^1$ distance between $F$,
the distribution function of $W$ with $\mathit{EW}=0$ and $Var(W)=1$, and the
cumulative standard normal $\Phi$ has the simple upper bound \[\Vert
F-\Phi\Vert_1\le2E|W^*-W|.\] This inequality is used to provide explicit $L^1$
bounds with moderate-sized constants for independent sums, projections of cone
measure on the sphere $S(\ell_n^p)$, simple random sampling and combinatorial
central limit theorems.
