  Let $W$ be a random variable with mean zero and variance $\sigma^2$. The
distribution of a variate $W^*$, satisfying $EWf(W)=\sigma ^2 Ef'(W^*)$ for
smooth functions $f$, exists uniquely and defines the zero bias transformation
on the distribution of $W$. The zero bias transformation shares many
interesting properties with the well known size bias transformation for
non-negative variables, but is applied to variables taking on both positive and
negative values. The transformation can also be defined on more general random
objects. The relation between the transformation and the expression
$wf'(w)-\sigma^2 f''(w)$ which appears in the Stein equation characterizing the
mean zero, variance $\sigma ^2$ normal $\sigma Z$ can be used to obtain bounds
on the difference $E\{h(W/\sigma)-h(Z)\}$ for smooth functions $h$ by
constructing the pair $(W,W^*)$ jointly on the same space. When $W$ is a sum of
$n$ not necessarily independent variates, under certain conditions which
include a vanishing third moment, bounds on this difference of the order $1/n$
for classes of smooth functions $h$ may be obtained. The technique is
illustrated by an application to simple random sampling.
