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 nonnegative 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.