  The Kolmogorov distances between a symmetric hypergeometric law with standard
deviation $\sigma$ and its usual normal approximations are computed and shown
to be less than $1/(\sqrt{8\pi}\,\sigma)$, with the order $1/\sigma$ and the
constant $1/\sqrt{8\pi}$ being optimal. The results of Hipp and Mattner (2007)
for symmetric binomial laws are obtained as special cases. Connections to
Berry-Esseen type results in more general situations concerning sums of simple
random samples or Bernoulli convolutions are explained. Auxiliary results of
independent interest include rather sharp normal distribution function
inequalities, a simple identifiability result for hypergeometric laws, and some
remarks related to L\'evy's concentration-variance inequality.
