  The theoretical prediction that trigonometric parallaxes suffer from a
statistical effect, has become topical again now that the results of the
Hipparcos satellite have become available. This statistical effect, the
so-called Lutz-Kelker bias, causes measured parallaxes to be too large. This
has the implication that inferred distances, and hence inferred luminosities
are too small. Published analytic calculations of the Lutz-Kelker bias indicate
that the inferred luminosity of an object is, on average, 30% too small when
the error in the parallax is only 17.5%. Yet, this bias has never been
determined empirically. In this paper we investigate whether there is such a
bias by comparing the best Hipparcos parallaxes which ground-based
measurements. We find that there is indeed a large bias affecting parallaxes,
with an average and scatter comparable to predictions. We propose a simple
method to correct for the LK bias, and apply it successfully to a sub-sample of
our stars. We then analyze the sample of 26 `best' Cepheids used by Feast &
Catchpole (1997) to derive the zero-point of the fundamental mode pulsators and
leads to a distance modulus to the Large Magellanic Cloud - based on Cepheid
parallaxes- of 18.56 +/- 0.08, consistent with previous estimates.
