  The evaluation of the $b$ value of the Gutenberg-Richter (GR) law, for a
sample composed of $n$ earthquakes, presents a systematic positive bias $\delta
b$ which is proportional to $1/n$, as already observed by Ogata \& Yamashina
(1986). In this study we show how to incorporate in $\delta b$ the bias
introduced by deviations from the GR law. More precisely we show that $\delta
b$ is proportional to the square of the variability coefficient $CV$, defined
as the ratio between {the standard deviation of the magnitude distribution and
its mean value.} When the magnitude distribution follows the GR law $CV=1$ and
this allows us to introduce a new procedure, based on the dependence of $b$ on
$n$, which allows us to {identify} the incompleteness magnitude $m_c$ as the
threshold magnitude leading to $CV=1$. The method is tested on synthetic
catalogs and it is applied to estimate $m_c$ in Southern California, Japan and
New Zealand.

Geophysics;Data Analysis, Statistics and Probability