  The Pearson distance between a pair of random variables $X,Y$ with
correlation $\rho_{xy}$, namely, 1-$\rho_{xy}$, has gained widespread use,
particularly for clustering, in areas such as gene expression analysis, brain
imaging and cyber security. In all these applications it is implicitly
assumed/required that the distance measures be metrics, thus satisfying the
triangle inequality. We show however, that Pearson distance is not a metric. We
go on to show that this can be repaired by recalling the result, (well known in
other literature) that $\sqrt{1-\rho_{xy}}$ is a metric. We similarly show that
a related measure of interest, $1-|\rho_{xy}|$, which is invariant to the sign
of $\rho_{xy}$, is not a metric but that $\sqrt{1-\rho_{xy}^2}$ is. We also
give generalizations of these results.
