  In this paper, we consider a scale invariant Cassinian metric and a Gromov
hyperbolic metric. We discuss a distortion property of the scale invariant
Cassinian metric under M\"obius maps of a punctured ball onto another punctured
ball. We obtain a modulus of continuity of the identity map from a domain
equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic
metric) onto the same domain equipped with the Euclidean metric. The
quasi-invariance properties of both the metrics under quasiconformal maps are
also established.
