Let \(D\) be a domain in \(\overline{\mathbb{R}^n}=\mathbb{R}^n\cup \{\infty\},\;n\geq 2\), and let \(x,y\in D\). The Apollonian distance of \(x,y\) is defined by \[ \alpha(x,y):=\max_{w,z\in\partial D}\log \frac{| x-w| \;| y-z| }{| x-z| \;| y-w| }. \] The author introduces the quantity \[ H_D(x):=\frac{\limsup_{y\to x}(\alpha(y,x)/\chi(y,x))}{\liminf_{y\to x}(\alpha(y,x)/\chi(y,x))}, \] where \(\chi\) is the chordal distance in \(\overline{\mathbb{R}^n}\). The Apollonian metric is conformal at \(x\in D\) if \(H_D(x)=1\). The quantity \(H_D(x)\) (which is proved to be invariant under Möbius transformations) measures the deviation of \(\alpha_D\) from being conformal at \(x\). The author finds various properties of \(H_D\) and computes it for some standard domains (ball, strip, annulus, sector). He also uses it to obtain sharp estimates between the Apollonian, hyperbolic, and quasihyperbolic metrics on such domains. Further work on the Apollonian metric has been recently carried out by P. Hästö and the author.