  The $\nu_{\mu} \to \nu_{\tau}$ and $\nu_{\mu} \to \nu_s$ solutions to the
atmospheric neutrino problem are compared with SuperKamiokande data. The
differences between these solutions due to matter effects in the Earth are
calculated for the ratio of $\mu$-like to $e$-like events and for up-down flux
asymmetries. These quantities are chosen because they are relatively
insensitive to theoretical uncertainties in the overall neutrino flux
normalisation and detection cross-sections and efficiencies. A $\chi^2$
analysis using these quantities is performed yielding $3\sigma$ ranges which
are approximately given by $(0.725 - 1.0, 4 \times 10^{-4} - 2 \times 10^{-2}
eV^2)$ and $(0.74 - 1.0, 1 \times 10^{-3} - 2 \times 10^{-2} eV^2)$ for
$(\sin^2 2\theta,\Delta m^2)$ for the $\nu_{\mu} \to \nu_{\tau}$ and $\nu_{\mu}
\to \nu_s$ solutions, respectively. Values of $\Delta m^2$ smaller than about
$2 \times 10^{-3}$ eV$^2$ are disfavoured for the $\nu_{\mu} \to \nu_s$
solution, suggesting that future long baseline experiments should see a
positive signal if this scenario is the correct one.
