The Atmospheric Neutrino Anomaly is shown through these groups of data. The up- and down- going electron neutrinos are approximately the same, but there is a considerable lack of up-going muon neutrinos (about half that of down-going ones).
This data agrees with our hypothesis that there should be noticeably fewer up-going muon neutrinos than down-going muon neutrinos. Our current best explanation for this is that these νμ's have oscillated into another flavor (most likely, the undetectable tau neutrinos). This oscillation would give us roughly the same number of up- and down- going electron neutrinos, while also giving us much fewer up-going than down-going muon neutrinos.
The larger number the asymmetry calculation yields, the greater asymmetry there is, and vice versa. Similarly, the greater the uncertainty, the more error there may be.
Here's the chain of reasoning:
What does all of this mean??
Neutrinos with mass are SOMETHING NEW, beyond the Standard Model of particle physics.
Neutrinos are also important for cosmology. Astronomers believe that the universe contains large amounts of DARK MATTER that we can't see - possibly more than 90% of the mass may be invisible. If neutrinos have tiny, but non-zero mass, they make up some (but not all) of the dark matter.