  Zero-field muon spin relaxation (ZF-$\mu $SR) data for dilute spin magnetic
systems have been widely interpreted with what is called a Kubo-Toyabe form
based on a Lorentzian distribution of local field components. We derive here
the proper magnetic field \textit{magnitude} distribution using independent and
uncorrelated \textit{component} distributions. Our result is then compared to
the previously accepted formula for ZF-$\mu $SR. We discuss the origins of the
magnetic field component and magnitude distributions. Further we found that
after rescaling the magnetic field, the differences that are amenable to
experimental examination are quite small, although the interpretations behind
them are quite different.
