Carbon stars as standard candles: II. The median J magnitude as a
  distance indicator
  We introduce a new distance determination method using carbon-rich asymptotic
giant branch stars (CS) as standard candles and the Large and Small Magellanic
Clouds (LMC and SMC) as the fundamental calibrators. We select the samples of
CS from the ($(J-K_{s})_0$, $J_0$) colour-magnitude diagrams, as, in this
combination of filters, CS are bright and easy to identify. We fit the CS
$J$-band luminosity functions using a Lorentzian distribution modified to allow
the distribution to be asymmetric. We use the parameters of the best-fit
distribution to determine if the CS luminosity function of a given galaxy
resembles that of the LMC or SMC. Based on this resemblance, we use either the
LMC or SMC as the calibrator and estimate the distance to the given galaxy
using the median $J$ magnitude ($\overline{J}$) of the CS samples. We apply
this new method to the two Local Group galaxies NGC 6822 and IC 1613. We find
that NGC 6822 has an "LMC-like" CS luminosity function while IC 1613 is more
"SMC-like". Using the values for the median absolute $J$ magnitude for the LMC
and SMC found in Paper I we find a distance modulus of $\mu_{0}=23.54\pm0.03$
(stat) for NGC 6822 and $\mu_{0}=24.34\pm0.05$ (stat) for IC 1613.
