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We introduce a new distance determination method using carbon-rich asymptoticgiant branch stars (CS) as standard candles and the Large and Small MagellanicClouds (LMC and SMC) as the fundamental calibrators. We select the samples ofCS from the ($(J-K_{s})_0$, $J_0$) colour-magnitude diagrams, as, in thiscombination of filters, CS are bright and easy to identify. We fit the CS$J$-band luminosity functions using a Lorentzian distribution modified to allowthe distribution to be asymmetric. We use the parameters of the best-fitdistribution to determine if the CS luminosity function of a given galaxyresembles that of the LMC or SMC. Based on this resemblance, we use either theLMC or SMC as the calibrator and estimate the distance to the given galaxyusing the median $J$ magnitude ($\overline{J}$) of the CS samples. We applythis new method to the two Local Group galaxies NGC 6822 and IC 1613. We findthat 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 LMCand 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.

Carbon stars as standard candles: II. The median J magnitude as a distance indicator