The remainder method for sample percentiles

A method called the Remainder Method is proposed for the calculation of sample quantiles of a given order, for example, quartiles, hexatiles, octatiles, deciles and percentiles assuming that all the observations are distinct. Proof is given for a special case of deciles. The criterion ‘equisegmentation’ is proposed, namely that the number of observations below the first quantile, that between the consecutive quantiles, and that above the last quantile are the same. The formulae for quantiles offered by the proposed method satisfy the equisegmentation property, and more interestingly provide the number of quantiles having integer ranks. Some open problems are indicated.