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A quantile summary is a data structure that approximates to$\varepsilon$-relative error the order statistics of a much larger underlyingdataset.  In this paper we develop a randomized online quantile summary for the cashregister data input model and comparison data domain model that uses$O(\frac{1}{\varepsilon} \log \frac{1}{\varepsilon})$ words of memory. Thisimproves upon the previous best upper bound of $O(\frac{1}{\varepsilon}\log^{3/2} \frac{1}{\varepsilon})$ by Agarwal et. al. (PODS 2012). Further, bya lower bound of Hung and Ting (FAW 2010) no deterministic summary for thecomparison model can outperform our randomized summary in terms of spacecomplexity. Lastly, our summary has the nice property that$O(\frac{1}{\varepsilon} \log \frac{1}{\varepsilon})$ words suffice to ensurethat the success probability is $1 - e^{-\text{poly}(1/\varepsilon)}$.

A randomized online quantile summary in $O(\frac{1}{\varepsilon} \log \frac{1}{\varepsilon})$ words