Modelling an empirical distribution by means of a simple theoretical distribution is an interesting issue in applied statistics. A reasonable first step in this modelling process is to demand that measures for location, dispersion, skewness and kurtosis for the two distributions coincide. Up to now, the four measures used hereby were based on moments.In this paper measures are considered which are based on quantiles. Of course, the four values of these quantile measures do not uniquely determine the modelling distribution. They do, however, within specific systems of distributions, like Pearson's or Johnson's; they share this property with the four moment-based measures.This opens the possibility of modelling an empirical distribution-within a specific system-by means of quantile measures. Since moment-based measures are sensitive to outliers, this approach may lead to a better fit. Further, tests of fit-e.g. a test for normality-may be constructed based on quantile measures. In view of the robustness property, these tests may achieve higher power than the classical moment-based tests.For both applications the limiting joint distribution of quantile measures will be needed; they are derived here as well. 1Consider a random variable x with mean p = E(x) and central moments A quantile measure for kurtosisThe (very familiar) moment-based measures for location, dispersion, skewness and kurtosis now are 0 the mean p 0 the variance p2 0 the third standardized moment PI = p3/p;'* 0 the fourth standardized moment pZ = p4/& They all exist provided E(x4) < 00. (Note that the symbol PI is used instead of the usual fi.) ~ ~~~ ~ ~ * moors@kub.nl QVVS. 1996 Published by Bluckwell Publishen, 108 Cowley Road. Oxford OX4 IJF. U K and 238 Main Sllec~. Cambndp. MA 02142. USA For the first three measures quantile-based alternatives are well-known. Defining quartiles Qi by P ( x < Q J 5 i/4, P(x > QJ 5 1i/4 for i = 1,2,3, they are given by 0 the median Q = Q2 0 the half interquartile range R = (Q3 -Q l ) / 2 0 Bowley's skewness measure S = (Q3 -2Q2 + Q I ) / ( Q~ -Q I )provided that Q3 # QI. MOORS (1986, 1988) presented a new interpretation of kurtosis as well as a quantile-based alternative for p2. Define octiles Ei by ' P(x < Ei) 5 i/8, P(x > Ei) 5 1i/8 for i = 1,2,. . . ,7. Then the quantile measure T for kurtosis readsprovided that €6 # E2. Note that Tis much less sensitive to outliers than a; it can be found even by graphical means. Furthermore, T exists even for distributions without finite moments; e.g. T = 2 for the Cauchy distribution.The quartet ( Q , R, S, T) can be seen as an alternative to ( p , p2, 81, P,). Like PI and p2, S and Tremain unchanged under linear transformations: these four quantities are location-scale-invariant. This is the main reason why in the sequel attention is focused on the pair (S, T). The rest of the paper is organized as follows. Sections 2 and 3 consider the Pearson system of distributions. Section 2 reviews its properties with emphasis on the moment measures 01 and P,; in Section 3 the quanti...