R. L. Plackett derived the maximum of the ratio of mean range to the standard deviation as function of the sample size, and gave the initial (symmetrical) distribution for which this maximum is actually reached. On the other hand, Moriguti derived the maximum for the mean largest value under the assumption that the distribution from which the maximum is taken is symmetrical. His mean value turned out to be one half of the value given by Plackett. In the following, these results will be generalized for an arbitrary (not necessarily symmetrical) continuous variate. The mean and the standard deviation of the largest value and the mean range will be given for two distributions: one where the mean largest value is a maximum, and another one where the mean range is a maximum. Obviously, a mean largest value can exist if and only if the initial mean exists. In addition we postulate in both cases the existence of the second moment.
E. J. Gumbel. "The Maxima of the Mean Largest Value and of the Range." Ann. Math. Statist. 25 (1) 76 - 84, March, 1954. https://doi.org/10.1214/aoms/1177728847