Normal theory of selecting the smaller variance from amongst k(?2) populations through likelihood comparisons was developed in Mukhopadhyay and Chou [10]. Hoel [7] had developed such a procedure for k = 2. We examine the situation in the case of two symmetric, not necessarily normal, population distributions. We proceed using the route of approximate sequential jF when the degrees of freedom are appropriately adjusted along the suggestions of Box and Andersen [4], that depend on the sample sizes as well as both second and fourth central moments. We then generalize it to the case when the underlying distributions have different shapes. We establish some theoretical properties of these procedures. In addition, through large sets of simulations for various non-normal mixture distributions, as well as normal distributions themselves, we conclude that the Box-Andersen version of the selection methodology withstands better the types of non-normality considered in this paper than the existing one derived from the normal-normal theory