In this article, we study the behavior of the coefficient of variation (CV) of a random variable that follows a symmetric distribution in the real line. Specifically, we estimate this coefficient using the maximumlikelihood (ML) method. In addition, we provide asymptotic inference for this parameter, which allows us to contrast hypothesis and construct confidence intervals. Furthermore, we produce influence diagnostics to evaluate the sensitivity of the ML estimate of this coefficient when atypical data are present. Moreover, we illustrate the obtained results by using financial real data. Finally, we carry out a simulation study to detect the potential influence of atypical observations on the ML estimator of the CV of a symmetric distribution. The illustration and simulation demonstrate the robustness of the ML estimation of this coefficient.